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where A £ K1 is a parameter, f(x) a given function, and u(x, t) the unknown function. The associated stationary equation of (2.1) reads as (^2+u2
=
Xf{x)
(u = 0
^
V a r G ( 0 > 1 ) i
at a; = 0,1.
(
21
)
We know that there exists a number 0 < A < oo such that for all A < A, (2.2) has a solution v\(x), called a steady state of (2.1), which depends continuously on A. The stability theory tells us that given a general function / G £ 2 (0,1), there is a Ao (0 < Ao < A), and a neighborhood U\ C L2(0,1) of the steady state v\ for any A < Ao, such that for any initial value
•^
Bifurcation Theory and Applications
u(x,t, if) of (2.1) converges to v\ in L2(0,1), i.e. lim \\u{,t,tp)vx\\L2
t—>oo
=0.
However, when Ao < A the stability will be lost. Then it is nature to study two questions: (1) from what values of A, the steady state v\ will lose its stability, and (2) if Ao is the point from which v\ loses its stability, then what behaviors of the solutions u(x, t) of (2.1) near v\ may occur for t —* oo. The problem (1) is related to the steady state bifurcations introduced in Chapter 1, and both problems (1) and (2) are central topics of dynamic bifurcation. We end this introduction with two remarks. First, let u = v + v\ in (2.1) and (2.2). Then the stability and bifurcation problems for (2.1) and (2.2) near v\ are equivalent respectively to the following problems:
1
v=0 v{x,0) = ip(x)
( 2i3 )
at a; = 0,1, Vz€(0,l),
r g + 2 U A V + u 2 = o v, e( o,D,
{2A)
at a; = 0 , 1 ,
[v = 0
near v = 0, where ip —
^(0,1)0^(0,1),
# 1 / 2 = #1(0,1),
H = L2(0,l),
where H2(0,1) and HQ(0, 1) are the usual Sobolev spaces. It is clear that Hi C Hx/2 C H are dense and compact inclusions. We define the mappings L\ = A + B\ : Hi —* H and G : Hx —> H by Av =  ^ ,
Bxv = 2vxv,
Gv = v2,
v e Hi.
(2.5)
Introduction to Dynamic Bifurcation
23
The equations (2.3) and (2.4) are then reformulated into the following operator forms respectively =L,u
+ Gu,
u€Hlt
( 2 6 )
«(0) = V, and (2.7)
L\u + Gu = 0.
We know that A : Hi —» H is a linear homeomorphism, B\ : Hi —> H a linear compact operator, and G : H\ —> i7 a compact operator. Moreover, we see that BA : ffi/a ^ H a bounded linear operator, G : Hi/2 —> H a bounded continuous operator, G(u) =
o(\\u\\1/2).
We notice that the equation (2.7) can be rewritten as uA1BuA~1Gu = 0,
u£H,
which has the same form as (1.13). In fact, many differential equations in Physics, Chemistry and Biology can be put into the abstract form given in (2.6) and (2.7). 2.2 2.2.1
Semigroups of Linear Operators Introduction
As an example, we continue to consider the equation (2.3). By the SturmLiouville theorem, for each A < A, there exist a sequence of eigenvalues {Afc} with Afc —> +oo as k —* oo and eigenvectors {cf>k} C Hi such that
{
d24> — k +2vx
4>k(0) = 0*(1) = 0, and {4>k} C H constitutes an orthogonal basis of H.
{2g)
24
Bifurcation Theory and Applications
Let u = Y^kLi xk4>k The equation (2.3) can be written as / I T = **** + **(«).
*= i,2,..,
(29)
l*fc(0) = Vfc, where V* = fo i> • fadx, Gk(u) = / ^ u2
u{x,t) = Y,e~Xkt^
t oo e Xk{t T)G u
Y, ~
•/o
fc=l
~
^ )^dT.
(2.10)
fc=l
We find from (2.10) a family of linear bounded operators T(t) : H —» H defined by CXI
OO
A
T(tty = 5> *Vfe&,
V t > 0 , V = X)Vfc^*.
(211)
fc=i fe=i
which enjoy the following semigroup properties: ' T(0) = id : £T ^ F , <
T(t + s)=T(t)T(s),
(212)
With the definition (2.11), the solution of (2.3) is written as u = T(t)ip + f T(t T)G(u)dT.
(2.13)
Jo More generally, if for a linear operator L : Hi —> H there exists a one parameter family of linear bounded operators T(t) : H —> H satisfying (2.12), then the solution of the nonlinear equation (d^=Lu
+
G(u),
( 2 i 4 )
I «(0) = V, can be expressed by (2.13). The family of linear bounded operators T(t) is called a semigroup of linear operators.
25
Introduction to Dynamic Bifurcation
2.2.2
Strongly continuous
semigroups
Definition 2.1 Let X be a Banach space, T(t) : X > X (0 < t < oo) a one parameter family of bounded linear operators. T(t) is called a strongly continuous semigroup of linear operators, if (1) T(0) = id : X > X the identity, (2) T{t + s) = T(t)T(s) for every *, s > 0, and (3) lim(_o T(*)a: = a; for each x eX. Let Xi be a Banach space, X\ C X a dense inclusion, and L : Xi —> X a bounded linear operator. We say that L : Xi —> X generates a strongly continuous semigroup T(t) : X —» X, if Lx = ]imT{t)x~X t—o i
= jT(t)* , dt t=o
VxeXj.
In this case, the operator L : Xi —> X is also called a generator of T(i). The following is the HilleYosida theorem. Theorem 2.1 A linear bounded operator L : Xi —+ X generates a strongly continuous semigroup T(t), t > 0 if and only if (1) L is a closed operator, i.e. if xn —» £0, Lxn —> 2/0 wi X , £/ten x n —> zo in X\ and LXQ = 3/0 • ^ T/iere are Ao > 0 and C > 1 suc/i t/iat (Ao, +00) C p(L), and (Aid  L )  "   < C(A  Ao)",
V A > Ao, n = 1,2, • • • ,
where p(L) is the resolvent set of L defined by p(L) = {A € C I (Aid  L)1 : X » X bounded) .
(2.15)
Remark 2.1 If L : Xi —> X is a linear completely continuous field, then L satisfies Condition (1) in Theorem 2.1. Condition (2) implies that there is a real number Ao > 0, such that for all eigenvalues A € C of L, ReX < Ao. A strongly continuous semigroup T(t) : X —» X generated by L : Xi —» X (Xi C X densely) has the following basic properties. Theorem 2.2 Let T(t) : X —» X (t > 0) be a strongly continuous semigroup generated by L : Xi —> X . Then
26
Bifurcation Theory and Applications
(1) For any UQ £ X\, the following initial value problem ( du
[ u(0) = u0, has a unique solution u(t) = T(t)u0 € Cl([Q, oo), X) n C°([0, oo), Xx). (2) If the equation (2.13) has a solution u(t) £ X\, then the solution u(t) satisfies (2.14). (3) For any xeX, f*T(s)xds € Xu and T(s)xds) = T(t)x  x.
L([
(4) For xeXi,_ T(t)x 6 Xi and T{t)x  T(s)x = f T(r)LxdT = f LT{r)xdT. Js Js 2.2.3
Sectorial operators and analytic semigroups
The strongly continuous semigroup T(t) : X —> X has a limitation that in general T{t)x $. Xi for t > 0 and x S X. However, for the semigroup of linear operators given by (2.11) the situation is different. Let Xx = H2{0,1) n ^ ( 0 , 1 ) , and X = L2(0,1). Then, t » T(t)x is analytic for 0 < t and x = 23^=1 xk4>k £ X, and oo
T(t)x
= Y^e~Xktxk
<=Xi,
Vt > 0, x e X.
(2.16)
fc=i
Moreover, for t h e operator L = A — B : X i —> X defined b y (2.5), where L = —C is a generator of t h e semigroup T(t) in (2.16), we can define fractional power operators Ca with domains Xa = D(Ca) (a G R ) as follows oo oo 1
{
fc=l r oo
< H^ii _
fc=l
\^\l»xl Lfc=l
oo
cax = YlKxk
J
I i/2
(2.17)
Introduction to Dynamic Bifurcation
27
provided that the eigenvalues of (2.8) satisfy 0 < A^, for any k = 1,2, It is clear that Xa C Xp (a > (3) is a dense and compact inclusion, the norms  •  and  • o are equivalent to those of H2(0,1) and L2(0,1) respectively. Furthermore, we have T(t):X^Xa
Vt>0, aeR1,
T(t)Cax = CaT(t)x
Vi£l o ,
nfi _ r/3 _ pot _ rjx+P
£a
Another important property is \\CaT(t)\\
Vt>0,
which follows from the following computation: £ a T(*)= sup \\£aT(t)x\\ lkllo=i
= sup
r oo
11/2
Tx^e^'xll
ll*llo=l U = 1 < max (A£e<
(A(
J Al
= )*) • e" Al *
CataeXlt.
The generalization of this case leads to the concepts of analytic semigroups and sectorial operators. Definition 2.2 A strongly continuous semigroup T{t) : X —> X is called an analytic semigroup, if t —» T(t)a; is analytic for 0 < t < oo and x £ X. An operator L : Xi —> • X is called a sectorial operator if L generates an analytic semigroup. Theorem 2.3 only if
A linear bounded operator L : X\ —• X is sectorial if and
(1) L is a closed operator. (2) There exist 0 < 6 < TT/2, a real number Ao, and a constant C > 1, such that SAo,0 = { A e C
I  a r 5 ( A  A o )  <  + ^}c/9(L),
and IKAidL)1^—^—
A — A o 
VA G SXote, A ^ Ao,
28
Bifurcation Theory and Applications
where p(L) is the resolvent set of L defined by (2.15). 2.2.4
Powers of linear operators
In this subsection we define fractional powers of sectorial operators. Let L : X\ —» X be a sectorial operator which generates an analytic semigroup {e*L}t>o, with the sector S\Oig C p{L) for some real number Ao < 0. For the operator L = — L, and a > 0, we define 1 P°°
£~a = f / ta~letcdt, r(a) Jo where T(a) = /0°°
(2.18)
f^e^dt.
Theorem 2.4 C~a (a > 0) is a linear bounded operator on X, which is one to one and satisfies C~a . CP = C(a+P),
Va,/3>0.
Definition 2.3 With C as above, we define Ca = inverse of C~a (a > 0), £° = id : X f X, Xa = D{Ca) = R(£a). It is easy to verify that if L = —C = —A + B : X\ —> X as defined by (2.5), then Ca and Xa defined by Definition 2.3 are coincident with (2.17). In fact, by (2.16), for x = Y%Li xk4>k tc
tL
OO
e~ x = e x = ^ eXktxkcj>k. fc=i By (2.18),
OO
fc=i
29
Introduction to Dynamic Bifurcation
Hence, we deduce that oo
fc=l
{
oo
oo
1
fc=i
fc=i
J
Theorem 2.5 .Le£ L : X\ ^> X be a sectorial operator, which generates an analytic semigroup T(t). If the sector S\oj C p(L) for some Ao < 0, then for the operators Ca (C = —L) we have (1) T{t) : X > Xa for each t > 0 and a € E 1 . (2) T{t)Cax = CaT{t)x V i e X a . (3) For each t > 0, the operator CaT(t) is bounded on X, and \\CaT{t)\\ < CaraeSt, (4) LetO
for some 6>0.
and x e Xa. Then r(f)xx < Mata\\Cax\\.
Theorem 2.6 Let L : X\ —> X be a sectorial operator with S\Ote C p(L) for some Ao < 0, and if B is a linear operator such that BC~a : X —> X is bounded for some 0 < a < 1, then —L + B is also a sectorial operator. Remark 2.2 As X\ c X is a compact inclusion, Xa C Xp is compact for a > (3. In this case, Theorem 2.6 implies that if L : X\ —> X is a sectorial operator, B : Xa —> X is linear bounded for some 0 < a < 1, then L + 5 is a sectorial operator. 2.3
Dissipative Dynamical Systems
In this section, we introduce briefly dynamical systems associated with partial or ordinary differential equations, which can be considered as an abstract operator equation on Banach spaces. More precisely, let X\ and X be two Banach spaces, and X\ C X be a dense inclusion. Consider an abstract operator equation
( «(0) = uo,
(2.19)
30
Bifurcation Theory and Applications
where F : X\ —> X is a continuous mapping. Let {S(t)}t>o be an operator semigroup generated by (2.19), which enjoys the following properties S(t) : X —> X a continuous (nonlinear) operator for each t > 0, 5(0) = id (identity on X), S(t + s) = S(t)S(s),
t,s>0.
Then the solution of (2.19) can be expressed as u(t,uo)
Vi>0.
= S(t)uo,
Definition 2.4 A set S C X is called an invariant set of (2.19) if S(t)E = E (V t > 0). An invariant set E C X of (2.19) is called an attractor if E is compact, and there exists a neighborhood U C X of E, such that for any •uo £ U we have dist(5(t)uo, E) —> 0 in Xnorm as t > oo.
(2.20)
The largest open set U satisfying (2.20) is called the basin of attraction of S, or we say that S attracts U. Especially, if the basin of attraction of E is X, the E is called a global attractor of (2.19). For a set S C X, we define the wlimit set of E by
u,(S) = f (JS(t)£, s>0
t>s
where the closure is taken in X. Likewise, when it exists, the alimit set of S C X is defined by
a(S) = f \JS(t)Z. s>0
t>s
Lemma 2.1 Suppose that for some subset E C X, E ^ 0, and for some t0 > 0, the set Ut>to5(t)S is relatively compact in X. Then the wlimit set w(E) is nonempty, compact, and invariant. If the sets S(—t)T, (t > 0) are nonempty and for some to > 0, Ut>t0S(—£)E is relatively compact, then the alimit set a(E) is nonempty, compact, and invariant. Definition 2.5 Let E C X be a subset and U an open set containing E. We say that E is absorbing in U if the orbit of any bounded set of U enters into E after a certain time (which may depend on the set). Namely, for any
Introduction to Dynamic Bifurcation
31
bounded set B C U, there exists a to = to(B) such that for all t > to(B), S(t)B C E. Lemma 2.2 Suppose that the operators S(t) are uniformly compact for t large, i.e. for any bounded set B there exists to such that Dt>toS(t)B is relatively compact in X. We also assume that there exists an open set U and a bounded subset BofU such that B is absorbing in U. Then the wlimit set of B, S = w{B) is an attractor which attracts the bounded sets of U, and it is the maximum attractor in U. Furthermore, if U is connected, then E is connected too. In the following, we consider the stability of the attractors. Let Sv(t) be a family of perturbation of the semigroup S(t), which depend on a parameter r), 0 < 77 < rjo. Let Xn C X (0 < 77 < rjo) be a family of closed subspaces with I)
Xrj is dense in X, and
Srj(t) : Xn —> Xn a continuous operator semigroup. Assume that the semigroup {<S'(£)}t>o has an attractor E which attracts a neighborhood U c X of S. We also assume that for each bounded interval /C(0,oo), sup sup dist(Sr^(i)u0, S(t)u0) —> 0 as r) —> 0, uoet/nx, t€i where the distance of two sets A and B C X is denned by dist(A, B) = sup inf x  y\\Xx€AyeB
(2.21)
Theorem 2.7 Under the above assumptions, if for each rj > 0, 5^(t) has an attractor T,v which attracts UnXv, where U is a neighborhood o/E,,nS independent off], then T,n converges to E in the distance (2.21): dist(S,,,I!)—»0
asrj^O.
32
Bifurcation Theory and Applications
2.4
Center Manifold Theorems
2.4.1
Center and stable manifolds in R"
We consider the system of ordinary differential equations
{
dx — =Ax + G1(x,y,X),
(2.22)
f
ft=By + G2{x,y,\),
where x e Rm, y e Rn~m (0 < m < n), A and B are the m x m and (n — m)x (n — m) matrices respectively, Gi(x,y, A) (i = 1, 2) are continuous on A, and Cr (r > 1) on (x, y) € Rm x M" m , and Gi(ar,y, A) =0(1*1,15,1)
(i = 1,2),
V A e R1.
(2.23)
The following is the center manifold theorem. Theorem 2.8 Suppose that all eigenvalues of A have nonnegative (resp. nonpositive) real parts, and all eigenvalues of B have negative (resp. positive) real parts. Then for the system (2.22) with the condition (2.23), there exists a Cr function, called the center manifold function, h(, A) : D —> Rnm,
D C Rm a neighborhood ofx = 0,
such that h(x, A) is continuous on A, and (1) h{Q1\)=Q,h'x{Q,\)=O; (2) the set Mx = {(x,y) \xeDcRm,
y = h(x,X)},
called the center manifold, is a local invariant manifold of (2.22); (3) if M\ is positively invariant (resp. negatively invariant), namely z(t,tp) € M\ (resp. z(—t,ip) G MA) V t > 0, then M\ is an attracting set of (2.22) (resp. a repelling set), i.e. there is a neighborhood U c M " of M\, as ip € U we have lim dist(z(t, ip), M\) = 0 t—>oo
(resp. lim dist(z(t,i/>), M\) = 0), t—>oo
where z(t, ip) = (x(t, ip),y(t, ip)) is the solution of (2.22) with the initial value z(O,ip) = ip.
Introduction to Dynamic Bifurcation
33
Property (1) means that the center manifold M\ C W1 is tangent to the eigenspace M.m of A at z = (x, y) = 0. Although, as we know, the local center manifold M\ may not be unique, the following theorem makes it applicable. Theorem 2.9 There is a neighborhood C / c R n of z = 0 such that every invariant set of (2.22) in U belongs to the intersection of all local center manifolds in U. In the following, we introduce the stable manifold theorem. Theorem 2.10 Let all eigenvalues of A have positive real parts, and all eigenvalues of B have negative real parts. Then there exist two unique manifolds Mu and Ms, called the unstable manifold and stable manifold of (2.22) at z = 0, which are characterized by
Mu = lz e Rn I lim S(t)z = o\ , Ms = lz G W1 I lim S(t)z = OJ , I
t—>oo
J
where S(t) is the semigroup generated by (2.22). Moreover, Mu and Ms are tangent to the eigenspaces of A and B respectively at z = 0: TZ=OMU = R m ,
TZ=OMS = K" m .
Therefore, the stable manifold Ms and the unstable manifold Mu are transversal at z = 0. Consider the following general system ' dx — < ^=By
=Ax\G1{x,y,z,\), + G2(x,y,z,X),
dz — =Cz +
(2.24)
G3(x,y,z,\),
where x G E m , y £ Rk, z € R n , Gi(x,y,z,\) = o(\x\, \y\, \z\) (t = 1,2,3). We have the following invariant manifold theorem. Theorem 2.11 Let the eigenvalues of A have positive real parts, the eigenvalues of B have negative real parts, and the eigenvalues of C have zero real parts. Then, the system (2.24) has three locally invariant manifolds Mu, Ms and Mc, which are tangent to the eigenspaces of A, B, C
34
Bifurcation Theory and Applications
respectively:
T'
A/Tu
u>=O»w
lUm
T1
— IK ,
H/f3
TD>^
= K ,
1W=QM
T1
A/fC
TO7"1
=K
J.w=oM
,
where Mu is the unstable manifold, Ms is the stable manifold, Mc is the center manifold. Moreover, if Gi (i = 1,2,3) are Cr (r > 1) then Mu, M", Mc are Cr. 2.4.2
Center manifolds for infinite dimensional
systems
Now, we introduce the infinite dimensional version of center manifolds theorems. Let H and Hi be two Hilbert spaces, and Hi C H be a dense inclusion embedding. We consider the nonlinear evolution equation given by tiG^AeR1,
f ^ = Lxu + G(u,A),
1 «(0) = u0,
(2.25)
where L\ : Hi —> H are parameterized linear completely continuous fields depending continuously on A £ R1, which are defined by < A : Hi —> H \ B\ : Hi —> H
linear homeomorphism, parameterized linear compact operators.
(2.26)
We first consider the case in which L\ : H\ —> H is a sectorial operator. Then, one can define fractional power operators L" for a £ l ' with domain Ha = D{L$). Furthermore, we assume that the nonlinear terms G(,A) : He —> H for some 0 < 8 < 1 are a family of parameterized Cr (r > 1) bounded operators continuously depending on the parameter A g l 1 , such that G{u, A) = o(Htf,),
(0 < 6 < 1), V A G R1.
(2.27)
We assume that the spaces Hi and H can be decomposed into •Hi=E$®E$, tiixnEi < oo, < H = [email protected], E2 = closure of E% in H,
(2.28)
Introduction to Dynamic Bifurcation
35
for A near Ao, where E^, E2 are the invariant subspaces of L\, i.e. L\ can be decomposed into L\ = C\ ® C2 such that for any A near Ao,
( L2 = L\\E\ . t,2 —> b2 ,
(2.29)
where the eigenvalues of C2 possess negative real parts, and the eigenvalues of L\ possess nonnegative real parts at A = Ao. Thus, for A near Ao, equation (2.25) can be written as
{%=C*1x + Gl(x,y,\),
(2.30)
where u = x + y £ Hi, x £ E$, y £ E%, G»(x,j/,A) = P{G(u, A), and Pi : H —» Ef are canonical projections. We have the following center manifold theorem. Theorem 2.12 Assume (2.26)—(2.28). Then there exists a neighborhood of Ao given by A — Ao < 5 for some S > 0, a neighborhood O\ C E$ of x = 0, and a C1 function h(,X) : O\ —> E^iO) depending continuously on X, where E^O) is the completion of E2 in the Hgnorm, with 0 < 6 < 1 as in (2.27), such that (1) ft(0,A) = 0, ^(0,A) = 0, (2) the set Mx = {(x,y) £H\x£Ox,y
= h(x,X) G E%(9)} ,
(2.31)
called the center manifolds, are locally invariant for (2.25), i.e. for each u0 £ Mx, ux(t,u0)£Mx,
V0
for some i(wo) > 0, where u\(t,uo) is the solution of (2.25); (3) if (x\(t),y\(t)) is a solution of (2.30), then there is a f3\ > 0 and k\ > 0 with kx depending on (x\(0),yx(0)) such that
WyxWhixxW^WnKkxeP**.
36
Bifurcation Theory and Applications
If we only consider the existence of the local center manifold, then the conditions in (2.29) can be modified in the following fashion. Let the operator L\ = Lx ® £2 > a n d £2 be decomposed into / fX
rX
ffi
rX
*2 — *"21 ^ ^22» £/ 2 = &21 ©  ^ 2 2 '
. ^
= ^0^2,
(2.32)
dim .E^i = d i m ^ < 00, , ^2i : ^2i ~* ^2i
are
invariant (i = 1,2),
such that at A = Ao [ eigenvalues of C\ : E^ —> E± have zero real parts, < eigenvalues of £31 : ^21 ~* ^21 have positive real parts, and ( eigenvalues of ££2 : ^22 ~* ^22
nave
(2.33)
negative real parts.
Theorem 2.13 Assume (2.26)—(2.28), (2.32) and (2.33). conclusions (1) and (2) in Theorem 2.12 hold true.
Then the
We now consider the case where L\ : Hi —> H generates a strongly continuous semigroup of linear operators. Assume that G(,X) : H —> H are Cr (r > 1) bounded continuously depending on A s R 1 , and G(u, A) = o(u JJ),
VAGK1.
(2.34)
Also, we assume that for any uo S Hi, the equation (2.25) has a unique solution u{t) £ Hi for all t>0. Theorem 2.14 Assume that L\ : Hi —» H generates a strongly continuous semigroup of linear operators, and the conditions (2.26), (2.28), (2.29) and (2.34) hold true. If the strongly continuous semigroup S\(t) generated by £2 satisfies
II5A(t)II < Kxea^,
(2.35)
for some constants K\ >l,a\>0, then there exists a C1 center manifold function h{, A) : B\ —> E2 such that the assertions (1)—(3) in Theorem 2.12 hold, in which the center manifold M\ is replaced by
MA = {(x,y) £H\x£BxcEi,y
= h(x,y) G E$ C H] .
37
Introduction to Dynamic Bifurcation
Remark 2.3 Theorem 2.14 can be proved by using the same method as in [Henry, 1981], which is useful in the dynamic bifurcation of nonlinear wave equations. 2.4.3
Construction of center manifolds
In later discussions, it is necessary to show how the center manifold is constructed. In this subsection we give a sketch of the proof for Theorem 2.12. Let pE:Ef^> [0,1] be a C°° cutoff function defined by H V ;
f 1
if x < e,
\ 0
if x > 2s,
for some e > 0. We denote CQ>l{E$,E${6)) = {h : E$ > E%{0) \ h{0) = 0, h is Lipschitz} . We need to find a function h € C°'x(^E^, E^B)) satisfying h()= [
J — oo
ec^Pe(x(T,))G2(x(T,),h(x(r,.)))dT
(2.36)
where x(t,xo) is a solution of the ordinary differential equation
{
f/r
=C>x
+ Mx)Gl{xMx),X),
(237)
x(0) = xo,
Then, it is easy to see that the function y(t,h(x0)) = h(x(t,x0)) = / J—oo
= /
J—oo
= f
J — oo
ec*TpeG2(x(T,x(t,xo)),h)dT ec*TpeG2(x(t
+ T,x0),h(x(t + T,x0)))dT
ec^t^pE(x(T,xo))G2(x(T,xo),h(x(T,xo)))dT
satisfies the equation f ^ = £$V + \ 1,(0) = h(x0).
Pe(x(t,xo))G2(x(t,xo),y),
38
Bifurcation Theory and Applications
Thus, (x(t,xo),h(x(t,xo))) is a local solution of (2.30), and the manifold given by (2.31) is locally invariant for (2.25). Hence, the existence of the center manifold of (2.25) is referred to as the existence of the fixed point of the mapping
F : C°>l{El,E${6)) —> C 0 ' 1 ^ , ££(
F(h)=f
(2.38)
J — oo
By the assertion (3) in Theorem 2.5 for a sectorial operator, one can prove that under the condition (2.27) the mapping F denned by (2.38) is a contraction operator for £ > 0 small. For the details of the proof, see [Temam, 1997; Henry, 1981], When the nonlinear terms G(, A) in (2.25) satisfy (2.34) and the linear part satisfies the condition (2.35), we can derive Theorem 2.14 in the same fashion.
2.5
Hopf Bifurcation
We consider the ordinary differential equations as follows dr
(2.39)
— =Ax + G(x,X), where i g l " (n > 2), A £ R1, A a parametrized n x n matrix
(
an(A) ••• a i n (A)\ ;
;
.
ani(A) ••• ann(X)J
and G : R" x R1 > R" a Cr (r > 1) mapping with G(x, A) = o(:r),
uniformly for A in any bounded set of R1.
(2.40)
Let all eigenvalues (counting the multiplicity) of A be given by /?i(A),
Introduction to Dynamic Bifurcation
39
f32(X), • • , /? n (A). Assume t h a t ' ft (A) = 02 (A) = a(A) + ip(A)
near A = A o ,
«(Ao) = 0, (241)
< da(X0) A»(Ao) ^ 0, Re/?j(Ao)^0
Vj>3.
(2.42)
Theorem 2.15 (Hopf Bifurcation). Assume (240)—(242). Then, equation (2.39) bifurcates from (re, A) = (0, Ao) a periodic orbit F(A), i.e. there exists a sequence of periodic orbits T\n of (2.39) such that
{
lim Xn = A o ,
lim d(Txn,0)=
n—>oo
lim max
\x\=0.
(2.43)
n—*oo x£Fxn
To see this theorem, with a coordinate transformation, we decompose the equation (2.39) into
{
 ^ =a(\)xi
£
+ p(\)x2 + G1(x1,x2,y, X),
—1 = p(X)xi + a(X)x2 + G2(xux2,y, X), at ^=By + G3(xl,x2,y,X),
(244) (2.45)
where B is an (n — 2) x (n — 2) matrix, which has the eigenvalues /? 3 (A),..,/3 n (A). By the center manifold theorem (Theorem 2.11), for (2.44) and (2.45) there is a Cr center manifold function h\ : U\ —> R
,
where £l\ C K2 is a neighborhood of (x\, x2) = 0, such that y(t) = hx{x1{t),x2(t)) is a solution of (2.45), where {x\(t),x2(t)) is a solution of (2.44). Thus, the bifurcation of periodic orbits of (2.39) is reduced to the existence of
40
Bifurcation Theory and Applications
periodic solutions of the following equations
{
—! = a(X)x! + p(X)x2 + Gi(xi,x2, hx(xi,x2), A), dx2
(2'46)
— = p(X)x1 +a(X)x2 +G2(xi,x2,h\(x!,x2),X). Prom (2.46), the Hopf bifurcation theorem can be easily derived from the PoincareBendixon theorem. 2.6
Notes
2.2 We refer the readers to [Pazy, 1983; Henry, 1981] for sectorial operators, and semigroups of linear operators. 2.3 For more detailed discussions on infinite dynamical systems theory, the readers are referred to [Temam, 1997; Vishik, 1992]. 2.4 The proofs of Theorems 2.82.11 can be found in [Chow and Hale, 1982; Hirsch et al., 1977; Iooss and Joseph, 1980]. The infinite dimensional center manifold theorems, Theorems 2.12 and 2.13, can be found in, among others, [Henry, 1981; Temam, 1997]. 2.5 The Hopf bifurcation theorem is introduced in many basic textbooks on ordinary differential equations and dynamical systems; see, among others, [Chow and Hale, 1982; Guckenheimer and Holmes, 1983]. PoincareBendixon theorem used in the sketch of the proof of the Hopf bifurcation theorem can be found in these references as well.
Chapter 3
Reduction Procedures and Stability
In this chapter, we focus on the following three topics (1) spectral theorems, (2) reduction procedures, and (3) asymptotic stability. These are crucial components of the bifurcation theory developed in this book.
3.1
3.1.1
Spectrum Theory of Linear Completely Continuous Fields Eigenvalues of linear completely continuous fields
Let H and Hi be two Hilbert spaces, and Hi C H be a dense and compact inclusion. A linear operator L : Hi —> H is called a completely continuous field if L = A + B : Hi > H, A : Hi —> H a linear homeomorphism, B : Hi —> H a linear compact operator. A number A = a + i/3 G C is called an eigenvalue of a linear operator L : Hi —> H if there exist x, y G Hi with x ^ 0 such that Lz = \z,
(z = x + iy), 41
(3.1)
42
Bifurcation Theory and Applications
and the space
Ex= U {x,yeH!  (L  A)"z = 0, z = x + iy} is called the eigenspace of L at A, and x, y £ E\ are called eigenvectors1 of L. Alternatively, (3.1) can be equivalently expressed as Lx = ax — j3y, Ly = /3x + ay. Definition 3.1 A linear mapping L* : Hf —» H is called the conjugate operator of L : Hi —> H, if (Lx,y)H
= {x,L*y)H,
V x,j/ G H±.
Here iJj c i? is a dense inclusion. A linear operator L : Hi —» H is called symmetric if L — L*. We remark here that in this definition, we first treat L as an closed and densely defined unbounded operator from H to itself with domain Hi. Then L* : H —> H make sense, and has domain H$ C H. In all applications presented in this book, however, we have Hi = Hf. Definition 3.2 We say that a linear operator L : Hi —> H has a complete eigenvalue sequence {Afe} C C if each eigenspace Ek at \k is finite dimensional, and all eigenvectors of L constitute a basis of H. We know that a linear completely continuous field L = —A + B : Hi —> H is a Predholm operator with index zero, and the spectrum of L consists of eigenvalues. Furthermore, the following properties are classical; see [Kato, 1995]. Theorem 3.1 Then
Let L : Hi —» H be a linear completely continuous field.
(1) the spectrum of L consists of eigenvalues; (2) each eigenvalue A € C is isolated, and the corresponding eigenspace E\ is finite dimensional; (3) L and its conjugate operator L* have the same eigenvalue, and the corresponding eigenspaces have the same dimension dimI?A — dim.E£; and throughout this book, eigenvectors are always referred to (generalized) eigenvectors.
Reduction Procedures and Stability
43
(4) if L is symmetric, then the eigenvalues of L are real. The following theorem is on the complete spectrum of symmetric operators. Although this theorem is essentially known, we shall give a proof because this is in a slightly different form from the classical one. We remark here that the proof presented here connects the eigenvalue problem of L with that of L~x o / : H\ —> H\ between the same space H\. With this in our disposal, Theorem 3.1 is obvious. Theorem 3.2 Let L : Hi —» H be a linear completely continuous field, and L be a sectorial operator. If L is symmetric, then L has a complete eigenvalue sequence {Afc} C R1, and all eigenvectors {
(3.2)
Thus, we have (L~1oIx,y)i = (L(L1oIx),Ly)H = {x,Ly)n (by the symmetry of L)
= {Lx, y)x 1
= {Lx,L(L oI)y)H = {x,(L1oI)y)i. We assume without loss of generality that all eigenvalues of L are positive. Hence, the operator L"1 o I : Hi —» Hi is symmetric and compact with respect to the inner product (3.2). For any x £ Hi with x ^ 0, {L~1olx,x)i
= (x,Lx)H = (l}l*x,LV*x)H >0.
Therefore, L~xol: Hi —» Hi has a complete eigenvalue sequence {pk} c R1
44
Bifurcation Theory and Applications
w i t h pk —> 0 a s k —• oo: pk
(3.3)
and the eigenvectors {fk} form an orthogonal basis of Hi under the inner • product (3.2). Hence the theorem follows from (3.3). Remark 3.1 It is still an unknown question if the complete spectrum theorem holds true for general symmetric linear completely continuous fields L : H1 > H. 3.1.2
Spectral theorems
We begin with finite dimensional linear operators. Let M be an n x n matrix, and M* its conjugate matrix. Let /?j (j = 1,... ,n) be all eigenvalues of M (counting multiplicities). Vectors £j £ R n (1 < j ' < n) are called eigenvectors of M if there exist 1 < kj < rrij with m,j being the multiplicity of /3j, such that {MPj)k^j=Q,
(3.4)
when 13j are real numbers, and f ( M  ^ ) f c i ( ^ + ^ + i ) = 0, \(M&+1)fc'&»6+i)=0,
(3.5)
when /3j = /3 J + 1 are complex numbers, with /3J+i being the complex conjugate of 13j. Let £,• = (£ji, • • • ,^jn)* be eigenvectors of M such that (3.6)
MP = PJ, where J is the Jordan form of M, and / & i 6 i  " £m \ P=
^
^

^.
(3.7)
V^ln ^2n • • • £,nn/ Then it is easy to see that Mt(Piy
= (pi)tj*,
(3.8)
45
Reduction Procedures and Stability
where J* is the transpose of J, which is also the Jordan form of conjugate matrix (transpose) M* = M*. Hence if we set ( P  1 ) *  (6, •••,1™), then £i, • • • ,£ n a re eigenvectors of M*,
P1
=
;
W
,
(3.9)
and
/^pn'jfc.e.)
&1&2
 * " . (0,
We note that equations (3.6) and (3.8) are equivalent to the classical Fredholm alternative theorem. Therefore, from (3.10) we get a theorem as follows, which is considered as a unified version of the Fredholm alternative theorem and the Jordan theorem. Theorem 3.3 Let f3j•, (j = 1, • • • ,n) be the eigenvalues of annxn matrix M. Then we can take the eigenvectors {£j £ Rn \ 1 < j < n} of M and eigenvectors {£,• € Rn  1 < j < n} of M* such that
&,l) = sij. The FredholmJordan theorem can be generalized to general linear completely continuous fields. For this purpose, consider a linear completely continuous field L — —A\ B : H\ —• H. As in the proof of Theorem 3.2, we assume that L has a compact inverse given by
L^IiA
+ B)1 :H>H,
where I : Hi —* H is the inclusion mapping. Therefore, the eigenvalue problem of L can be equivalently written in the following form (L'1  Pk)m^k = 0 ,
(3k = X^\
for some m > 1.
(3.11)
46
Bifurcation Theory and Applications
Let H
H = H®C=<^2
I
fc
J
with inner product ( u .u)# = ((wi.vi)j/ + (u2,U2>i/) + i((u2,^i)H (UI,V2)H)
for u = ui + iu 2 , w = vi + iv2 € H
(3.12)
(2) H can be decomposed into the following direct sum
(
H = E1®E2, Ei = the closure of span {ipk  k > 1} in H, E2 = {veH\(v,
(3.13)
Vfc>l};
(3) Ei and E2 are invariant spaces of L~l, and lim \\Lnv\\^n
=0
\/veE2;
n*oo
(4) Let Pi,f3k in the order
£ C be eigenvalues of L~x (counting the multiplicity) \Pi\ >  / 3 2  > ••• >  / H V»i, • • • , 4>Z G H ® C be the
corresponding eigenvectors of (L~i)* = L*" 1 , and let E^ = span {VI,,^}, (E*k=
then there is an eigenvalue (3k+i & C of L~l with /3jt+i = Pfc+i, ow^ l/3fc+i < )8fc.
Reduction Procedures and Stability
47
Remark 3.2 By (3.12) and (3.13), for any u G # , we have the generalized Fourier expansion u = ^2 uWk + v,
v£E2,
uk = (u,
(3.15)
k
In particular, if the operator L = —A + B : H\ —> H has a complete eigenvalue sequence {Afe} c C with eigenvectors {tpk} C H\, then we have the following complete Fourier expansion oo U = ^UkVk, k=l
Uk = {u,
Remark 3.3 Let X be a Banach space, X* its dual space, and B : X  » X a compact linear operator with its conjugate operator B* : X* —» X*. Then the above spectral theorem holds true as well for the eigenvalue problem Bx = \x. Proof of Theorem 3.4 We proceed in several steps as follows. STEP 1. It is known that the space H can be decomposed into direct sums of invariant spaces of L~x as follows
'H=
E1®E2,
H = E{®E* , 1 22 Ei = the closure of span {ip^  k > 1} in H,
(3.16)
E± = the closure of span {ip*k \ k > 1} in H. By the spectral radius theorem, we have f lim L n H^ / n = 0 ,. I n"°° [ lim^ L* n u^ /n = 0
Vv£E2, Vv G E\.
(3.17)
Assertion (3) follows. STEP 2. Let {£i, • • • ,£OT} c Hx
sponding to an eigenvalue A £ C, and {T?I,• • • ,rfM} be the eigenvectors of L* corresponding to an eigenvalue p 6 C. Set & = &i +1&2,
Vj = Vji + iVj2> 1 < ^ < "i, 1 < j < M.
We shall prove that if A ^ p, (Zik,rijr)H
= 0,
V f c , r = l , 2 , \<£<m,
1< j < M,
(3.18)
48
Bifurcation Theory and Applications
and (v,
(v*,
eE%
(3.19) (3.20)
(tek,Z*r)H = 5ej6kr.
For simplicity, we consider only the case where m = M = 2, and the geometric multiplicities of A and p are one; while the general case can be proved in the same fashion. Then we have L^i = A^i, 1
L ^
1
= p ^,
L~% = A  ^ 2 + £ 6 , 1
1
L* rll=p r,t+1r&.
(3.21)
(3.22)
for some e,y G R 1 . When A ^ p , it follows from (3.21) and (3.22) t h a t (£i, »72)# = (^11 + ^12 > V21 + iV22)H
= \(L1t1,r,Z)A = *{Zi,L
V2/H 1
=
*P~ (ZI,V2)H
= 0, *P~1(ZI,V2)H
(ZUV*2)H =
= 0, which imply that (£ifc,*72r)ff = 0 ,
k,r = 1,2.
Similarly, we can induce from (3.21) and (3.22) that <&,»?;>£ =°>
&,V])H
= 0,
1 < t, J < 2.
Thus, we have proved (3.18), and (3.19) can be obtained in the same fashion from (3.11) and (3.17). STEP 3. When restricted to the generalized eigenspace with a particular eigenvalue, the problem is then reduced to Theorem 3.3. Hence Assertion (1) holds true. 4. P R O O F OF ASSERTION (4). It is clear that the numbers defined by (3.14) satisfy 0 < p < HL"1!!. By the spectral radius theorem, if STEP
49
Reduction Procedures and Stability
p > 0 then L~l must have at least an eigenvalue /3 G C. Let fij e C (j > 1) be the eigenvalues of L"1 in the order IAI > Iftl > • • • > l&l > &+i > • • • ,
(323)
and {^j} C H
u= Y^, UJ^J + u> j>fc+i
Urn \\Lnv\\1/n
n—>oo
= 0.
For simplicity, we assume that the eigenvalues (3j are simple, and other cases can be proved in the same fashion. Then we have
\{Lnu,u)6\= / ^
^UjiPj + L~nv, Y,
\j>k+l
=
i>k+l
Yl aj^ + (Lnv,u)H ,
j>k+\
Vrfi+A I
H
(3.24)
where i>k+l
Then by (3.23) and (3.24) we obtain that \/3k+i\ — pk+iThe proof of the theorem is complete.
•
Remark 3.4 By the proof of Theorem 3.4 we see that if ipk is an eigenvector of L corresponding to /3jt, and tpj is an eigenvector of L* corresponding to f3j with /3fc 7^ /3j, then (
50
Bifurcation Theory and Applications
L* respectively corresponding to the same eigenvalue /?, such that
Lip2 = Py>2 +£
(3.25)
_L*
(326)
<™»*{*l *! = * 3.1.3
Asymptotic
'
properties of eigenvalues
We consider a class of linear completely continuous fields L = — A + B : Hi —> H. We assume that A is symmetric, and there exists a real eigenvalue sequence {pk} C R1 and an eigenvector sequence {tfk} C Hi of A:
{
A
Pkfk,
0 < pi < pi < • • • ,
(3.27)
lim pk = oo,
k—^oo
such that {pk} is an orthogonal basis of H. It is easy to see that A is a sectorial operator, and we can define the fractional powers as follows oo
oo
Aax = J2 PZxk
Vx
fc=l
Let
i
(3.28)
k=l
oo
X&H
= Yl xWk e H.
X
oo
"^
= Y,xwk,Y.pk*xl<°°\' fc=l
fe=l
J
(329)
51
Reduction Procedures and Stability
endowed with the inner product oo
=J2plaxkyk.
(x,y)Ha
fc=i
We also assume that B : Hy —> H is bounded for some 0 < 7 < 1.
(3.30)
By Theorem 2.6, the operator L = A + B satisfying (3.27) and (3.30) is sectorial. The eigenvalues of L have the following asymptotic property. Theorem 3.5 Let L = A + B :HX>H satisfy (3.27) and (3.30). If L has an infinite number of eigenvalues Xk — ok + ipk £ C, then ak —> —oo when k —> oo, and for the number 7 < 1 as in (3.30) we have lim r ^ L Proof. satisfy
=
o,
V 7 < 5 < 1.
(3.31)
Let Zfc = Zfc + iyk be the eigenvector corresponding to Xk, and
{
 Axk + Bxk = akxk + pkyk,  Ayk + Byk = pkxk + akyk.
yo.oZj
We infer from (3.32) that ak
(Axk,A2exk) + (Ayk,A2eyk) (Bxk,A2exk) + (Byk,A2eyk) (xk,A™xk) + (yk, A»yk) (xk, A26xk) + (yk, A2°yk) _ INIg + i  (Bxk,A™xk)  (Byk,A™yk)
=
~ Pk
_
~
'
iwii
{Bxk,A29yk){Byk,A™xk)
'
Iklli
where 0 < 9 < 1/2 is a constant to be specified later. By (3.30) we get (B^,A 2 V>l
52
Bifurcation Theory and Applications
Hence W <MJi*»,
,3.33,
\\zk\\o
K l a K  a
II**K + ,  «BI IWWWI» '
h B
IWI! + , + lBIIMIrll»lla. • IMS '
(3 34)
'
<3 35)
'
Since L is a sectorial operator, if the number of eigenvalues A^ of L is infinite, then limfc_oo a^ = limfc_oo ReA^ = —oo. We derive from (3.35) that +e
\\zk\\e
,oo
(fc>oo).
(3.36)
We take 6 > 0 as follows
\e + \ = i
2 < [0 = 7
if7>J,
1 ifO<7<.
(337)
Since 0 < 7 < 1, we obtain from (3.37) that 0 < 6 < \, and f i + 0>max{ 7 ,20}, < ?
[+0>min{7,20}.
(338)
We take z fc  e+ i = 1, V k G N. Then by (3.38) we have 1 > N7,
1 > INfcllM
It follows from (3.36) that \\zk\\g —* 0 (fc —» 00), which implies that Zk —> 0 in Hi+0. Because if a «—> iJ^ is compact for all /? < a, by (3.38) we infer that 2 I l * f c   7  I l * * l l 2 » — > 0 , fc  > o o .
(3.39)
Based on (3.33), (3.34) and (3.39), there is a K > 0 large enough such that
53
Reduction Procedures and Stability
for all k>K, 2\\B\\  N  7  N  2 e
\Pk\ =
,
'
'
,
(by (3.37)) 2\\B\\\\zk\\2e 25 1 20 z IIZL.II ll z *:i 9llllztll *:l e " '
+
By t h e interpolatio n of space s (see [Temam, 1997]),
H2e =
[Hi+e,He]e,
where e is determined by (1  e){\ + 9) + eO = 29, i.e. £ = 1  20. We have the interpolatio n inequality • I _ it ^ / i I I , 20  _   l  2 0 2fc20 S ^02fci + 0Zfc0 i
where Ce > 0 is a constant. Thus, we obtain from (3.40) tha t
J^L<2fl.Cfl,
Vk>K,
which implies (3.31). The proof is complete. 3.1.4
D
Generic properties
We present in this subsection a density theorem on operator s with complete spectrum, and a genericity theorem for simple eigenvalues. The spectra l completeness is a very interesting and difficult problem. By Theorem 3.2 we know tha t if L : Hi —» H is a symmetric sectoria l operator , then L has a complete eigenvalue sequence. But, for genera l sectoria l operators , we don't know if the spectra l completeness theorem is still true. However, we have the following density theorem on operator s with complete spectrum. Let L = A + B : Hi > H. We assume tha t A : Hi —» H is a sectoria l operator , A has a complete eigenvalue sequence {A^} C C, ReAfc > 0 V k > 1, B:Hi*H
satisfies (3.30).
' '
'
54
Bifurcation Theory and Applications
Let S(HUH) = {L:Hi^H SA(HlfH)
= {Le
satisfies (3.41) and (3.30)} ,
S(HUH)
\A:Hi+H\a
fixed}
.
Then we have the following density theorem for completion spectrum. Theorem 3.6 There exists o dense set D C S{H\,H) (resp. D C SA{HI,H)) such that each operator L £ D has a complete eigenvalue sequence. Remark 3.5 In fact, it is reasonable to conjecture that each L € S(Hi,H) satisfying (3.41) and (3.30) has a complete eigenvalue sequence. Proof of Theorem 3.6. Let L = A + B £ S(Hi,H), and {
^UHPi. *=1
As PnB : H\ —> H is finite rank, the operator Ln = —A + PnB has a complete eigenvalue sequence for any n € N. In addition, since B : Hi —» H is compact, we have Yanio\\BPnB\\s{Hun)=Q. Thus we obtain
J m ^ L  Ln\\S(HltH) = JLjm, IIB  PnB\\s{HuH)
= 0.
Therefore, the proof of the theorem is complete.
•
Now we consider the genericity of simple eigenvalues. An eigenvalue A € C of L : Hi —> H is called simple if the eigenspace E\ has dimension (l if A = real, dim^A = < [ 2 if A = complex. Let the eigenvalues {Afc} of a linear operator L G S(Hi,H) be given in the following order Ai < A2 < • • • <
Afc <  A f c + i  < • • •
.
The following is the genericity theorem for simple eigenvalues.
Reduction Procedures and Stability
55
Theorem 3.7 For any integer m > 1, there is an open and dense set O c S(Hi,H) (resp. O C SA(Hi,H)) such that each L G O has at least m eigenvalues, and the first m eigenvalues {\j \ j = 1, • • • , m} are simple. Proof. By the classical theory of linear operators, each isolated eigenvalue depends continuously on the operators [Kato, 1995]. Therefore, the set O of all linear operators having at least m eigenvalues is open and dense in S(HUH) (resp. in SA(HUH)). Let O C O be the set of all linear operators whose first m eigenvalues are simple. Obviously, O is an open set. We shall prove that for any L = —A + B G O, there is a sequence Ln = —A + Bn G O, which converges to Lin S(Hi, H). Let {fk} be the eigenvectors of L. Without loss of generality, we assume that the first eigenvalues of L have multiplicity two, i.e. Ai = A2, and L
Lif2 = A1V2 + ipi.
(3.42)
By (3.15), for any u £ H, u = J2k ukfk + v, v G Ei We define a linear operator Tn : Hi —> H by T nu =  u i v i . n Then we infer from (3.42) that the first two eigenvalues Ai and Ai + ^ of the operator Ln = L + Tn are simple, i.e. Ln(p2  nipi) = \i(
56
3.2
Bifurcation Theory and Applications
Reduction Methods
We consider the following nonlinear evolution equation
 g = L>« + G (.,A), I »(0)  «„,
(343)
where L\ : H\ —> H is a parameterized linear bounded operator, G(,\) : ffi > ff a C (r > 1) mapping depending continuously on A 6 I 1 , and Hi c H is a dense and compact inclusion. 3.2.1
Reduction procedures
Reduction to center manifolds To investigate the stability and bifurcation of (3.43), it is crucial to reduce it to the center manifold. We start with a canonical reduction of (3.43). Let L\ = — A + B\ : Hi —» H be a linear completely continuous field, and G(, A) : Hi —> H be a C°° operator, which can be expressed as oo
(3.44)
G(u,\) = YJGn(u,X), n=k
for some k > 2, where Gn : Hi x • • • x Hi —» if is an nmultiple linear m a p p i n g , a n d Gn(u,X)
= Gn(u, • • •
,u,X).
Let /3i(X) G C be the eigenvalues (counting the multiplicity) of L\, and ej(A) and e*(A) be the eigenvectors of L\ and L*x, respectively, corresponding to A (A). Let {hi} be a canonical orthogonal basis of H, and let oo
oo
3= 1
3= 1
Then {ej} and {e*} generate two infinite matrices (in 6 i   A ^ = U12 62 • • • I
f =
(in Hi • • • \ ^1*2 & • • • .
Let L and £* be two infinite matrices induced by L\ and L*x respectively, £ = (ay), C* = (aijY,
a,ij 
(L\hj,hi)H.
57
Reduction Procedures and Stability
Definition 3.3 The eigenvectors {e^} and {e*} of L\ and L*x are called canonical if under an orthogonal basis they satisfy
d = ^J, £T = r A
(345)
where J = J\ is the Jordan matrix
h .
J = Jx=
,
V°
(3.46)
/
and Jjt are the Jordan blocks defined by //? £k Jk=
"'
0 \ "" '••
if
£k
/?jfc+i = • • • = /3jfc+mfc = /?, and by
/ a p p a
ek 0 0 £fc
Jfe=
0 \
•••
e* 0
a
\ 0
P
o ' £k
—p
aj
if /3jfc+i = • • • = f3jk+mk =a + ip. It is easy to see that the Jordan matrix J is independent of the choice of the orthogonal basis {hi} C H. By Theorem 3.4 and Remark 3.4, if {e,} and {e*} are the canonical eigenvectors of L\ and L\, then ( =0
^ f 0
iti^j,
if i = j .
(3.47)
58
Bifurcation Theory and Applications
Let {ei} and {e*} be canonical. By (3.15), for any u e H,
u = 'Y2xkek+v,
v £ E2 .
k
Then, by (3.45)—(3.47), the equation (3.43) can be expressed as
/zA
/x{\ 1 * 2
^
= JA
*2
/gi(u,X)\ +
= Lxv + PG(u, A),
veE2,
(3.48)
(3.49)
where L\ = L\\E2 '• ^2 —* E2, P '• H —> ^2 the projection, and flfc(tt,A)=,
\N
(ek,e*k)H
(G(u,X),el)H.
Now, we return to consider the reduction of (3.43) to the center manifold. Assume that the eigenvalues ft (A) satisfy
{
<0
if A < Ao,
(3.50) = 0
ifA = A0,
>0
ifA>A 0 ,
(Re^{\0)>0 [ReP^Xo) < 0
l
(3.50)
m<j<m + n, m + n<j.
Let there be a center manifold of (3.43) near A = Ao with the center manifold function given by S(.,A):fi—>££, (3.52) where fl C Em is a neighborhood of u = 0, and Em = ij^Xiei(X)
(x1,,xm)€Rm\,
(u,e*(\))H = 0, 1 < * < m l .
Ei = \ueH
Thus, under the conditions (3.44) and (3.50)—(3.52), the equations (3.48) and (3.49) are reduced to the center manifold in the following form dx = JmXX + g(x,X), at
1r
(3.53)
59
Reduction Procedures and Stability
where Jm\ is the Jordan matrix corresponding to the first m eigenvalues /3 1 (A),.,/3 m (A)ofL A ,
,gm(um + $(um,A),A))*,
g(x,X) = (gi(um + $(um,\),\),
gi(u, A) are as in (3.48), and um = J2f=i xier In some applications, the stability and bifurcation of (3.43) at (a;, A) = (0, Ao) is determined by the first approximation of (3.53), given by dx — = JmXX + Fk(x,X),
(3.54)
where k > 2 is given in (3.44), and Fk is a ^multilinear vector field given by
(Fk(x,X) = (fki(x,X),Jkm(x,X)y,
e
<355>
(«^<^HJ>4 'i ' Second approximation
We see that the first approximation (3.54) of (3.53) is independent of the center manifold function (3.52). There are many cases where we need to consider the second approximation of (3.53). Assume that the linear operator L\ : Hi —> H has a complete eigenvalue sequence {/3j(A)} C C In this case, the equation (3.43) is equivalent to the system of (3.48). First, we compute the strongly continuous semigroup T(t) generated by L\. Let u = Y^=\ xk(t)ek be a solution of the linear equation AW
I dt
I «(0) = V,
'
(3.56)
where tp = Yl'kLi fk^k Then, the equation (3.56) is equivalent to ( dx_ \ dt ~ for x = (xi(t),X2(t), itly as follows.
JxX
'
(3.57)
•••)*. The solution of (3.57) can be calculated explic
60
Bifurcation Theory and Applications
When /^(A) = • • • = /3 J+m (A) = /3(A) = real, and L\ej+k
= (3j(\)ei+k
£j = 0, ej+k ^ 0, k > 0,
+ ej+kej+ki,
we have k
xi+k(t) = ip^e?* + J2m £i+*tefj+eiept.
(3.58)
l=i '
When j3j+k = Pj+k+1
= a + ip, k = 0, • • • , 2m, m > 0, and = Pjej+k +
L\ej+k L\ej+k+i
£j+kej+k2,
= Pjej+k+i
+£j+kej+ki,
we have
f
I xj+k
fc
1
+ tar,+fc+1 = ^+ke^^
+ ^  £ , + , t'^ + / _ i e (+*/')*,
<
f=1
«•
(cs.oy)
[ V"j+fc = Vj+fc + ifj+k+iThe semigroup T(i) : H —» i7 is given by oo
r(i)^ = ^ ^ ( f ) e / t , fc=i
(3.60)
with xfe(t) given by (3.58) and (3.59). It follows from (3.58)—(3.60) that if Re/3j < 0 (j = 1,2, • • •), then there exist M > 1, 5 > 0 such that \\T(t)\\<MeSt,
Vi>0.
(3.61)
We are now in position to describe the second approximation of (3.53). For simplicity, we only consider the case where f3j are real with L\ej — /3,e., (j = 1,2, •••), and /?i = • • • = (3m Then, the second approximation of (3.53) is given by /x[\
//3(A)
{':} ={
0 \
/Xl\
••
i \+F2(x,\) + F3(x,\)+F(x,\), (3.62)
where F2 and F3 are as in (3.55), F(x, A) = {fx{x, A), • • • , fm(x, A))*, and MXA)=
U^ \ek,ek)H
£ ^*^^.je=l
(363)
Reduction Procedures and Stability
61
Here OO
a
ije=
j
Z)
Tm
ft
\le
e*\
(G2{ej,ee,X),e*n)H
x (G 2 (e*,e n , A) + G 2 (e n ,e*, A),e*k)n, and G2(u, w) is the 2multiple linear operator as in (3.44). We prove (3.62) and (3.63) in the following. By (3.53) we have
^
=/3i(A)*fc + 5fc ( f > j e j + (s,A),Aj
(3.64)
where x = YliiLi x%£i The center manifold function OO
*(z,A)= Y,
*n(a:,A)c
n=m+l
satisfies (2.36), i.e. *n(i,A)=/
e^p £ (G(z(r,x)+$,A), e ;)^r,
(3.65)
where z(i, x) = X)£li zs(*i 2;)e2 satisfies
( z4(0) = a:*. Hence Zi(t, x) = Xie0lt + o(\x\).
(3.66)
Inserting (3.66) into (3.65), by (3.44) we get m
x
*n(rr, A) = Y, ^ i,e=i
\
J
0
°°
e^^Tdr{G2(euee,
A),e*n)H +o(\x\2). (3.67)
62
Bifurcation Theory and Applications
Thus, from (3.64) and (3.67) it follows that
I
+ Yl a'ljiXiXjxA + o(z3), i,j,e=i
J
which yields the second approximation (3.62) and (3.63). Prom the above discussions, it is easy to obtain the following theorem, providing an approximation of the center manifold function. This approximation will be used in applications in later chapters of the book. Theorem 3.8 Under the conditions (3.50) and (3.51), if L\ is a sectorial operator and G satisfies (3.44), then the center manifold function $(x, A) can be expressed as * ( * , A) = (£$)1P2Gk(x,
A) + O(\Re(3(\)\ • \\x\\k) + o(\\x\\k),
(3.68)
where £% «s given by (2.29), P2 : H —> E2 the canonical projection, x e E$, and f3(X) = (/3i(A), • • • ,/3m(A)) the eigenvalues of C\.
3.2.2
Morse index of nondegenerate singular points
We say that v G Hi is an equilibrium point, or a steady state of (3.43), if v satisfies Lxv + G{v,X)^0.
(3.69)
In order to investigate the stability and dynamic bifurcation of (3.43), it is necessary to consider the geometric properties of singularities of (3.69). To this end, we shall use the LyapunovSchmidt method to reduce the equation (3.69) to a finite dimensional algebraic equation, then consider the singularities of the reduced equation. Let the operator G(, A) : Hi —> H be compact and satisfy (3.44). Then, for any uo £ Hi, the derivative operator DUG(UQ, A) : Hi —> H is a linear compact operator.
Reduction Procedures and Stability
63
Definition 3.4 A singular point UQ e H\ of (3.69) is nondegenerate, or is regular, if the derivative operator Lx + DuG(u0, X):H1—^H
(3.70)
of L\ + G at UQ is a linear homeomorphism. Definition 3.5 Let UQ £ H\ be a nondegenerate singular point of (3.69), and for all eigenvalues of the operator defined by (3.70), Re0i ^ 0. We define the Morse index of uo by k = number of eigenvalues of (3.70) having positive real part. By Theorems 3.11 and 3.12, if the Morse index k = 0 of a nondegenerate singular point UQ of (3.69), then uo is a locally asymptotic stable equilibrium point of (3.43). If the Morse index k > 1, then the singular point u0 is called a saddle point of (3.43). It is known that a saddle point u0 of (3.43) with Morse index k hasfcdimensionalunstable manifold and a stable manifold with codimension k in H. Let (3.50) and (3.51) hold, and { e j and {e*} are canonical eigenvectors of L\ and L^. Near A = Ao, the equation (3.69) has the decomposition as follows
/*i\ (gi{x + $,\)\ Jmx : + : =0,
\W
(3.71)
\gm(x + $,\)J
£x$ + PG(x + $,\) = 0,
(3.72)
where Jm\ is the Jordan matrix corresponding to the first m eigenvale E ues /?!,••• ,/3m of Lx, gk{u,X) as in (3.48), x = YX=ixiei m = span{ei, , e m } , $ G E^ = {u e Hx  (u,e*)H = 0, 1 < i < m}, and Cx = LX\E± : Ei —> Ei, the closure of E^ in H, P : H —> E^ the projection. By (3.51), the linear operator Cx is invertible. By the implicit function theorem, near (x,\) = (0, Ao) there is a solution of (3.72) $ = $(z,A),
x€Em
(3.73)
64
Bifurcation Theory and Applications
Putting (3.73) in (3.71), we get the following reduced equation of (3.69) /*i\ Jmx
:
/ s i ( x + *(x,A),A)\ +
\xj
:
= 0.
(3.74)
\gm(x + $(x,\),\)J
Under the conditions (3.50) and (3.51), the equation (3.69) may cause bifurcation from (0, Ao). By Theorem 1.9, if x(X) is a bifurcation solution of (3.74), then w(A) = x(X) + $(z(A), A) is a bifurcation solution of (3.69). The following theorem is useful to verify the regularity of a bifurcated solution of (3.69). Theorem 3.9 Let u(X) = x(X) + $(x, A) be a bifurcation solution of (3.69) from (0,Ao). Then u(X) is a nondegenerate singular point of (3.69) if and only if x(A) is a nondegenerate singular point of (3.74). Proof.
The derivative operator of (3.74) at x\ = x(X) is given by Jmx + Dxg + Dsg o Dx$\x=Xx
: Em —> Em.
(3.75)
On the other hand, the derivative operator (3.70) of (3.69) at UQ = x\ + $(XA,A) is invertible if and only if the following equations have no nonzero solution u = (y,w); (JmX + Dxg) • y + D$g w = 0,
(3.76)
(Cx + D^PG) • w + DXPG y = Q,
(3.77)
where y G Em, w 6 E^, and the derivative is taken at tto = x\ + $(x\,X). Since uo is small near Ao, by (3.44), £)$PG is also small. Hence the operator
B = Cx + D*PG{u0, X) :H!—>H is invertible. Thus it follows from (3.77) that w = B~1oDxPGy.
(3.78)
Putting (3.78) in (3.76) we obtain (Jmx + Dxg  D*g o B'1 o DXPG) • y = 0. We deduce from (3.72) that £>**•!/= (  5  1 o DXPG) • y.
(3.79)
Reduction Procedures and Stability
65
Therefore the operator (3.76) is invertible if and only if (3.79) has no nonzero solution in Em. The proof is complete. • In general both the vector fields in (3.53) and (3.74) are not the same because the center manifold function is different from the implicit function. Since the Morse index of a singular point is also characterized by its stable and unstable manifolds, if the Morse index of a singular point XQ of (3.53) is k for A near Ao, then under the conditions (3.50) and (3.51), the Morse index of singular point UQ = XQ + $(xo, A) of (3.69) is k + n, n as in (3.51). But, this assertion is not generally true for the equations (3.69) and (3.74). However, for the firstorder and secondorder approximations of (3.74), we have the following result. The equation (3.74) can be rewritten as oo
JmXX + Y,Fp(X>X)=0> p=k
( 3 ' 8 °)
where k > 2 is given as in (3.44), x G R m , and Fp is a pmultiple linear function of x. The following equation
Jmxx+
k+Nl
J2 Fp(x,X)=0
(3.81)
P=k
is called an iV'order approximation of (3.74). Theorem 3.10 Each nondegenerate singular point of (3.81) uniquely corresponds to a nondegenerate singular point of (3.69) near (0, Ao). If XQ is a nondegenerate singular point of (3.81) with N = 1, and XQ has Morse index k (k > 0), then the corresponding singular point u0 of (3.69) has Morse index k + n, where n as in (3.51). The proof of Theorem 3.10 is trivial. For N = 2, if the first m eigenvalues of L\ are real at Ao, then Theorem 3.10 holds true also, because the vector field in (3.81) differs with that of (3.53) in a sufficiently small quantity near (0, Ao). We remark here that the reason why we are concerned with the topological structure of singular points of (3.81) rather than that of (3.53) is that the computation of the implicit functions is relatively easier than that of the center manifold functions.
66
Bifurcation Theory and Applications
3.3
Asymptotic Stability at Critical States
In the attractor bifurcation theory to be addressed in the later chapters, we shall see that the asymptotic stability of equilibrium points at critical states, i.e. at A = Ao in (3.50) and (3.51) with n = 0, is very important. This section is devoted to the problem. 3.3.1
Introduction to the Lyapunov stability
Definition 3.6
Let v e Hi be an equilibrium point of (3.43).
(1) We say that v is stable in H, if there is a neighborhood U C H of v, such that for any wo £ U the solution u(t, uo) of (3.43), if it exists, satisfies \\u(t, UQ) — V\\H —> 0 as uo —> v, V t > 0. (2) v is called locally asymptotically stable in H, if lim \\u(t,u0) — V\\H = 0,
V w0 € U.
t—>oo
lfU = H, then v is called globally asymptotically stable. (3) v is called exponentially stable if there are constants a,fi > Q such that \\u{t,uo)v\\H
V t > 0.
The following theorem is a classical stability result; see [Henry, 1981]. T h e o r e m 3.11 Let L : Hi —> H be a sectorial operator, G : Ha —> H be Cr (r > 1) for some a < 1, and v G Hi be an equilibrium point of (3.43). If the spectrum a of L + DG(v) satisfies that Rea < e, for some s > 0, where DG(v) is the derivative operator of G at v, then v is locally asymptotically stable with exponential decay. The following stability theorem is essentially known, which is useful on the stability of nonlinear wave equations. T h e o r e m 3.12 Let L : Hi —> H be a linear bounded operator, which generates a strongly continuous semigroup T(t), and G : H —> H be continuous satisfying \\G(u)\\H < C\\u\\kH,
for some k> 1.
(3.82)
Reduction Procedures and Stability
67
If there are constants M > 1, 5 > 0 such that \\T(t)\\<MeSt,
V*>0
(3.83)
then u = 0 is an equilibrium point of (3.43), which is locally asymptotically stable with exponentially decay. Proof.
The solution of (3.43) reads u(t) = T(t)u0 + f T(tJo
T)G(u)dr.
It follows from (3.83) that IMjj<Me"«off + Af / Jo
es^\\G{u)\\HdT.
Let u(t)i/ = e~stz{t). Then we infer from (3.82) that z(t) < M\\uo\\H + MC I Jo
eS{k^rzk{T)dT,
which implies that Z ( * ) < M   « O   H + M I  sup
zk(r).
0
Thus, we infer that z(t) is bounded for z(0) = u 0 in a sufficiently small neighborhood of u = 0. Therefore, u — 0 is locally asymptotically stable with exponentially decay. The proof is complete. • Remark 3.7 by
Theorem 3.12 is also valid if the condition (3.82) is replaced G{u) = O(UJJ),
3.3.2
Finite dimensional
u G H.
(3.84)
cases
We first consider the 2D system given by
f^=G1(x1>x2),
{ f
(3.85)
68
Bifurcation Theory and Applications
where G = {GUG2) and G(0,0) = 0. Assume that G  f + g, and div/=A OXi
+
 ^
= 0
OX2
.
(3.86)
Let /•X2
V(xi,x2)=
/Xl
fi(0,x2)dx2
f2(xi,0)dx1. Jo Jo We have the following theorem, which is useful in the stability, of hyperbolic equations at critical states. Theorem 3.13 Assume the condition (3.86). If there exists an open set Q, C M.2, with 0 € f2, such that x = 0 is a unique singular point of G in £1, and (1) V{x) > 0, V x E il, x ^ 0, (2) fi92  /2S1 < 0 , V i e f i , and (3) div g < 0, V x € Q,, x ^ 0, then x = 0 is asymptotically stable in 0 for (3.85). Proof.
It follows from (3.86) that 8V
dV •5— = J l i
ox2 Hence, by condition (2) we find
7j— = —J2
ax\
dV(x(t)) _dV dV + ~^^dx~1Gl dx~22 = hgi + /1S2 < 0, V x(t) e ft, x(t) a solution of (3.85), which implies, by condition (1), that V(x) is a Lyapunov function. Hence, x = 0 is stable. On the other hand, by (3.86) and condition (3), we know that (3.85) has no periodic orbits in O. Then the theorem follows from the PoincareBendixson theorem. • This theorem can be generalized to the higher dimensional systems as follows. Hr •— = G{x), x G R 2n , n > 1, (3.87) OX
w h e r e x = ( z i ,    ,xn,yx,
,yn), a n d G ( 0 ) = 0 . L e t
G{x) = JVH(x)+g{x),
69
Reduction Procedures and Stability
where H is a Hamiltonian function with H(0) — 0, and
'0DIn the same fashion as used in Theorem 3.13, we can obtain Theorem 3.14
If there is an open set O C R2n, with 0 6 0 such that
(1) x = 0 is a unique zero •point of G in O, (2) H{x) > 0 V xeO andx^O, and
(3) E?=i [Wl'9i + I f 5n+i] < 0, V x G fi, x # 0, then H(x) is a Lyapunov function of (3.87), and x = 0 is asymptotically stable in O. Now, we consider the system as follows
{
dxi —— = an^i + ai 2 x 2 + Gi(x), dx
^ 3 ' 88 ^
—1 = a2ixi + a22X2 + G2(x), where Gi(x) = o(x), 2 = 1,2, and
j
a i l + a 2 2 = 0
'
[ CtuCt22  " 1 2 0 2 1 > 0.
(3.89)
The eigenvalues of matrix (ay) are given by /3± = ±Waua22
 ai2«2i •
It is known that the equilibrium point x = 0 of (3.88) must be one of the three cases: a center, a stable focus, an unstable focus. The following theorem can be used in the Hopf bifurcation, which is a corollary of the PoincareBendixon theorem. Theorem 3.15 Let O c R2 be a neighborhood of x = 0. Under the condition (3.89), the following assertions hold. (1) If div G = 0 in O, then x = 0 is a center; (2) If div G < 0 (^ 0) in O, then x = 0 is a stable focus. (3) If div G > 0 (^ 0) in O, then x = 0 is an unstable focus.
70
Bifurcation Theory and Applications
3.3.3
An alternative principle for stability
We consider the equation given by diL
(3.90)
— =Lu + G{u)
where L : H\ —> H is symmetric, therefore all eigenvalues of L are real. Let the eigenvalues {fa} of L satisfy
{ Set
H fa
=0
Pj < 0
Eo = {ueH1\Lu
if 1 < i < m,.
(391)
if m + 1 < j < oo. = 0},
Ejt = {u e Hi I (u,v)H
=0,VveE0}.
The following theorem is crucial in the stability and bifurcation of the RayleighBenard convection and the Taylor problem which will be discussed in Chapter 10. Theorem 3.16 Let L : H\ —> H be symmetric with spectrum given by (3.91), and G : H\ —> H satisfy the following orthogonal condition (G(u),u)H
= 0,
Vueft.
(3.92)
Then one and only one of the following two assertions holds true: (1) There exists a sequence of invariant sets {Tn} C EQ of (3.90) such that 0^rn,
lim dist(rn,0) = 0.
n—>oo
(2) The trivial equilibrium point u = 0 of (3.90) is locally asymptotically stable under the Hnorm. Furthermore, if (3.90) has no invariant sets in EQ except the trivial one {0}, then u = 0 is globally asymptotically stable. Proof.
We proceed in the following four steps.
1. It is easy to see that Assertions (1) and (2) in Theorem 3.16 cannot be true at the same time. STEP
71
Reduction Procedures and Stability
By (3.91) and (3.92), direct energy estimates imply that the solutions u of (3.90) satisfy that °° J2 A N 2 < 0 ,
• jt\\u\\2H = 2
IMIIf
(393) (3.94)
where u = w + v&H = E0®
EQ,
oo
i=m+l m
w = y^Uj € Ei = EQ. It is easy to see that for any
V t! < t2 and
(3.95)
Hence limt_Kx> u(t, v) exists. STEP 2. For any tp G Hi, we have lira  K * , v )   = lim t—>oo
\\v(t,
t—>oo
Then the cjlimit set, which is an invariant set, satisfies that w(
((p) is an invariant set, for an ip G w() we have « ( « » Cw(y)) C 5,5
Vi>0.
Hence if ip = v + w e EQ ® Eo with v ^ 0, then by (3.93), for any t > 0,
IKt,^)<W = <J, a contradiction. Namely, for any
Ao:
(3.96)
72
Bifurcation Theory and Applications
STEP 3. If Assertion (2) is false, then there exists u n e H\ with u n  » 0 as n —> oo such tha t 0 $ w(wn) C £0, and
lim dist(w(u n ),0) = 0 .
n—>oo
Namely, Assertion (1) holds true. STEP 4. If Assertion (1) is not true, there exist a neighborhood U C H of 0 such tha t for any <j) £ U, lim u(t l ¥ >)=0 .
t—»oo
Namely, Assertion (2) holds true. The rest par t of the proof is trivial, and the proof is complete. • Remark 3.8 If all eigenvalues j3j < 0 (1 < j < 00) of the linear symmetric operato r L : Hi —* H, then under the condition (3.92), u = 0 must be a globally asymptotically stable equilibrium point of (3.90). In addition, under the conditions of Theorem 3.16, (3.96) is true. Likewise, we have Theorem 3.17 Under the condition (3.91), if G : H\ + H satisfies that (GU,U)H < 0 for all u ^ 0 sufficiently small, then u = 0 is a locally asymptotically stable equilibrium point of (3.90). 3.3.4
Dimension
reduction
We consider the equation (3.90) where the linear operato r L : Hi —» H is not necessar y to be symmetric. Assume tha t the conditions (3.44) hold true, and the eigenvalues {(3k} of L satisfy tha t f i *A = O
ifl
[ Rej3j < 0
if m + 1 < j < 00.
(3.98)
As the reduction of (3.43) to (3.53), we can reduce the equation (3.90) to an mdimensional ordinar y differential equation ^=JmX
+ g(x),
x€Rm,
(3.98)
73
Reduction Procedures and Stability
where Jm is t h e J o r d a n m a t r i x corresponding t o / ? i ,    ,/3m, a n d g = (fll," i£m),
sw
G
* =
Wi+
e
ra( (fr *(fH)' *)H'
Here $ is the center manifold function. By the center manifold theorem (Theorem 2.12) we know that if x = 0 is a locally asymptotically stable equilibrium point of (3.98), then u = 0 is locally asymptotically stable for (3.90). Equation (3.98) can be expressed as .
oo
£ = Jmx + Y, FP(X)> fc > 2 as in (3.44).
(3.99)
p—k
We take the iVt/lorder approximation of (3.53) as follows H^T1 dx — = Jmx + £ Fp(x),
N>1.
p=k
(3.100)
Theorem 3.18 Assume that for some N > 1 there are constants C, r > 0 and integer M < k + N such that \
Ik+Nl
/ ^ Fp{x) + Jmx,x) <C\x\M, \ p=k I
V x < r.
(3.101)
Then, x = 0 is a locally asymptotically stable equilibrium point of (3.99). Proof.
Since Zplk+N /
(
F
P(X) = o(af ^  ^ we have
oo
V
\p=k+N
\
r
F (x) x ) < 
\x\k+N+1
I
for all \x\ < r\, where r\ > 0 is sufficiently small. It follows from (3.101) that for any x < r\,
I
\
C
M ^ •Tp(X), F (T\ T — plrl (I JTmxT _i_ \ Y 2_^ x \) < ^ ——  , A \ I P=k
which implies that the theorem holds true.
•
74
3.4
Bifurcation Theory and Applications
Notes
3.1 The spectral theorem, Theorem 3.4, is very useful in the bifurcation theory developed in this book. It is used, together with the construction of center manifold functions and the attractor bifurcation theory, to solve various problems in Physics, Mechanics, Chemistry, Biology, and Engineering. Theorem 3.4 was first introduced in [Ma and Wang, 2004a; Ma and Wang, 2004b]. The asymptotic properties of eigenvalues stated in Theorem 3.5 is given here for the first time. 3.2 Reduction to center manifolds for infinite dimensional systems is a basic tool for studying dynamic bifurcations. The main new ingredient presented here is the first and second order approximations of the center manifold functions, and its applications to various bifurcation and stability problems including applications to superconductivity, to the RayleighBenard convection, and to the Taylor problem; see [Ma and Wang, 2005a; Ma and Wang, 2005e; Ma and Wang, 2005b]. Note that Theorem 3.8 was introduced by the authors in [Ma and Wang, 2004b]. 3.3 This section is based on the authors' recent work. In particular, the global stability theorem, Theorem 3.15, is crucial for studying the stability and bifurcation of the Benard convection and the Taylor problem [Ma and Wang, 2003a; Ma and Wang, 2004d; Ma and Wang, 2005d].
Chapter 4
Steady State Bifurcations
4.1
4.1.1
Bifurcations from HigherOrder Nondegenerate Singularities Evenorder
nondegenerate
singularities
Consider a parameter family of nonlinear operator equations (4.1)
Lxu + G(u,X) = 0,
where L\ : Hi —> H is a completely continuous field, and G(, A) : Hi —> i7 is a C°° operator having the Taylor expansion as (3.44). Let the eigenvalues (counting multiplicity) of L\ be given by {/?i(A), /?2(A), • • • } with A(A) £ K1 (1 < i < m) such that
{
<0
if A < Ao,
(4.2) = 0
if A = Ao,
> 0
if A > Ao,
V 1 < i < m,
(4.2)
(3j(Xo)^0 Vm + l < j . (4.3) Let {ei, • • • , e r } and {ej, • • • , e*} C Hi be the eigenvectors of L\ and L*x at A = Ao respectively: LXoej = 0, L£oe* = 0 , 1 < j < r. (4.4) Here r < m is the geometric multiplicity of the eigenvalue /3i(A0). We set a
hjk
=
(Gk(eji,
where k > 2 is given by (3.44). 75
,ejk,X0),e*)H,
76
Bifurcation Theory and Applications
Definition 4.1 Assume (4.2) and (4.3). The steady state solution u = 0 of (4.1) is called kth order nondegenerate at A = Ao, if x = (xi, • • • , xr) = 0 is an isolated singular point of the following system of rdimensional algebraic equations r
E
4^i^=°>
l<*
(4.5)
The following two theorems establish bifurcations from evenorder nondegenerate singular points. Theorem 4.1 Let the operator L\ + G : Hi —> H satisfy (4.2) and (43). If there is an even number k > 2 such that u = 0 is a kth order nondegenerate singular point of (41) at X = Xo, then (u, A) = (0, Ao) must be a bifurcation point of (41), and there is at least one bifurcated branch on each side of X = Xo. This theorem establishes the existence of a transcritical bifurcation, and the following is its global version, in the same spirit of the Rabinowitz global bifurcation theorem. Let E+ = {{u, X) G Hi x R 1  L\u + G(u, A) = 0, u ^ 0, A > Ao} , I T = {{u, A) G # i x R1  Lxu + G(u, X) = 0, u ^ 0, A < Ao} . Theorem 4.2 Let Lx : Hi > H satisfy (3.41), and G(, A) : Hi » H be compact. Let S c S (resp. E c S ) be the connected component of S + (resp. S ~ ) containing (0, Ao). Under the conditions of Theorem 41, one of the following assertions holds true. (1) E is unbounded, (2) £ contains points (0,Ai) with Ai > Ao (resp. Xi < XQ) such that there are some eigenvalues f3j(X) of L\ at X = Ai with f3j(Xi) = 0, or (3) There exists a point (VQ, /J.) € Hi x R 1 with fi > Ao (resp. /i < Xo) such that
{
0
ifX>fi
(resp. A < n),
(vo,fi) i/A = /i, Ti(A) + T2(A) if Ao < A < \i (resp. p < A < A o ;, tufeere T ^ A Q ) = (0,A 0 ), T2(A0) ^ (0,A 0 ), Ti(X) ^ %, andY2{X) + 0. Remark 4.1 Assertion (3) in Theorem 4.2 implies that there is a bifurcation of (4.1) from (vo,p), and there are at least two bifurcated branches
77
Steady State Bifurcations
Fi(A) and ^(A) on this side of A < /x or \x < A, as shown in Figure 4.1. Also H
v0
/
!
^\
x
Fig. 4.1 Bifurcation from (vo, p) •
we mention that this bifurcation as shown in Figure 4.1 occurs frequently in hydrodynamic equations, which will be investigated in the forthcoming book on the hydrodynamic stability and bifurcation by the authors.The type of bifurcation shown in Assertion (3) is called saddlenode bifurcation. Proof of Theorem ^.1. We proceed with the LyapunovSchmidt method, together with the Brouwer degree theory. 1. Consider (4.1). Let {ttfi(A), • • • ,wm(X)} c Hi and {wl(X), • • • , w^,(A)} C Hi be the eigenvectors of L\ and L\ respectively, corresponding to eigenvalues given by (4.2), i.e. STEP
(L A A(A)r^(A)=0, (LXft(A)r'<(A) = 0, for some rij > 1, m, > 1 (1 < i < m), such that (wi,w*)H = 5ij. By the normalization of the LyapunovSchmidt reduction given by
78
Bifurcation Theory and Applications
(3.74), we reduce the equation (4.1) to the following equation xeRm.
(4.6)
,gkm(x,\)Y,
(4.7)
Jmxx + g(x,X)=0, Here g(x, A) = Fk(x, A) + f(x, A) is given by Fk{x,X)
= (gkl(x,X),fc
/ ( i , A) = o(a: ).
(4.8)
for some k > 2, where m
9ki{x, A) =
5Z
x
i i ' ' ' x ^ (Gfc(wji. • • • . wjk.
X),w*)H.
Since the Jordan matrix J m A has /?i(A), ,/3m(X) as its eigenvalues which satisfy (4.2), we have the index formula f(l)m ifA
(4.9)
We proceed in three steps to prove (4.9). STEP 2. We first consider the case where Jm\0 = 0, i.e. r = m, the geometric multiplicity equals the algebraic multiplicity at A0By assumption, Fk : R m —> R m is a nondegenerate fcth order multilinear mapping. There exists a number (3 > 0 such that
\Fk(ax,Xo)\>P\a\k,
V x G R m , \x\ = 1.
(4.10)
It follows from (4.8) and (4.10) that for any r > 0 sufficiently small, Fk(x, Ao) + tf{x, Ao) ^ 0,
Vx
GM
m
, x = r, 0 < t < 1.
By the homotopy invariance property of the Brouwer degree, we have ind(J mAo +sO,A 0 ),0) =deg(J m A o +gXo,BR,0)
(4.11)
= deg(F fc (.,A 0 ),B R ,0), where BR = {x G R m  a; < R}. By Theorem 1.5, we have deg(Ffc(,Xo),BR,0) = even,
if k = even.
(4.12)
79
Steady State Bifurcations
Hence (4.9) follows from (4.11) and (4.12). S T E P 3 . C A S E WHERE m = 3, r = 2 AND k = 2. To present the
main idea of the proof, we proceed with the special case where the algebraic multiplicity m = 3, the geometric multiplicity r = 2 at A = Ao, and Fk(x, A) is a second order multilinear mapping. Let L\Bwi = 0, L\ow2 = 0, L\ow3 — w2. By (3.25) and (3.26), we see that the dual eigenvectors of {wi,W2,wz} satisfying {W^W^H = hj are given by
LIM = 0, L*Xowl =wl, L*xM = 0. Therefore, we have ei =w1, e2 =w2,
el = i y * , e*2 =w%,
(4.13)
and
=
Jm\ox
/ 0 0 0 \ (Xl\ 001 u
\0 0 0/ \xj
=
/ o \ i3 ,
\o)
m = 3.
(4.14)
By k = 2, /
FM^Ao) =
3
3
b x x
3
hxx
J2 ij * i> J2 l i i, \*J = 1
i,j=l
>
Yl % i i,j=l
where &fj =
\ '
h XiX
(G2{wi,wj),w})H.
)
( 4  15 )
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Bifurcation Theory and Applications
Let T : K 3 —> R 3 be a linear homeomorphism defined by
/I 0 0\ fXl\ Tx=\00l\
,
\x2
VOIO/ VW
/°\
TJ3AOZ = 1 0
I.
Then we derive from (4.14), (4.15) and (4.7) that T(J3\0 + 9\0)x = T(J3Xo + F2)x + Tf(x, Ao), Eh^XiXj x3 + E
3
J=1
\,
(4.16) (4.17)
bfjXiXjJ
2
(4.18)
Tf(x,\0)=o(\x\ ).
By assumption, (x\, X2) = 0 is an isolated singular point of (4.5) with r = 2 and k = 2, which, by (4.13), reads 2
J2aeijxixj=0, a
t=l,2,
ij = {G2{ei,ej),el)H
= b}j,
a*j = {G2(ei,ej),eZ))H
= b3ij.
Thus, we infer from (4.17) and (4.18) that for any 0 < t < 1, x = (xi,X2,xs) = 0 is an isolated zero point of the following equation / E L = 1 ahX*Xi + tX* E?=l(fci3 + Hi)Xi + *&33^\ £ j j = i a?.x i x i + te3 Ef=i(&« + hli)*i + ^33^3 + *Tf(x, Ao) = 0. V
x
3 + tElj=lbijxixj
)
(4.19) Based on the homotopy invariance of Brouwer degree, it follows from (4.16)(4.19) that deg(T( J3Xo +gXo), BR, 0) = deg(tf, BR, 0),
(4.20)
81
Steady State Bifurcations
where R > 0 is sufficiently small, and /EL=i
a
ijxixi\
*(*) = E U i 4 ^ ' • According to Theorems 1.2 and 1.3, it follows from (4.16)(4.19) that deg(T(J3A0 + 9xo),BR, 0) = ind(T, 0) • deg(J3Ao + g, BR, 0) = (l)deg(J3Ao +9,BR,0),
(4.21)
and deg(K, BR, 0)  deg(^, S f i n l 2 , 0 ) ,
(4.22)
where
/V2
A^!,^; I
2
a1 a;a; A 2
I •
By assumption, {x\,x
(4.23)
Finally, the index formula (4.9) follows from (4.20)(4.23). STEP 4. GENERAL CASE. Let the Jordan matrix Jm\0 have the following form
/° \o
Jl
°\
/01 0
00 1
'••
JJ \0
m
The space R can be decomposed into
JmXoV = 0,
Vj/6lr.
0\
1
0/
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Bifurcation Theory and Applications
Let P : R m > R r be the canonical projection, T : Rm * R m given by //
0\
/0 .. 0 1\ 1 ••• 0 0 %= . . . . .
7\ .
T= \0
,
\ 0 ••• 1 0 /
TJ
It is clear that T is a linear homeomorphism, and TJm\ohr = °,
TJ OT A 0 K— = *mr the identity,
:
where ^ 1 ; . . .
J t
(: 
ajij^Ji '"" ^
 r)
{
,
a r e
^
i n
(4.24)
(4.25)
(45)
As in the proof of (4.20)(4.23), we can deduce from (4.24) and (4.25) that deg(JmAo +gXo,BR,0) = (l)m~r deg(T(JmXa + gxo),BR,0) = (l)m~r deg(TJmAo + TFk, BR, 0) =
(l)mrdeg(PTFk\Rr,BRnRr,0).
Since PTFk\w : W —* M.r is an even mapping, and by assumption x = (xi, • • • ,xr) = 0 is an isolated singular point of PTFk\Rr, we have ind(P2TfcR,,0) = deg(PTFk\Rr,BR
nRr,0)
= even. Thus (4.9) follows. The proof of this theorem is complete.
•
Proof of Theorem J^.2. It is known that the LeraySchauder degree theory is valid for the completely continuous fields L\ + G : H\ —> H, provided that L\ satisfies (3.41). The eigenvalues /3j(A) of L\ with Re/3, > 0 are finite, and if L\ : H\ —> H is invertible, then we have ind((L A + G), 0) = deg((L A + G), BR, 0) = (1)",
(4.26)
where n is the sum of all algebraic multiplicities of the real eigenvalues /3 i (A)>0ofL A .
83
Steady State Bifurcations
By similar properties for the LeraySchauder degree theory as those for the Brouwer degree used in the proof of Theorem 3.13, we can show that under the conditions of Theorem 4.2, the topological degree of L\ + G at A = Ao is an even number, i.e. deg((L Ao +G),BR,0)
= (l)n+mr = even.
deg(PTFk\Ur,BRnW,O) (4.27)
where W, TPFk are the same as in (4.25). Then the rest of the proof of this theorem is routine using (4.26) and (4.27). The proof is complete. • Remark 4.2 In fact,in the proof of Theorems 4.1 and 4.2, we have shown the following result for the LeraySchauder degree. Let L\ + G : H\ —> H be a completely continuous field, and G(u, A) = w °(ll lffi) If there are {wi,•• ,wm} C H\ such that L^Wi = 0,
1 < i < m, Hi > 1,
and u = 0 is afcthorder nondegenerate singular point of L\o + G(, Ao) for some k > 2, then deg((L Ao + G(, Ao)), BR, 0) = (1)" deg( JXom +g,BRn
E, 0)
where n is the sum of all algebraic multiplicities of the real eigenvalues /?j(A0) > 0 of L\o, E = span {ei, • • • , e m } , JXom the Jordan form of L\o restricted on E, and g = — PG\E : E * E, and P : H —> E the canonical projection. 4.1.2
Bifurcation at geometric simple eigenvalues: r = 1
In this subsection, we address the case where the geometric multiplicity of the eigenvalue is one, i.e. r = 1 in (4.4). Definition 4.2 Let T(X) C Hi be a branch bifurcated from (uo,Ao) of (4.1). F(A) is called regular if for any A — Ao > 0 sufficiently small each singular point v\ € F(A) of (4.1) is nondegenerate. Theorem 4.3 the number
Assume the conditions (42), (4.3) and r = 1 in (4.4). If a = (Gk(e1,\0),e*1)H^0,
k>2,
(4.28)
where Gk is given by (3.44), then the following assertions hold true.
84
Bifurcation Theory and Applications
(1) For the case where k is even in (428), there exists a unique bifurcated branch of (41) on each side of Xo. (2) For the case where k is odd and m is even, if a < 0, then (41) has no bifurcation from (0,Ao), and if a > 0 then there are exactly two branches bifurcated from (0, Ao) on each side of XQ. (3) For the case where k is odd and m is odd, (41) has no bifurcated branch on X < Ao (resp. on X > Xo) and has exactly two branches on X > Ao (resp. on X < Xo) if a < 0 (resp. a > 0). (4) Each branch F(A) bifurcated from (0, Ao) for (41) is regular, and the singular points u\ G T(A) can be expressed as ux = ± la" 1 /?!(A) • ••(3m(X)\1/(k~l) 1
+ 0 (a /?i(A)/3 r o (A)
Proof.
Let {wi,•••,
I LXow1
w m}
ej
(4.29)
1/(fc 1)
 ).
and {w*, • • • , w*^} C H\ satisfy
= 0, LXow2
= wi,
,LXowm
=
\ Llow*m = 0, L ^ i i C . ! = < ,  •  , L*Xowl
wmi, =w*2.
(4.30)
From (3.25) and (3.26), we see that f =0
iii^j,
<""'•"'> { ^ 0 B .  i
(4 3I)
'
In this case, the bifurcation equations (4.6) can be written in the following form //MA) 1 0 . . 0 /32(A) 1 ••• ' . ^0
0
0 0 .
0 ••• Pm(X)J
where g(x, A) is the same as in (4.6).
\ /
M M +9(x,X)=0, ^
m
'
(4.32)
85
Steady State Bifurcations
We see from (4.30) that ei = wi and e\ = w^. By (4.7), (4.8) and (4.31), we infer from (4.32) that •x2 =
f31(\)x1+O(\x\k),
' x m = /3 m _ 1 (A)rr m _ 1 +0(x / £ ),
a(X)xk1 = f3m(X)xm +
fel
(4.33)
m
O(^2'£\x1\i\xj\ki),
i=0 j=2
where a(Ao) = a. It follows from (4.33) that near A = Ao, a(X)xk11 = (l) m /?i(A) • • • 0 m (A) + od^(A) • • • /3m(A)).
(4.34)
Thus, Assertions (1)—(3) and (4.29) follow from (4.2), (4.28) and (4.34). Assertion (4) can be derived from the following determinant of the first order approximation of the Jacobian of (4.32) and Theorem 3.9: / ft (A) det
0
1 /?2 (A)
0 ••• 1 •••
0 \ 0
W " 1 0 0 ••• /3m(\)J = 0i"0ro(irafcc*1 = (1  fc)0i • • • 0m + o(0i • • • (3m\) ^0
(by (4.34))
for A ^ Ao (by (4.2)).
The proof of this theorem is complete. 4.1.3
Bifurcation with r = k = 2.
Let r = 2 in (4.4), Gfc be bilinear (i.e. k = 2), and a n = (G 2 (ei,ei,A),ei)tf, a22 = (G 2 (e 2 ,e2,A),ei)H, 012 = (G 2 (ei, e 2 , A), e*)^ + (G 2 (e 2 , ei, A), ej)#, 6ii = (G 2 (ei,ei,A),e;)K, {G2(e2,e2,\),e^)H, b22 = bi2 = (G 2 (ei, e 2 , A), e ^ ^ + (G 2 (e 2 , eu A), e>.
•
86
Bifurcation Theory and Applications
We have the following bifurcation theorem. Theorem 4.4 Let the conditions (42), (43) and r = 2 in (44) hold. If u = 0 is a secondorder nondegenerate singular point of L\ + G at X = \Q, and the two vectors (an,ai2, a i3) and (611,612,^13) are linearly independent, then we have the following assertions. (1) There are at most three branches of (41) bifurcated from (0,Ao) on each side of X = Ao. (2) If the bifurcated branches on A > Ao (resp. on X < Xo) are regular, then the number of branches on this side is either 1 or 3. (3) On any given side, the number of branches is 3, then these 3 branches must be regular. (4) If the number of branches on one side is 2, then one of them is regular. Proof.
We proceed in several steps as follows.
S T E P 1. We only have to prove the case where m = r = 2. For the case where m > r = 2, as in the proof of Theorem 4.3, the mdimensional bifurcation equations (4.6) can be treated as if they are twodimensional equations, and the proof given below for the case where m = r = 2 can then be applied as well. As m = r = 2 in (4.2) and (4.4), the bifurcation equations (4.6) can be rewritten as follows
f /3i(A)xi + aux\ + ai2xix2 + a22x\ + o{\x\2) = 0, \ /32(A)x2 + bxlx\ + b12x1x2 + b22x\ + o{\x\2) = 0,
(4.35)
where x = (xi,x2). 2. We now show that there is an £ > 0 such that for any C°° function (fi(x), f2(x)) with /i(x) +1/2(2;)! small near x = 0, the following equations STEP
anxl + a12iia;2 + ^ z ! + fi(x) + o(x2) = 0, 2
buxl + b12Xlx2 + b22x\ + f2{x) + o(\x\ ) = 0,
(4.36) (4.37)
have at most four solutions in \x\ < e. Since the second order terms in (4.36) and (4.37) are secondorder nondegenerate, at least one of the coefficients a n , 022, ^n and i>22 is not zero. Without loss of generality, we assume that a22 ^ 0. Then we obtain from
87
Steady State Bifurcations
(4.36) that X
*
=
~~2a~ Xl
±
2^~ V ^
2
~ 4 a i i a 2 2 ) x i ~ 4a 22/i( a: ) + °(N 2 ))
which, by the implicit function theorem, implies that
^s^ss*
(4 38)

where A = y ( a i 2 4ana 2 2)a: i + g(xi) + o(\xx\2) ff(xi)»0
if
/i(s)>0 .
Inserting (4.38) into (4.37), we have ax\ + b22g(xi) + Aal2f2(x) + o(\x\2) = ^jfttiA,
(4.39)
where a = 4a 22 (a 2 2&n  011622) + 2ai2(ai 2 6 2 2  022612), f3 = 2(022612  012622)We infer then from (4.38) an d (4.39) tha t px\ + /iiOn)z 2 + h2{Xl) + o(\Xl\4) = 0,
(4.40)
where p = a2  f32(a2n  4 a n a 2 2 ) , h1(x1),h2(x1)^0
if
/ i ( x ) , / 2 ( x )  » 0.
It is easy to verify that the condition p =f= 0 is equivalent to the following conditions
6 22 4+^ ± +W0,
z± = Z ^ i i ^ S Z ^^ _ ^O 2 2
( (4.40)
By the assumption of this theorem, conditions in (4.41) ar e valid. Therefore, p ^ 0, which implies that (4.40) has at most four rea l solutions satisfying that xi > 0 as fx(x), f2(x) > 0. On the other hand, each solution of (4.40) correspond s to one of the signs ± in (4.39) and (4.38). Hence equations (4.36) and (4.37) have at most four rea l solutions near x = 0. Namely, Assertion (1) is proved.
88
Bifurcation Theory and Applications STEP
3.
P R O O F OF ASSERTION
(2). Let FA : M2 » R2 be defined by
p ix\ _ fPiWxi + a i i x i + a\ix\xi + 022^2 + O (M 2 A M ' \/32(X)x2 + bux\ + b12x1x2 + b22x\ + o(\x\2)) ' By Theorem 3.9, the singular points z(\) of (4.35) near x = 0 is regular providing the bifurcated branches of (4.1) from (0,Ao) is regular. Hence, we have ind(Fx,z(\)) = ±l
(4.42)
for  A — Ao 7^ 0 sufficiently small. In addition, we know that if A ^ Ao, ind(FA,O) = sign/?i(A)ft(A) = 1,
(4.43)
and, by (4.9), we have Y^ ind(F A , ^(A)) = even,
(4.44)
\zt\<e
for any e > 0 small. Hence Assertion (2) follows from (4.42)—(4.44) and Assertion (1). 4. P R O O F OF ASSERTIONS (3) AND (4). We know from Step 2 that the solutions of (4.36) and (4.37) are one to one corresponding to those of (4.40). Let x0 6 i 1 be a solution of (4.40) near x = 0. Then (4.40) can be expressed as STEP
(x  xo)kg(x)
= 0,
k>\.
(4.45)
We shall prove that a solution z0 £ R2 of (4.36) and (4.37) corresponding to XQ is nondegenerate if and only if k = 1 in (4.45). Obviously, if k > 2 in (4.45), under a perturbation there are more solutions of (4.36) and (4.37) near ZQ. Suppose zo is degenerate, i.e. detDF(zo) = 0, where DF(zo) stands for the Jacobian matrix at F, and f o u i j + ai2xix2 + a22x\ + fi(x) + o(a;2), \ bnxl + 612X1X2 + 622X2 + / 2 (x) + o(x 2 ).
(4.46)
89
Steady State Bifurcations
Under a suitable coordinate system, DF(z0) must be one of the following three forms
(4.47)
DF{zo)=(^ DF(zo)=(°0
°Qy o),
(4.48) (4.49)
for some a ^ 0. Since / i , f2 are C°° sufficiently small function, and the quadratic form in F are secondorder nondegenerate, for both the cases of (4.47) and (4.48) it is known that with a linear perturbation the equation (4.36) and (4.37) will yield one more solution near z0, which implies k > 2 in (4.45). We proceed for th case of (4.49). Near z0, F can be expressed as
p(y) = {ay2 + ; u y J + c ; * y m + c ; i + ° ^
duVx + di2yiy2 + d22yi + o{\y\i),
[
(4.50)
where y = x — ZQ. The zero point y = 0 of (4.50) stands for the zero point ZQ of F. Now we consider the perturbation equation <*yi = (cnyf £J/i = ~(dnyl
+ c12y!y2 + c22yl) + o(y 2 ), 2
+ d12ylV2 + d22y\) + o(\y\ ),
(4.51) (4.52)
for e small. By the implicit function theorem, we obtain from (4.51) a solution y2 = /(yi). Then (4.52) implies that eyi = (dnyf + d12yif(yi)
+ d22f2(yi)) + o(\yi\2).
(4.53)
B y t h e s e c o n d  o r d e r n o n d e g e n e r a c y of F(y), t h e r e exist a ^ O a n d m > 2 such t h a t (4.53) b e c o m e s eyi=ay^
+ o(\yi\m).
(4.54)
We infer from (4.54) that equations (4.51) and(4.52) has two solutions (0,0) and (yi,f(yi)), where
90
Bifurcation Theory and Applications
and £ is taken such that e • a > 0. Thus, we prove that if z0 is a degenerate zero point of (4.35) and (4.37), then the exponential k > 2 of the corresponding solution XQ in (4.45). We are now in a position to prove Assertions (3)—(4). If (4.35) has three bifurcated solutions (resp. two bifurcated solutions) from (0, Ao), then the corresponding solutions Xi ^ 0, i = 1,2,3, (resp. Xj ^ 0, j = 1,2) of (4.40) can be expressed as (a;  xi)(x  x2){x  x3)xgi(x)
= 0,
2
(resp. (x — xi) (x — X2)xg\{x) = 0, which implies that Assertion (3) (resp. Assertion (4)) holds true. The proof of this theorem is complete. 4.1.4
•
Reduction to potential operators
We consider the case where the equation (4.1) is reduced to an equation with the first fcmultiple linear operator being potential. We assume that the eigenvalues of L\ in (4.2) satisfy /?i(A) =    = / M A ) = / ? ( A )
nearA = A0,
(4.55)
and m = r the geometric multiplicity. Let the reduced equations (3.74) be expressed as P(\)xi+Fi1(x)+o(\x\ki)=0,
(l
(4.56)
where fci > k, k > 2 is the same as in (3.44) of Chapter 3, and 771
X
Fid ( ) =
Yl
a
himxh
•' • xi^ •
We also assume that the operator F^ = (F^ ,•••, FJ^) is a potential operator, i.e. there is a function V(x) such that
(4.57) By the Krasnoselskii theorem, (x,X) = (0,Ao) must be a bifurcation point of the following equations, which are the nth order approximation of (4.56), n = kik + l: P(\)Xi
+ Flkl (x) = 0 ,
(1 < t < m).
(4.58)
91
Steady State Bifurcations
We have the following theorem. Theorem 4.5 Let the conditions (4.2), (4.3) and (4.55)—(456) hold. If (458) has r bifurcated branches Fj(A) (1 < i < r) on a given side of A = Ao, which satisfy that for zx 6 Fj(A), ind(/?(A)id + F f c l I ZA)^O
(4.59)
then the equation (41) has at least r bifurcated branches on the side. Proof. Since the vector field Fkl is fcimultiple linear, the solutions zx = (xi(A), • • • ,x m (A)) G I\,(A) have the form
Xi{\)
1
= c^CA)! ^!),
m
Yla^ °' t=l
( 46 °)
where on are constants independent of A. Since zx S Tj(X) is an isolated zero point of (4.58), by (4.60) there exist constants 5 > 0 and a > 0, such that z\ is a unique singular point of (4.58)
in Bs(z\), and \p(\)x
+ Fkl (a:) > a I / J M I ^ * 1  1 ' ,
V i e dBs(zx),
(4.61)
for  A — Ao  ^ 0 small, where
Bs(zx) = [x G R m  \x  z A  < 5/3(A)1/
P{\)x + Fkl (x) + tf{x) ^ 0, where f(x) = o(\x\kl). obtain that
V 0 < t < 1, a; G dB5(zx),
By the homotopy invariance of Brouwer degree we
deg(/3(A)id + Fkl + /,B s (z\),0) = ind(/3 id + Fkl,zx).
(4.62)
We infer from (4.59) and (4.62) that the equation (4.56) has at least a • solution near zx for all A — Ao ^ 0 small. The theorem is proved. Corollary 4.1 Assume (4.2), (4.3) and (4.55)—(4.57). If x = 0 is an isolated zero point of Fkl with hi = odd, and the bifurcated branches of (458) are finite, then (41) has at least 2m bifurcated branches on both sides of A = AQ .
92
Bifurcation Theory and Applications
Proof. Because there are at least m pairs of bifurcated singular points, which are the minimax points of the functional   x  2 + V(x), the minimax points must have a nonzero index. The proof is complete. •
4.2 4.2.1
Alternative Method Introduction
Let X be a Banach space, I : I  » I a linear compact operator, and h and G : X —> X be continuous mappings. We consider the bifurcation of the following types of equations x  \{L + h)x + G{x) = 0,
(4.63)
x  Xh(x) Lx + G{x) = 0,
(4.64)
x  XLx  h(x) + G(x) = 0,
(4.65)
x  \h{x) + G(x) = 0.
(4.66)
For this case, the classical bifurcation theory is not accessible. In this section, we present a method, called the alternative method, to deal with the bifurcation of the equations (4.63)—(4.66). To show how the alternative method works, we discuss the bifurcation in the following assumptions L,h : X —> X
linear compact operators,
G : X —» X
compact,
G(x)=o(\\x\\). Consider the following parameterized linear operators id\iL\2h:XxIxI—>X,
7 = [0,l].
Assume that Ai = 1 is not an eigenvalue of L, and Ai = p1 is a unique eigenvalue of L in [l,oo) with odd algebraic multiplicity. In the parameter space (Ai, A2) € I x I, the index of id — AiL — A2/1 at x = 0 is shown as in Figure 4.2. By the homotopy invariance of the LeraySchauder degree, there exists a curve 70 C / x I, which divides the square I x I into two domains U\ and
93
Steady State Bifurcations
1
_,
J.
ui
X
X /°
/ y * o '^o*
y'
•
P
*i
^ ^
(i. *")
*i
D
Pig. 4.2
17%, such that
• ^
x T
xhn\
J
1
ind(id — X\L — A2fi, 0) = <
\ 1
if(Ai,A 2 )€C/ 1 ,
if(Ai,A 2 )Gl7 2 ,
which implies that • ACA ^ T M.j.nn\ I1 md(id — \\L — A2/i + G, 0) = < ^ \ 1
if(Ai,A 2 )et/i, if(A 1 ) A 2 )et/ 2 .
Thus, one can see that if the curve 70 is the case as shown in Figure 4.2(a), then the number Ao in the intersection of 70 with the diagonal of / x / is a bifurcation point of (4.63). If 70 is the case as shown in Figure 4.2(b), then the number A* in (1,A*) G 70 is a bifurcation point of (4.64). Obviously, the curve 70 must pass through either the diagonals {(A, A) I 0 < A < 1} or the line segment {1} x [0,1]. Hence we can claim that (i) there must be one of the two equations (4.63) and (4.64) having a bifurcation point Ao £ [0,1]; (ii) at least one of the three equations (4.64)— (4.66) has a bifurcation point in [0,1]. In the following subsections, we shall discuss the alternative method for the case where h : X —> X may be a nonlinear operator.
94
Bifurcation Theory and Applications
4.2.2
Alternative bifurcation theorems
We consider the following equations x + L(x, A) + h(x, A) + G(x) = 0,
(4.67)
x + L(x, 1) + h(x, A) + G(x) = 0,
(4.68)
x + L(x, A) + h(x, 1) + G(x) = 0,
(4.69)
x + h(x, A) + G{x) = 0.
(4.70)
Suppose that £(•, A) : X —* X is a linear compact operator with L(x, 0) = 0, h : X x M.1 —> X a compact operator with /i(:r, 0) = /i(0, A) = 0, G : X —> X a compact operator with G{x) = o(a;). We say that Ao £ R 1 is an eigenvalue of L : X x R 1 —> X, if there exists x € X with x 7^ 0 such that a; + L(a;,Ao) = 0. Theorem 4.6 If\ = lis not an eigenvalue of L, and the sum of algebraic multiplicities of eigenvalues of L in (0,1) is an odd number, then we have the following assertions: (1) At least one of the two equations (467) and (468) has a bifurcation point in [0,1]. (2) At least one of the three equations (468)—(470) has a bifurcation point in [0,1]. (3) If \ = 0 is a bifurcation point as above, then there is a bifurcated branch on the side of A > 0. The following theorem is a global version. Let S = {(x, X)eXx(0,oo)\x
+ L{x, 1) + h(x, A) + G{x) = 0, x ^ 0} .
Theorem 4.7 Let the conditions of Theorem 46 hold true. Assume that the following equation has no bifurcation point in [0,1] for any R>1, x + L{x,X) + h(x,XR)+G(x)=0
(4.71)
and the set of bifurcation points of (468) is bounded in [0, oo); then there exists a bifurcation point Ao > 0 of (468) such that one of the following two assertions holds true.
95
Steady State Bifurcations
(1) The connected component Eo of £ containing (0, Ao) is unbounded in X x (0, oo), or (2) So contains a nonzero solution of the equation x + L(x, 1) + G{x) = 0.
(4.72)
Proof of Theorem 46. By assumption, there is a constant R > 0 small such that for 0 < r < R we have deg(id + Lx + G, £ r , 0) = deg(id + Lu Br, 0) 
(if.
where L\ =£,(, 1), and /? the sum of algebraic multiplicity of all eigenvalues of L\ in (0,1), which is an odd number. Hence, deg(id + Li + G,jBr,0) =  l ,
V 0 < r < R.
(4.73)
On the other hand, we know that deg(id + G,5 P ,0) = l.
(4.74)
Suppose that 0 ^ (id + L\ + h\ + G)(dBr) for r > 0 small, otherwise X = 1 is a bifurcation point of (4.67). We also assume that deg(id + I a + / i i + G , j B r , 0 ) = l,
V 0 < r < S,
(4.75)
for some 5 > 0. Let Ht = id + i i + h{, t) + G. Then Ho = id + Lx + G, Hi=id + Lx + hl + G. By the homotopy invariance, it follows from (4.73) and (4.75) that for any 0 < r < 5, there are xr e dBr and 0 < Ar < 1 such that xr + L{xr, 1) + h(xr, Xr) + G(xr) = 0. Obviously, the numbers Ao = lim Ar,
Ai = lim Ar,
T—>0
r+0
are the bifurcation points of (4.68) in [0,1], and if Ao = 0, the bifurcated branch is on the side of Ao < A. If (4.75) is false, then we have deg(id + L i + / i i + G , 5 r , 0 ) ^ l ,
V 0 < r < 5.
(4.76)
96
Bifurcation Theory and Applications
Let ( Ht = id+ L(;t) + h(;t)+G, I Ho = id + G, [ Hi = id + Li + /ii + G.
(4.77)
Thus, in the same fashion as above, it follows from (4.74) and (4.76) that (4.67) has a bifurcation point Ao in [0,1], and if Ao = 0 the bifurcated branch is on Ao < A. The assertions (1) and (3) are proved. We know that if (4.68) has no bifurcation point in [0,1], then (4.76) holds. By (4.77), if (4.70) has no bifurcation point in [0,1], it follows from (4.74) that deg(id + hi + G, Br, 0) = 1,
(4.78)
for all r > 0 small. Thus, one can derive from (4.76) and (4.78) that (4.69) must have a bifurcation point Ao in [0,1], and if Ao = 0 the bifurcated branch is on Ao < A. The proof is complete. • Proof of Theorem 4..7. By assumption, from Theorem 4.6 we see that the equation (4.68) has a bifurcation point in [0,1]. Let {Afc} C [0, 00) be the set of bifurcation points of (4.68). Based on the conditions, there is a number 0 < A, such that {A^} C [0, A). Denote by Ck the connected component of £ containing (x, A) = (0, Afc), and
C0 = {JCk, k
C = C 0 U({0}x[0, J R]), QR = BRX [0,R], which are as shown in Figure 4.3. Assume that Co is bounded in X x [0,00). Then, there is a R > 1 such that Co e QR, and Co n 8QR = (f> provided Assertion (2) of this theorem is false. Since (4.71) has no bifurcation point in [0,1], we take e > 0 sufficiently small such that id + Li + G and id + L\ + h(, R) + G has no nonzero points in Be. Let D = dQR\{B£ x {0} UB £ x {R}).
97
Steady State Bifurcations X \ B
R
QR
^
^
^
^
y
_^
C
^
/
/ /
/ c kk
Fig. 4.3
If Assertion (2) is false, C and DUj (7 = S\C 0 ) are two disjoint closed sets. Therefore, there exists a relative open set V C X x [0,R] such that
CcV, F n 7 = 0, VnD = 0, which implies that (see Figure 4.3), VO,VRCBE,
dVn(Zu({O}x(O,R)))=0,
(4.79)
where V\ is the section of V at A £ [0, R]. By the homotopy invariance and the excision property, from (4.73) and (4.79) it follows that
det(id + Ll + h(,R) + G,VR,0) = deg(id + Z 1 +G,V o ,0) = 1.
(4.80)
98
Bifurcation Theory and Applications
Because VR C Be and id + L\ + h{, R) + G has no nonzero point in BE, we have
deg(id + L1 + h(;R) + G,VR,O)=deg(id + L1 + h(.,R)+G,B£,O).
(4.81)
On the other hand, by assumption, we have deg(id + L(, A) + h(, XR) + G, B£, 0) = const.,
V 0 < A < 1.
Hence deg(id + Li + h(, R) + G, Be, 0) = deg(id + G, Be, 0) = 1,
(4.82)
a contradiction to (4.80) and (4.81). The proof is complete.
4.2.3
General
•
principle
Let X, Y be two Banach spaces. We consider the equation given by (4.83)
Ax + h{x,X)=0. Suppose that
h : X x R 1 —> Y is a compact operator with h(0, A) = h(x, 0) = 0, A = L + G : X —>7a completely continuous field, i.e. L : X  » 7 a linear homeomorphism, G : X —> Y a compact operator with G(x) = o(a;). T h e o r e m 4.8 Under the above assumptions, at least one of the following assertions holds: (1) The equation (483) has a bifurcation point in [0,1], and if Ao = 0 is a bifurcation point, then there is a bifurcated branch on the side of X > 0. (2) For any compact operator f : X —> V with /(0) ^ 0, there is a constant S > 0 such that for all \e\ < S the equation A(x) + h(x, 1) = ef{x) has a solution x{e) £ X with x(e) —> 0, as £ —> 0. Proof.
We first assume t h a t deg(>l + / i (  , l ) , f l r , 0 ) = 0 ,
V r > 0 small.
(4.84)
99
Steady State Bifurcations
By assumption, we have deg(A, Br, 0) = deg(L, Br, 0) ± 0.
(4.85)
Assertion (1) follows from (4.84) and (4.85). If (4.84) is false, then
deg(A + h(,l),Br,0)^0,
Vr > 0 small,
which implies the assertion (2). The proof is complete.
D
The above theorem is a general principle of the alternative method. The following theorem gives a global version. Assume that (Ai) There exists a real number Ai and a nondecreasing function r\ > 0 of A > Ai such that for any s > 0 there is / G X, \\f\\ = e, and A{x) + h(x, A) = / ,
A > Ai
has no solution in Brx. (A2) There is a real number Ao < A and e > 0 such that for any 0 < t < 1 the equation
Ax + th(x,X0) = 0 has no nonzero solution in Be. Let £ = {(x, A) e X x R  Ax + h(x, A) = 0, x ^ 0} . Theorem 4.9 Under the assumptions of Theorem 48 and (A\)—(A^), the equation (483) has a bifurcation point in [Ao, Ai]. Moreover, if the set of bifurcation points of (483) is bounded in [Ao,oo), then there exists a bifurcated point A* € [Ao,oo) of (483) such that the connected component C of S containing (x,X) = (0,A*) satisfies at least one of the following assertions (1) C is unbounded in X x [Ao,oo). (2) C contains a solution (XQ,\Q) of the equation A(x0) + h(x0, Ao) = 0,
x0 ± 0.
The proof of Theorem 4.9 is similar to that of Theorem 4.7.
100
4.3
Bifurcation Theory and Applications
Bifurcation from Homogeneous Terms
In Section 4.2, we consider the bifurcation from a nonlinear term. In this section, we shall continue to investigate the problem. Let X, Y be two Banach spaces. We consider the bifurcation equation given by Ax + XBx + G(x, A) = 0,
x € X,
(4.86)
where A, B, G\ : X —> Y are continuous operators such that I A(Xx) = XkA(x) s [ B(Xx) = XpB(x)
for A > 0 and some k > 0, for A > 0 and some p > 0 with p ^ k,
I G (XMPlx, X) = o (\X\k^kpA { ) x / N \G(\1'<*»x,\1)=o(\\\k'
(4.87) '
K
iik>p, ( 4  88 ) if p>k,
for a fixed x € X with x ^ 0. Let x0 € X with xo ^ 0 be a solution of the equation A(x0) + B{x0) = 0.
(4.89)
It is easy to verify that a;(A) = A 1 /' fc ~ p )x 0 (A > 0) is a solution of the following equation A(x) + XB{x) = 0,
0 < A < oo.
(4.90)
Moreover, as p > k, x(X) = X~1^p~k^xo is a bifurcation solution of (4.90) from (x, A) = (0,+oo); and as k > p, x(X) is a bifurcation solution from (x,X) = (0,0). Prom this simple example, we are motivated to prove the following more general result. Theorem 4.10 Let xo ^ 0 be a solution of (489). Under the conditions (487) and (488), if A, B : X + Y is differentiable at x0 e X, and the derivative operator DA(x0) + DB(x0)
:X^Y
is a linear homeomorphism, then the following assertions hold: (1) If p < k, the equation (486) has a bifurcated branch (a;(A),A) (A > 0) from (x,X) = (0,0), and the solutions in the branch near X — 0 can be
101
Steady State Bifurcations
expressed as
( x(X) = A1/(p~fc)x0 + \1/{pkh(\)
VA > 0,
1 z(X) is continuous on X, and z(0) = 0. (2) Ifp < k, the equation (486) has a bifurcated branch (x(A),A) (A > 0) from (x, A) = (0, +oo), and the solutions in the branch near A = +oo can be expressed as ( x(X) = XVtrVxo ]
+ X~l^pkh{X)
VA > 0,
lim z(X) = 0.
Proof. We first proceed with the case of 0 < p < k. Let the solutions of (4.86) near A = 0 be expressed as X(X)
= A ^  ^ r r o + z(X)).
(4.91)
Putting (4.91) into (4.86), by (4.87) we find A{xo+z) + B{xo+z) + Xk^k^G(x1'^~'P\xo+z),Xsj
=0. (4.92)
By the differentiability of A and B at i 0 , near z = 0 the equation (4.92) can be written as (DA(x0) + DB(xo))z + g(z, A) + o ( z ) = 0,
(4.93)
where g(z,X) = Afc/
+ z), A) .
Based on (4.88) we find that g(z,0) = 0 for all z £ X, which implies that g : X x R —> Y is differentiable on z at (z, A) = (0,0), and Dxg(0,0) = 0.
(4.94)
By the implicit function theorem, it follows from (4.93) and (4.94) that there exists an unique continuous function z = z(X) with z(0) = 0 satisfying (4.93). Hence (4.91) is a bifurcation solution of (4.86). Now, we consider the case where k < p. Let (3 = X~l. Then the equation (4.86) is equivalent to (3A(x) + B{x) + f3G(x, (3~1) = 0.
(4.95)
102
Bifurcation Theory and Applications
Let the solution of (4.95) near f3 = 0 be of the form xU3)=pVtoQ(xo
+ z(J3)).
(4.96)
Inserting (4.96) in (4.86) we have A(xo + z) + B(xo+z)+pkttrVG(j31ttrV(xo
+ z),f3^ = 0. (4.97)
In the same fashion as above from (4.97) we can derive Assertion (2). The proof is complete. D Remark 4.3 If the mapping G : X —* Y is independent of A, then the condition (4.88) implies that G(x)=o(\\x\\k).
Remark 4.4 The condition that A > 0 in (4.87) is not essential. Actually, one can also discuss the case where A < 0 in some concrete situations. For the case where k — p in (4.87), the bifurcation of (4.86) is not simpler as k j= p except the linear case. We know that if A + XB is a linear completely continuous field, then the bifurcation point Ao must be an eigenvalue of Ax + XBx = 0. For the nonlinear case, the similar result holds. An operator K : X —> Y is called essential at x = 0 if for any bounded sequence {xn} C X with K(xn) —> 0 (n —» oo), {xn} has a convergent subsequence. Obviously, if K is a completely continuous field, then K is essential at x = 0. Theorem 4.11 Assume the condition (487) with p = k. Let the operators A + XB are essential for X € R1, and G(x, A) = o(xfe). If Xo £ R1 is a bifurcation point of (486) then the equation Ax + XQBX = 0
(4.98)
has a nontrivial solution in X. Proof. Let (x(A0 + t),X0 + t) be a bifurcation solution of (4.86) from (0, AQ). Then we have lim x(A0 +1) = 0.
Steady State Bifurcations
103
Denote by p(t) = \\x(\o+t)\\ z(t)=x{X0+t)/\\x{X0+t)\\. From (4.86) we obtain A(z) + X0B{z) + tB(z) + {3kG{f3z, A) = 0 which implies HmA(z(t)) + XoB(z(t)) = 0. Since A + XB is essential at x = 0 and \\z{t)\\ = 1, it follows that z(t) —> zo ^ 0 in X, and Zo is a solution of (4.98). The proof is complete. • 4.4
Notes
4.1 This section is based on the authors' recent work [Ma and Wang, 2004a; Ma and Wang, 2004b]. The main theorems in this section are Theorems 4.1 and 4.2, which establish the existence of steady state bifurcation when the parameter crosses an eigenvalue of the linearized problem with even multiplicities. These theorems are proved by calculating the topological degree of the equation, and the bifurcation is obtained when the degree changes as the parameter crosses a critical value. In the classical Rabinowitz and Krasnoselskii theorems, when the eigenvalue of the linear problem has odd multiplicity at the critical parameter, the degree, which are either +1 or —1, changes sign as the parameter crosses the critical value, leading to bifurcation. This is essentially a linear theory as the degree is calculated, using the homotopy property of the degree, by calculating the degree of the linearized operator. When the eigenvalue has even multiplicity, there is no change of the degree for the parameter on the two side of the critical value. The key idea in this article is based on the observation that for a class of nonlinearities, called fcth (k > 2) order nondegenerate singularities, the degree at the critical parameter is even, creating the discrepancy of the degree, and leading to bifurcation. To carry out this idea, the spectral theorem, Theorem 3.4, plays a crucial role. 4.2 This section is due to [Yu and Ma, 1989; Ma, 1990]. 4.3 Theorems 4.10 and 4.11 are introduced here for the first time.
Chapter 5
Dynamic Bifurcation Theory: Finite Dimensional Case This chapter addresses the dynamic bifurcation theory for finite dimensional systems, developed by the authors based on the notion of bifurcation called attractor bifurcation. Infinite dimensional version of the theory will be presented in the next chapter. The (finite dimensional) theory, together with center manifold reduction procedures, can be applied to both finite dimensional dynamical systems and infinite dimensional dynamical systems; see applications in Chapters 810. 5.1
Introduction
A classical example of dynamic bifurcation is the famous Hopf bifurcation. In fact, dynamic bifurcation occurs very often in many scientific and engineering fields. In this section, we give two examples, and briefly describe the basic principle of dynamic bifurcation. 5.1.1
Pendulum in a symmetric magnetic field
We consider a pendulum in a vertical plane forced by a symmetric magnetic field, friction, and gravity; see Figure 5.1. Let the length of this pendulum string be / = 1, and one end of string be tied with a small iron ball with massTO= 1. The small ball moves with friction on a vertical unit circle. On both sides of the small ball there are two symmetrical magnetic plates attracting it, which have the same magnetic magnitude. The distances between the magnetic plates and the downward vertical center of the pendulum are equal, and denoted by r » 1. Intuitively, if the magnitude A of magnetic field on each side plate is smaller than some critical value Ao, i.e. A < Ao, under the action of friction 105
106
Bifurcation Theory and Applications fc.«
r
»
r
i e \'= ]
F
l
^^^J__^^^
Pig. 5.1
F2
Pendulum with in a symmetric magnetic field.
and gravity, the pendulum will gradually stop at position 8 = 0, where 9 is the angle between the string of the pendulum and the downward vertical axis. However, as the magnitude A exceeds Ao, two new symmetric equilibrium positions ±6\ ^ 0 (A > Ao) will appear, and the position where the pendulum settles down is determined by its initial state. This is a typical dynamical bifurcation problem. The motion of pendulum satisfies the following initial value problem: d2e TIT
„
,d9
,
„
= kjT  g sin 6 + f cos 6, at at* 0(0) = ai,
e'(o) = a2, where k > 0 is the damping coefficient, g the gravity constant, and / the magnetic force. By the Coulomb law and r » 1,
f = F2F1 A
A 2
~ ( r  sin 6») 4r A sin 9 =
(r 2  sin 2 6I)2
~ A sin 6,
(r + sin0) 2
107
Dynamic Bifurcation Theory: Finite Dimensional Case
where A is proportional to the magnitude of magnetic field, and A = 4Ar~3. Thus, the motion equation can be approximatively written as jfl
J2Q
—r = —k— q sin 9 + A sin 9 cos 6. y dt2 dt Let xi = 9, X2 = 9'. Then, the motion equation is in the form
{
dxi
dx2 —— = — kx2 — g sin xi + A sin xi cos X\, dt with the initial condition xi(0) = au
(5>1)
(5.2)
x2(0) = a2.
By the Taylor expansion Asinxi cosa;! — ^sinxi = (A — g)xi — ~(4A — g)x\ + o(a;i 3 ), the system (5.1) near x = 0 is expressed as
(
dx\
(53)
f
^ = (A  g)Xl  kx2  i(4A  g)xl + When A < g, the matrix
o(\Xl\3).
(5.4) has two eigenvalues . A± =
k ± y/k2  A(g  A) 2 '
which have negative real parts. In this case the equilibrium point x = 0 of (5.3) is locally asymptotically stable. When A = g, the equations (5.3) are as follows dX! 1
!
±l = kx2 gxl
(5 5) +
o(\x1\*).
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Bifurcation Theory and Applications
The eigenvalues of (5.4) are Aj = 0, A2 = — k, i.e. The system (5.5) is at the critical state. We shall apply Theorem 3.13 to show that x = 0 is a locally asymptotically stable equilibrium point of (5.5). Let / = (/i, / 2 ), g = (gi,g2) be given by /i=z2,
h = —^gx\ + o{\x1\3),
9i = 0 ,
g2 = kx2.
Obviously x = 0 is an isolated singular point of (5.5), and div/ = 0, divg = k < 0, fi92 ~ /251 = kxl < 0,
V{x1,x2)=lxl+lgx\ + o{\x1\i). It it easy to see that the conditions of Theorem 3.13 are satisfied. Hence x = 0 is locally asymptotically stable for (5.5). When X > g, the system (5.1) bifurcates two new equilibrium points (xf,x%) = (±ex,0) from (0,g), where ex=cos1{g/\). It is easy to verify that (±#,\,0) are two locally asymptotically stable equilibrium points of (5.1). The above discussion can be summarized as the following theorem. Theorem 5.1 There exists a neighborhood U C R2 of x = 0, such that the following are true. (1) If A < g, x = 0 is locally asymptotically stable for (5.1) in U as shown in Figure 5.2(a), and (2) If A > g, the system (5.1) bifurcates from (0,g) two equilibrium points (±#A,0) which are asymptotically stable in U. Moreover, the open set U is decomposed into two open sets U\ and U%, as shown in Figure 5.2(b),
U^ul + Uf, U{nul = $, OedUlndul with (0A,O) e U\, (—6»A,0) £ Ul, such that
lim (o:i(t,a),a;2(t,a)) = (0A,0) t—•OO
lim (x1(t,a),x2(t,a)) = (0X,0)
t—*oo
ifa= (QI,Q 2 ) € U\, if a = {a^a?) e Ul,
Dynamic Bifurcation Theory: Finite Dimensional Case
where (xi(t,a),X2(t,a))
109
is the solution of (5.1) and (5.2).
*2
(a)
 *2
U
2
)
(b) Fig. 5.2 (a) If A < p, a; = 0 is an attractor; (b) If A > g, the system bifurcates to two attractors.
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Bifurcation Theory and Applications
5.1.2
Business cycles for Kaldor's model
In this subsection, we consider business cycles in the following Kaldor model:
{
dx —
7
=I(x,y,X)S(x,y,X),
(56)
I =/(*,*, A),
where x is the total social income, y the total social capital, A the industrial technique parameter, I(x, y, A) the investing function, and S(x, y, A) the saving function. For each technique parameter A, the social business has an equilibrium state (xo,yo), xo > 0, y0 > 0, which depends continuously on A. According to economic laws, in a neighborhood of (xo,yo), I and S satisfy that dl dl OS dS 7j > 0, j j  < 0, 7j > 0, 7, > 0. ox ay ox oy By these conditions, in a neighborhood of (xo,yo), I and 5 take the form as follows (I = ax(xxo)ax(yyo)3, (5.7) \S = b\(x  x0) + c\(y  2/0) + Py{x  x0)3 + jx(y  y0)3, where a\,b\,c\ > 0, depending continuously on A. The parameters here a, /3,7 > 0 are constants, a\ stands for investing intention, and 6A stands for saving intention. Based on (5.6) and (5.7), the Kaldor model is given by
{
 £ = k\(x  x0)  cx(y  2/0)  Sx(y  y0)3  Py(x  x0)3,
f
j = ax(x  xo)  ax(y  yof, at where k\ = a\ — b\, 5 = a + j . The eigenvalues of the matrix \ax
0J
are given by A± =
k\± y/k\  4axc\
•
(5.8)
Dynamic Bifurcation Theory: Finite Dimensional Case
111
Near k\ = a\ — b\ = 0, the eigenvalues of A\ are complex number and Re\± = 7}k\. Hence if k\ < 0, the equilibrium point (xo,2/o) of (5.8) is locally asymptotically stable. If k\ — 0, the eigenvalues A± = ±i\/4a.\c\. We shall prove that (io,2/o) is locally asymptotically stable. To see this, we note that the divergence of the nonlinear terms in (5.8) is nonpositive in a neighborhood of (xo,yo). Namely, for any (x — XQ)2 + (y — yo)2 < £ with (x,y) ^ (xo,yo) and with e > 0 sufficiently small, — [6x(y  y0)3  /3y(x  x0)3} + Q^[ctx(y  yo)3} = [3/3y(x  x0)2 + 3ax{y  y0)2]  S(y  y0)3 <0. Then the local asymptotic stability of (xo,yo) follows from Theorem 3.15. Let Ao € 1 be the critical technique parameter satisfying
{
<0 =0
ifA
>0
if A > Ao.
(5.9)
Under the condition (5.9), by the Hopf bifurcation theorem we know that if A > Ao, the equation (5.8) bifurcates from (xo,yo) to an attractor, which is homeomorphic to a circle S1; see Figure 5.3. From the economics point of view, the S1 attractor bifurcation represents periodic fluctuations of the economic system. When the industrial technique level parameter A < Ao, the investing interest and the social income are lower. Therefore the investment intention is weaker than that of social savings, namely ax < &A Thus, k\ = a\—b\ < 0, and in this case the social economy fluctuates around the equilibrium point (xo,yo)', as shown in Figure 5.3(a). When the technique level is promoted to exceed the critical value Ao, i.e. A > Ao, the new industrial technology brings the investing interest higher, and in this case the investment intention is stronger than that of social savings. Therefore k\ = a\ — bx > 0; thus the business is transferred from a stable equilibrium state to the form of periodic fluctuation, as shown in Figure 5.3(b), which is that we see today.
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Bifurcation Theory and Applications
(a)
(b)
Fig. 5.3 (a) When A < Ao, (xo,yo) is stable; (b) when A > Ao, the system bifurcates to a cycle attractor.
5.1.3
Basic principle of attractor bifurcation
The two examples above show that a parameter family of vector fields have an attractor bifurcation if some eigenvalues cross from left to right the imaginary axis leaving the other eigenvalues on the left side, provided that the equilibrium point at the critical parameter is locally asymptotically stable. In later discussions we can see that these are basic conditions for attractor bifurcations. To illustrate the basic principle behind attractor bifurcations, we consider the following equations ^=\X ^
+ Gl(x,y),
= y + G2(x,y),
(5.10) (5.11)
where x £ R m , y € R"~ m (1 < m < n), and Gi(x,y)
=o(\x\,\y\),
i = 1,2.
It is shown that near A = 0, the flow of (5.10) and (5.11) in a small neighborhood of (x,y) = 0 is squeezed into an mdimensional surface E\, which is called the center manifold. The mdimensional surface Y,\ can be expressed by a function y = h(x,X). The flow in E.\ has the same topological structure as the flow in R m of the following equations, which are the projection from Y,\ into R m : (5.12) £=\x + Gy(x,h(x,\)). at It suffices then to consider the dynamic bifurcation for (5.12). When
Dynamic Bifurcation Theory: Finite Dimensional Case
113
A < 0, near x = 0 the linear term Aa; determines the dynamical behavior of (5.12), and therefore the flow of (5.12) converges to x = 0. When A = 0, assume that x = 0 is locally asymptotically stable for (5.12) at A = 0, i.e. (IT
(5.13)
— =G1(x,h(x,0)). Then when A > 0 is sufficiently small, (5.12) can be rewritten as (IT
— =G 1 (x,/ J (x,0))+A 2 r + X(x,A),
(5.14)
where K(x,X) = Gi(x,h(x,X))  Gi{x,h{x,Q)) > 0 as A > 0. Hence, the flow of (5.14) is a superposition the flow of (5.13) and flow of the following system dx — = Xx + K{x,X)
(A>0).
(5.15)
The flow of (5.15) is outward from x = 0, and this flow of (5.13) is inward to x = 0. Near x = 0, the linear term dictates the flow, and for x away from x — 0, the nonlinear term G\(x, h(x, 0)) dominates the flow structure. Thus the inward flow and outward flow squeeze an attractor near x = 0, as shown in Figure 5.4.
/
Rm
Fig. 5.4 Schematic showing the existence of bifurcation attractor.
114
5.2 5.2.1
Bifurcation Theory and Applications
Attractor Bifurcation Main theorems
We consider the dynamic bifurcation for the finite dimensional systems given by dx — ~ Axx + G(x,X),
AeR, x£Rn
(n>l),
(5.16)
where G : I " x I 1 > R" is C r (r > 1) on a; 6 Kn, and continuous on AeR 1 , G(x,\)=o(\x\),
VAeR1,
(5.17)
the operator
(
a u (A) ••• ai n (A)\ ani(A) ••• ann(X)J
is an n x n matrix, and ay (A) are continuous functions of A. Let all eigenvalues (counting multiplicities) of A\ be given by
^(A),...,A,(A). We know that the eigenvalues /?i(A) (1 < i < n) are continuous with respect to A. Definition 5.1 (1) We say that the system (5.16) bifurcates from (x,X) = (0,A0) an invariant set Q,\, if there exists a sequence of invariant sets {fi.\n} such that 0 ^ £l\n, and lim An = Ao,
n—*oo
lim d(Q\ ,0) = lim max Ircl = 0.
(2) If the invariant sets Q\ are attractors of (5.16), then the bifurcation is called attractor bifurcation. (3) If the invariant sets Cl\ are attractors of (5.16), which are homotopic equivalent to an mdimensional sphere, then we say that the system (5.16) has an 5 m attractor bifurcation at (0, Ao).
Dynamic Bifurcation Theory: Finite Dimensional Case
115
Assume that
{
<0
ifA
= 0
if A = Ao,
>0
ifA>A 0 ,
Re0j(\o) < 0
forl
for m +1 < j < n.
(5.18)
(5.19)
The eigenspace of A\ at Ao is then given by m oo
£0 = U L ) { ^ r I (A, &(Ao))fc* = °}i=lk=l
It is known that dim EQ = m. The main results in this section are the following 5 m attractor bifurcation theorems for the finite dimensional system (5.16). Assume that (5.17)(5.19) hold true, and x = 0 is locally Theorem 5.2 asymptotically stable for (5.16) at X = Ao. Then we have the following assertions. (1) The system (5.16) bifurcates from (0, Ao) to an attractor Q\ for A > Ao and near Ao, with m — 1 < dimf^A < rn, which is connected when m>2. (2) The bifurcated attractor Q,\ is a limit of a sequence of mdimensional annulus Mk with M^ +1 C Mk, i.e. Cl\ = n ^ M f e . Especially, if Sl\ is a finite simplicial complex, then Cl\ is a deformation retract of mannulus; hence Q,\ has the homotopy type of Sm~1. (3) For any x\ E £l\, x\ can be expressed as x\ =z\+o(\z\\),
Z\GE0;
(4) If the number of singular points of (5.16) in Q.\ is finite, then we have the following index formula:
En x6
A
 J/ / A ^s s f2 tf™= odd, md((Ax + G),x) = \ 1 ° if m= even.
Theorem 5.3 Assume that (5.17)(5.19) hold true and x = 0 is globally asymptotically stable for (5.16) at A = Ao. Then for any bounded open set U C Rn with 0 € U, there is an e > 0 such that if Ao < A < Ao + e, the attractor Q\ of (5.16) bifurcated from (0, Ao) attracts U\T, where F d " is the stable manifold ofx = 0 with dimF = n — m. Especially, if (5.16) has
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Bifurcation Theory and Applications
global attractors near A = Ao, then e is independent of the bounded open sets U C l " .
Remark 5.1 Theorem 5.2 can be regarded as a united version of both the steady state and dynamic bifurcations. When m = odd, the Sm~lattractor Q.\ must contain singular points of the vector field A\ + G, which is an assertion of the Krasnoselski theorem. When m = 2 and /3i(A0) = /?2(^o) = h this theorem shows the existence of periodic orbits, as in the classical Hopf bifurcation. In addition, one advantage of the theorem here is the stability of the bifurcated periodic solution, which is in general an important, but difficult problem, and which is not available in the classical Hopf bifurcation theorem. Remark 5.2 In general, the attractors QA in Theorem 5.3 are not global. For example, the system
satisfies the conditions of Theorem 5.3 with Ao = 0, and for any A > 0 sufficiently small, Q\ contains two singular points X12 ~ iA 1 / 2 of the function \x — x3(l — e~1^x ). However it is easy to see that for each A > 0 small, Q.\ is certainly not a global attractor since there are two more singular points xifi ~ ±A~1//2, bifurcated from infinity, which are not in Q,\. 5.2.2
Stability of attractors
In order to prove the attractor bifurcation theorems, we first introduce a stability theorem of attractors, which is crucial not only in the proof of Theorems 5.2 5.3, but also in the proof of perturbation stability theorems for infinite dimensional systems in Ch.6. This theorem is also a supplement of Theorem 2.7 where the existence of attractors for the perturbation dynamical systems is given as an assumption. To proceed, we start with a technical lemma on stability of extended orbits for vector fields. Let v G C r ([/,R") be a vector field where U C Rn is an open set. A curve 7 C U is called an extended orbit of v, if 7 is a union of curves
7= U 7 i 8=1
such that either 7* is an orbit of v, or 7, consists of singular points of v, and if 7, and 7 i + i are orbits of v, then the wlimit set of 7* is the alimit
Dynamic Bifurcation Theory: Finite Dimensional Case
117
set of 7i+i, w(x) = a(y),
Vz eji,
y€ 7»+i'
Namely, endpoints of 7, are singular points of v, and the starting endpoint of 7J_I is the finishing endpoint of 7$; see Figure 5.5.
PI Fig. 5.5 An extended orbit.
Then we have the following stability lemma of extended orbits. The result stated this lemma has been proved in Step 2 of the proof of Lemma 4.5 in [Ma and Wang, 2002]. Here we only state the result as a lemma. Lemma 5.1 Let Vk £ Cr(U, R") be a sequence of vector fields with limfcKjoVA; = v0 £ Cr(U,M.n). Suppose that jk C U is an extended orbit of Vk with uniformly bounded length, and the starting points p\ of 7^ converge to pi, then the extended orbits 7fc of Vk converge, up to taking a subsequence, to an extended orbit 7 of VQ with starting point p\. Remark 5.3 The stability lemma of extended orbits is useful for orbit analysis of vector fields, which is a basic tool to solve some problems in fluid dynamics, such as the boundary layer separation and the interior separation of fluid flows; see [Ma and Wang, 2004c; Ma and Wang, 2003b; Ma and Wang, 2004f; Ghil et al, 2001; Ghil et al., 2003; Ma and Wang, 2005c]. The following is the stability theorem of attractors. Theorem 5.4 Let vn £ Cr(U, R n ) (r > 1) be a sequence of vector fields such that Yvoin^ooVn = VQ 6 Cr(U, R"). Let £0 C U be an attractor of VQ and D C U be the basin of attraction for So • Then the following assertions hold true. (1) For each n sufficiently large, vn has an attractor S n C D, and lim d(E n ,Eo) = lim sup dist(x, So) = 0 .
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Bifurcation Theory and Applications
(2) For any bounded open set O C D there is an N sufficiently large such that O is in the basin of attraction of S n for all n > N. Proof. Let Do CC D be a bounded open set with S o C A) Then the wlimit set for VQ of Do is the attractor So, i.e. So = w(Do).
(5.20)
To complete the proof, by Lemma 2.2 and Theorem 2.7, it suffices to prove that
{
lim d(ujn(Do),Ho) = lim sup dist(x, So) = 0, ^«<«»>
w n (A>)= n u t > r S n (0A>, r>0
(521)
~~
where Sn(t) is the operator semigroup generated by vn. By Lemma 2.1, wn(D0) is an invariant set of vn. If (5.21) is false, then at least one of the following two cases must occur: (a) There exist a number 5 > 0 and points pn g uin(Do) such that Vn —* Po £ So (n —» oo) and the extended orbits F n c wn(Do) of un starting at p n with bounded length satisfy that d(r n ,S 0 ) = sup dist(x, So) > 6.
(5.22)
(b) There exist extended orbits Tn C uin{Do) of vn and a number 5 > 0 such that inf inf x  y > 5
Vn 6 N.
(5.23)
For the case (a), by Lemma 5.1, the extended orbits Tn converge to the extended orbit To of vo with the starting point p0 £ So r n —• To
(n > oo).
(5.24)
It follows from (5.22) and (5.24) that d(T0, So) = sup dist(x, So) > 5 > 0. (5.25) zero On the other hand, So is an invariant set of VQ. Hence To C So, a contradiction with (5.25). If only the case (b) occurs, then by the definition of wlimit sets, we infer from (5.20) and (5.23) that there exist a number $i > 0 and points
Dynamic Bifurcation Theory: Finite Dimensional Case
119
Pn S Do with pn —» po £ Do (n —> oo) such that the orbits xn (£) of u n with the initial value xn(0) = pn satisfy that dist(in(t),S0)>5il
Vt>0.
(5.26)
By Lemma 5.1, from (5.26) we derive that the (extended) orbit xo(t) of i/o starting at po € Do satisfies dist(zo(*)> So) > Si,
V 0 < « < oo.
Here the inequality holds true for all £ S [0, oo) is due to the fact that for any time T > 0, the arclengths of xn(t)
an(T) = f \vn(t)\dt Jo
is uniformly bounded independent of n. It is a contradiction with (5.20). Hence the equality (5.21) holds true. The proof is complete.
5.2.3
•
Proof of Theorems 5.2 and 5.3
It is easy to see that Theorem 5.3 is a direct corollary of Theorems 5.2 and 5.4. Hence we only need to prove Theorem 5.2. We proceed in several steps as follows. S T E P 1. By the canonical reduction procedure, the dynamic bifurcation problem of (5.16) is reduced to the dynamical bifurcation problem of the following equations (527)
^=Jxz+g(z,\),
where z e M™, J\ is the Jordan matrix corresponding to the eigenvalues /3i(A), • • • , /3m(A), which satisfy (5.18), and gi(x,y,X)=o(\z\)
VAeM.
For simplicity, we assume that Ao = 0, i.e.
{
< 0
if A < 0,
= 0
if A = 0,
> 0
if A > 0,
1 < i < m.
(5.28)
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Bifurcation Theory and Applications
STEP 2. PROOF OF ASSERTION (1). Consider (5.27), and let v\ = J\ +9(',ty By assumption, x = 0 is asymptotically stable for (5.16) at Ao = 0. Consequently z = 0 is an attractor of v0. By Theorem 5.4, there exist constants r and Aj > 0 such that for all 0 < A < Ai, the set Br = {z e R m  \z\ < r} is an absorbing set of v\, and the wlimit set
Ax = u\{Br) c B r
(0 < A < Ax)
(5.29)
is an attractor of v\ in some open set D C M.m (Br c D). In addition, by the stable manifold theorem (Theorem 2.10) and (5.28) the unstable manifold W£ of v\ contains an open neighborhood of x = 0 in R m for all A (0 < A < Ai). Prom (5.29) we see that W%cA\CBr,
V0
By the definition of unstable manifolds, we obtain that A\\W%
C D CW1 is an attractor of (5.27) in D\{0},
which implies that Cl\ is an attractor of (5.16),
<
m — 1 < dimf^A < m,
(53°)
o ?nx, lim d(fl\,0) = lim sup a; = 0, A.O+ \>o+ x € n A
where £l\ is denned by Qx = {(x,y) eWl\x£Ax\WZ,y
= h(x, A)} ,
and h(x,y) is the center manifold function. Here m — 1 < dimfi,\ follows from Corollary 2 on p.46 in [Hurewicz and Wallman, 1941]. Therefore, Assertion (1) is proved. STEP 3. PROOF OF ASSERTION (2). Let SA =  4 A \ W " be the attractor
of (5.27) in D\{0}. Obviously, there is an rx > 0 sufficiently small such that the disk Brx C W%, and the annulus Rx = Br\Br;< is absorbing in £>\{0}. Hence, by Lemma 2.2, the attractor SA of (5.27) in £>\{0} is given by
(5.31) T>0 t>T
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Dynamic Bifurcation Theory: Finite Dimensional Case
Since £ A C R\ is invariant under the action of the semigroup {S\(t)}t>o, for any t0 > 0 £A C Sx(t0)Rx.
(5.32)
It follows from (5.31) and (5.32) that there is a sequence {£„} with tn+i > tn > 0 and tn —> oo (n —> oo) such that ( Sx(tn+1)RX
c
Sx(tn)Rx,
I EA = lim 5A(tn)i2A = n n—*oo
\
n=l
Sx(tn)Rx.
We know that S\(t)R\ is homeomorphic to an mannulus for any t e i 1 . Hence OA is a limit of a sequence of mannulus. Let £ A be a finite simplicial complex. We shall prove that EA is a deformation retract of an mdimensional annulus. Let M C Km be an mdimensional smooth manifold with boundary. For each point x € dM we define z(x, s) — the point z G M, which lies on the inward normal line starting at x, and the arc length from z to x is s {s > 0). Obviously, z(x, 0) = x. We take a sequence of smooth mdimensional annulus {Mn} in R\ = Br \ Brx such that
{
SA C M n + 1 c Mn c i?A, Vn > 1, Mn are deformation retracts of R\, and
(5.33)
lim Mn = EA. Moreover, the sequence {Mn} enjoys the following properties n—>oo
i) for any point x € dMn, there exists a number An(a;) > 0 such that for all x,y G 9M n , x ^y, the line segment ^x = {z(x,A) 0 0, z(x, Xn(x)) G dMn+1, if An(a:) = 0( which implies x G 9M n D dMn+1).
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Bifurcation Theory and Applications
These properties can be achieved by letting the smooth manifold Mi = R\ shrink along its inward normal direction, and by taking properly the intercepted manifold Mi (i = 2,3, • • •), inductively. Thus, for any x G dM\ we can define a curve
Lx =
U ^»'
xi=x
n=l
Xn+i = z Xn A x
( ' «( "))
>
Namely Lx is the union of the line segments tXn, where the endpoint xn+\ of £x is the starting point of £Xn+1Since SA is a finite simplicial complex, the length Lx is finite for any x G dM\\ otherwise the number of simplexes in £A can not be finite. It is easy to see that L x n L y = 0,
Vz ^ y , x,y£
dM1:
and by (5.33) the endpoint qx of Lx = Dn°=1£Xn satisfies lim yn = qx G
SA
(yn £ 4 n ) ,
n—»oo
since Lx has finite length. On the other hand, we see that Ma = £AU( ( J L x ) . \xedMx
)
Then we define a mapping H : Mi x [0,1] —> Mi by H{y,t) = < [p(y,t), y G Lx, where p(y, i) is the point p G Lx such that the arc length along L x from y to p is t x r(y) where r(y) is the length of the arc in Lx from y to qx eT,\, the endpoint of L x . It is clear that H is continuous, and H(,0) =
id:Mi>Mlt
^(.IJIMX^EA,
tfoi = i,i:EA>£A, where i : EA —> Mi is an inclusion mapping. Hence SA is a deformation retract of Mi = RA Assertion (2) is proved. STEP 4. PROOF OF ASSERTION (3). By (5.30) we see that the attractor Qx of (5.16) is in the center manifold MA for 0 < A < Ai. Since the
Dynamic Bifurcation Theory: Finite Dimensional Case
123
eigenspace Eo is tangent to M\ (0 < A < Ai) at x = 0, Assertion (3) follows. STEP 5. P R O O F OF ASSERTION (4). By the Brouwer degree theory and the conditions (5.18) and (5.19), the degree of the vector fields in (5.16) satisfies that
VAGCA^AI),
(5.34)
for some r, Xi > 0. Since £l\ is the maximum attractor of (5.16) in Br\{0}, singular points of A\ + G\ in Br are in Cl\. Then we have
all nonzero
deg((Ax + G),Br,0) = l
deg((A A + G), Br, 0) = ind((Ax
+ Y,
+ G), 0) +
ind((Ax + G),Xi).
(5.35)
On the other hand, by (5.18) and (5.19)
{
1
if m = even,
—1 if m = odd, for 0 < A < Ai. Hence Assertion (4) follows from (5.34)(5.36). The proof of Theorem 5.2 is complete.
5.2.4
Structure of bifurcated
(536>
attractors
Pitchfork bifurcation A related interesting question is to address under what conditions the attractors Q\ in Theorem 5.2 are homeomorphic to an (m — l)dimensional sphere. Here we consider the case where m = 1, which corresponds to the classical pitchfork bifurcation, and is a general version of the example of pendulum in a symmetric magnetic field given by Theorem 5.1. Theorem 5.5 Assume that conditions (5.17)(5.19) with m = 1 hold true, G(x, A) is analytic at x = 0, and x = 0 is locally asymptotically stable for (5.16) at A = Ao. Then there exists an open set U c K™ with 0 e U, such that if A > Ao the system (5.16) bifurcates from (0,Ao) exactly two equilibrium points x\,X2 € U, and the open set U is decomposed into two open sets U\ and U%, satisfying the following properties
(i)V = u\ + ul,uinui = %,
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Bifurcation Theory and Applications
(2) o e dU\ n dUl, Xi eU{(i = l, 2), (3) for any ye U{ (i = l,2), lim x(t,
t—KX>
where x(t,ip) is the solution of (5.16) with x(O,
A\x + G(x, A) = 0, x e Rn.
(5.37)
By the LyapunovSchmidt reduction, the bifurcation equation of (5.37) near Ao is as follows 0i(\)z + g(z + $(z),\) = O, z e l 1
(5.38)
where g(z, A) = P\G{x, A), Pi : R n —» E\ the cannonical projection, E\ the eigenspace corresponding to f3\, and $(z) is the implicit function defined by (3.73) of Ch.3. Because G is analytic, $(2) and g(z + $(z),\) are analytic at z = 0. Hence there exist a\ ^ 0 and k > 2 such that g(z + $(z), A) has the Taylor expansion as follows g(z +
$(z),\)=axzk+o(\z\k).
By assumption, x = 0 is asymptotically stable for (5.16) when A < Ao. Therefore if A < Ao, the system (5.37) has no nonzero solutions near x = 0, which implies that k is an odd number, and a\ < 0 for A near Ao. By (5.17), the equation (5.38) is given by
0i(\)z + axzk + o(\z\k)=O, which obviously bifurcates from (0, Ao) to exactly two singular points *i.2(A) = ±/3! (A)/^! 1 /^"!) + 0 (/3 1 (A) 1 /(*D). By Theorem 5.2, the system (5.16) bifurcates from (0, Ao) to an attractor A\ with dim A\ < 1. It is clear that A\ consists of exactly two singular points Xi{\)  (zi(\),$(zi)) £ R" (i = 1,2). By the stable manifold theorem (Theorem 2.10), there is an (n — l)dimensional stable manifold Ws of (5.16) at x = 0 dividing the open set U into two parts U\ and U% such that Xi G Ulx and X{ attracts U{ (i = 1,2). This proof is complete. •
Dynamic Bifurcation Theory: Finite Dimensional Case
125
Remark 5.4 It is still an open problem whether the analytic condition of vector fields is sufficient for the bifurcated attractor £l\ of (5.16) being homeomorphic to an (m — l)dimensional sphere when m > 2. Minimal attractors It is easy to see that a subset T\ C Cl\ may be an attractor as well. The minimal attractor T\ contained in £l\ is called the bifurcated minimal attractor of (5.16). In general, the bifurcated attractor Q\ may have no minimal attractors other than itself. We are interested here in the existence problem of minimal attractors. Here we always assume that the conditions of Theorem 5.2 hold. If the bifurcated attractor Q\ of (5.16) is homeomorphic to a sphere Sm, we denote it by £l\ = Sm for simplicity. Theorem 5.6 Under the conditions of Theorem 5.2, we assume that m = 2 and D,\ = S1 contains singular points of (5.16) which are all nondegenerate. Then the following assertions hold true. (1) The number of singular points on£l\ = S1 is 2k for some integer k > 1, and there are exactly k singular points {x^  1 < i < k} C £l\ such that each Xi is a bifurcated minimal attractor of (5.16). (2) There is an open set D C R™ which can be decomposed into k open sets Di (1 < i < k) such that fl\ U {0} c D, D = X)*=1 A , A n Dj =
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Bifurcation Theory and Applications
Fig. 5.6 The steady states x\ and X2 are minimal attractors, and the circle represents the bifurcated attractor Q\.
0 U = E l i U i , UinUj =
Dynamic Bifurcation Theory: Finite Dimensional Case
127
assumption we can derive that £l\ = £ = S2; otherwise by the PoincareHopf index theorem, Q\ must contain more than two singular points. Since Q\ = S2 contains only two nondegenerated singular points, by Theorem 5.2, their indices are +1. Therefore, the two singular points are either attractors or repellors in $l\ = S 2 (since Q\ contains no periodic orbits). Due to the PoincareBendixon theorem, we can deduce that one of the two singular points is an attractor and another is a repellor, furthermore, the attractor attracts fl\/{xi}, where x\ is the repellor in Cl\. Hence the attractor attracts a neighborhood of x = 0 except the stable manifolds of x\ and x = 0. Thus Assertions (1) and (2) are proved. When the vector field G{x, A) in (5.16) is odd, the two singular points in Cl\ have the same eigenvalues. Hence they have the same local topological structure, which implies, by the above conclusion, that fix must contain a periodic orbit. The proof is complete. • 5.2.5
Generalized Hopf bifurcation
Now, let us consider more general bifurcation. Let the eigenvalues of A\ satisfy
{
< 0 (or > 0)
if A < Ao,
(5.39)
=0 ifA = A0, Vl 0 (or < 0) if A > Ao, Re(3j{\0) ^ 0 V m + 1 < j < n. (5.40) It is known that if m = odd, the system (5.16) must bifurcate from (0, Ao) a singular point. When m = 2, the Hopf bifurcation amounts to saying that if /?i(A) = /32{X) with 7m/3i(A0) ^ 0, then under the conditions (5.39) and (5.40) the system (5.16) bifurcates from (0,Ao) a periodic orbit. Our next question is to see whether (5.16) bifurcates from (0,Ao) an invariant set assuming only (5.39) and (5.40) with m = even are valid, i.e. without the asymptotic stability assumption. In general, as we shall see later, this statement is not true. However, we can still derive a generalized version of the Hopf bifurcation as follows; see also the discussion in Section 6.3. Under the conditions (5.39) and (5.40) we know that there exists an mdimensional center manifold of (5.16) at A = Ao Mcm = {(x,y) G W1  x £ Q c M.m, y = h(x,X0)} ,
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Bifurcation Theory and Applications
which is invariant under the flow of (5.16) at A = Ao. We say that the center manifold M™ is stable (resp. is unstable) if the wlimit set (resp. the alimit set) of M™ is w(Mcm) = 0
(resp. a(M™) = 0).
Theorem 5.8 Let the conditions (5.39) and (5.40) hold. If the center manifold M™ of (5.16) at \ = \0 is stable or unstable, then (5.16) must bifurcate from (0, Ao) an invariant set T\ with A ^ Ao, and T\ has the homotopy type of Sm~1 provided T\ being a finite simplicial complex. Especially, if m = 2 and there are no singular points in F\, then F\ must contain a periodic orbit. The proof of Theorem 5.8 is similar to that of Theorem 5.2; we omit the details. Remark 5.5 If m = 2 in (5.39) and (5.40) and ft (A) = /?2(A) with /mft(A 0 ) 7^ 0, then the center manifold Mc of (5.16) must be one of the three cases: i) stable, ii) unstable, iii) containing infinite periodic orbits which implies the bifurcation of periodic orbits. Hence, the Hopf bifurcation is included in Theorem 5.8. In the following, we give an example which shows that under only conditions (5.39) and (5.40), bifurcations to invariant sets may not occur. We consider the parameterized vector field given by
{
Xxi — x2 — A x2, Xx2 + x\  1xxx\ + \2xl.
(5.41)
It is clear that the vector field (5.41) has no bifurcated singular points from (a;, A) = (0,0). Actually, we see from (5.41) that x2vx 
Xlv2
= {x\  x\f  X2{x\ + x\) < 0,
for all {x\,X2) ^ 0 and A ^ 0. In addition, by using the polar coordinate system, it is easy to show that (5.41) has no bifurcated periodic orbits from (0,0). The topological structure of vector field (5.41) can be schematically illustrated by Figure 5.7(a)(c).
Dynamic Bifurcation Theory: Finite Dimensional Case
129
Fig. 5.7
5.3
Invariant Closed Manifolds
Motivated by the 5 m attractor bifurcation, we study in this section invariant closed manifolds and their stability of a vector field. 5.3.1
Hyperbolic invariant
manifolds
A closed manifold is a compact manifold without boundary. Let v € Cr(D,,Rn) (r > 1) be a vector field and fi C Rn be an open set. Let M C fl be an mdimensional manifold with 0 < m
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Bifurcation Theory and Applications
Definition 5.2 We say that an invariant manifold M of a vector field v e Cr(Q.,Rn) is /istable if there exists a neighborhood O C C r (fi,R") of v such that for any u 6 O, u has a unique invariant manifold M\ which is homeomorphic to M and d(Mi,M) = max dist(a;,M) > 0 as u —> v.
(5.42)
A point #o € M is called C r (r > 1) if there is a Cr neighborhood O C M of xo • Let M be an m— dimensional invariant manifold of a vector field v € Cr(O,]Rm) and x0 G M is a C 1 point. Then there exists an (n — m)dimensional normal space NXo of M at xo We call the vector field u Nx0 °f the projection of u on Nx the normal vector field of v at XQ, and XQ is an origin of NXo. Obviously, a Cr (r > 1) manifold M C 0 is invariant for vector field u 6 Cr(Q.,Wn) if and only if the normal vector field of v at each point xo G M satisfies that vNxo(x0) = 0.
(5.43)
Definition 5.3 Let v £ Cr(fi,]Rn) (r > 1) and M C fi be an invariant manifold of v. We say that M is hyperbolic if (1) M is Cr manifold; (2) for each point xo £ M, the normal vector field VNXO of v is regular at the origin Xo, i.e. the Jacobian matrix DVNXO{XQ) of VJVXO at XQ is nondegenerate; (3) the real parts of all eigenvalues of DVNXO (XO) are nonzero. The concept of hyperbolic invariant manifolds is a natural generalization of hyperbolic singular points and closed orbits. Similar to the stable manifold theorem for singular points, we have the following stable manifold theorem for hyperbolic invariant manifolds. Theorem 5.9 Let M be an mdimensional hyperbolic invariant manifold of v £ C r (Q,R. n ). Then there exist two unique manifolds Wu and W3 with dim Wu = m + ki, dim Ws = m + k2, fci + k2 = n — m, called the unstable manifold and stable manifold of M, which are characterized by Wu = {z G Rn I lim dist(S(t)z,M) = 0}, Ws = {zeRn\
t—»oo
lim dist{S{t)z,M) = 0},
t—too
where S(t) is the semigroup generated by the vector feied v. Moreover, Wu and Ws are Cr and transversal on M.
Dynamic Bifurcation Theory: Finite Dimensional Case
131
Proof. Since M is hyperbolic, by Theorem 2.10, for each point xo 6 M the normal vector field VNXQ of v has the unstable manifold W"o and stable manifold W*Q with dim W^o = &i,
dxmW*0=k2,
ki+k2=nm,
and W£o, W£o are Cr and transversal at xo. It is clear that the sets
wu= U w?0,
w s = U W^
xoeM
xoeM
are the desired manifolds by this theorem. The proof is complete.
•
Prom the stable manifold theorem (Theorem 2.10) we know that for each point XQ £ M, the normal space NXo can be decomposed into a direct sum of two spaces NXo = Ei® E2,
dimEi = k\,
dimE2 — k2,
such that WXo and W£Q are tangent to E\ and E2 at the origin XQ respectively. Moreover, E\ and E2 are the eigenspaces of DVN^^XQ), and the eigenvalues Ai, • • • , X^ on E\ and /?i, • • • ,(3k2 on E2 of DVNXQ (XO) satisfy that ReXi > 0 (1 < i < hi),
Reft < 0 (1 < i < k2).
(5.44)
By Theorem 5.9, we can see that there is a tubular neighborhood O c ! l of M such that f LO(U) n U = Wu n U, a{U) nU = W3nU, and < \ M = LJ(U) na(U) nU,
(5.45) J
where u>(U) and a{U) are the wlimit and alimit sets generated by v and Wu, W3 are the unstable and stable manifolds of M. In fact, (5.45) can be also achieved from (5.43) and (5.44). Let u 6 Cr(Q,R") with u  v\\c > 0 sufficiently small. Then there exists a sufficiently small neighborhood U of M such that the normal vector field UNX of U on Nx with x £ M has a unique zero point ZQ S U PI A^, and the Jacobian matrix DU^^ of L/jvx at ZQ is nondegenerate: (UN.(ZQ)=0,
{ detUNx (z)
Z0£NX,
X£M,
^0,\fzeNxDU.
(5.46)
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Bifurcation Theory and Applications
Obviously, the below set M = {zxeU\
UNx (zx) = 0, x £ M}
(5.47)
is a manifold homeomorphic to M. Although in general M is not invariant for u, we can expect that there is a unique invariant manifold of u around M which is homeomorphic to M. It is the subject in the next subsection. 5.3.2
S1 attractor bifurcation
In this section, we shall prove that the bifurcated attractor Q,\ of (5.16) from an eigenvalue with multiplicity two is homeomorphic to a circle S1. Let o b e a twodimensional Cr (r > 1) vector field given by vx(x) = \xG(x,\), for xeR2.
(5.48)
Here
G(x,X)=Gk(x,X)+o(\x\k), where Gk is afcmultilinearfield,which satisfies CMk+l
< (Gk(x,X),x) < C2\x\k+1,
(5.49)
for some constants Ci > C\ > 0, k = 2m + 1, and m > 1.
Theorem 5.10 Under the condition (5.49), the vector field (5.48) bifurcates from (x, X) = (0,0) on X > 0 to an attractor Cl\, which is homeomorphic to S1. Moreover, one and only one of the following is true. (1) Q\ is a periodic orbit, (2) Q\ consists of only singular points, or (3) Q\ contains at most 2(k + 1) = 4(m + 1) singular points, and has AN + n (N + n > 1) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero, as shown in Figure 5.8 for N = 1 and n = 2. Proof.
We proceed in the following five steps.
STEP 1. Obviously, (5.49) implies that x = 0 is asymptotically stable for (5.48) at A = 0. Hence, by Theorem 5.2, the vector field v\ bifurcates from (x, X) = (0,0) to an attractor Q\ on A > 0, which has the homology type of a circle S1.
Dynamic Bifurcation Theory: Finite Dimensional Case
"pj I
""
\T
"
133
lp6 "
i
Fig. 5.8 Q\ has AN + n (N = 1 and n = 2 shown here) singular points, where pi, P4 are saddles, P3, P6 are nodes, and p2, ps are singular points with index zero.
2. Let ^A have no singular points. Then, Q.\ must contain at least a periodic orbit. We need to show that Q,\ contains only one periodic orbit. Take the polar coordinate system (xi,x<2) — (r cos6,r sin9). Then the vector field v\ becomes STEP
d r
^rcos^i+sin^2 d6 cos 6v2 — sin 6v\'
(55Q)
We see that cos 6vi = Xr cos2 6 — cos 6gi (r cos 0, r sin 9, A), sin 0^2 = Ar sin 2 6 — sin 0
t = l,2.
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Bifurcation Theory and Applications
By (5.49), (5.50) is rewritten as d r
dO
X
_
r2m(cQS 9
m
+
g i n Qgk2)
+
2m l
r  {sm9gklcos9gk2
o ( r 2m)
+ O{r))
'
{(5.52)
' '
Based on (5.49), we have C\ < cos6gki(cos6,sin#, A) + sin#<7fc2(cos#,sin0, A) < C2.
(5.52)
On the other hand, by assumption, Q\ contains a periodic orbit for any A > 0 sufficiently small. Hence 0 < C < sin0gki(cos9,sin9,\)
 cos0gk2(cos6,sin9, A) + O(r),
(5.53)
for any 0 < 9 < 2TT and some constant C > 0. The condition (5.53) amounts to saying tha t the orbits of v\ ar e circula r aroun dx = 0. Let r(9,r 0) be the solution of (5.51) with initial value r(0,r 0 ) = ro . Then we have the following Taylor expansion r2m(9,r0)
= r20m + R(9)o(\r0\2m),
R(0) = 0.
(5.54)
+ o(r^m),
(5.55)
It follows from (5.51) and (5.54) tha t
= 2n(a\brlm) where ,2TT
a=
L
h=
L
!
m+o(r)d9> m+o(r)d9>
a{9) = cos9gki + smdgk2, (3{6) = sin6gki  cos9gk2. Pro m (5.55) we see tha t the periodic solutions of v\ near x = 0 corre spond to positive solutions of 2n(aX  br2m) + o{rlm) = 0.
(5.56)
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Dynamic Bifurcation Theory: Finite Dimensional Case
By (5.52) and (5.53), a > 0 and b > 0. Therefore, (5.56) has a unique positive solution near r = 0:
(
> \ i/2m
for any A > 0 sufficiently small. Thus, Q\ has a unique periodic orbit. 3. We claim that if Cl\ contains either finite number of singular points or a circle of singular points, and if it contains finite number of singular points, then there are at most 2(k + 1) of them near x = 0. In fact, if STEP
9i(x,X) = £i g2 (x, A)
12'
then ft A has a cycle of singular points. Otherwise, by (5.49), the number of singular points of v\ is finite. The maximal number of singular points for v\ is determined by the following equation Xx~Gk(x,X) =0.
(5.57)
Since Qk is a fcmultilinear vector field, the singular points of (5.57) must be on the straight lines xi = zx\, where z satisfies z = gM (*l,S2,A) =
5fci(zi,Z2,A)
^llil^. gki(l,z,X)
(5.58)
The number of solutions of (5.58) is at most k + 1. Since k =odd, the number of solutions of (5.57) is at most 2(fc + 1). STEP
4. Let £l\ contain a cycle S1 of singular points, then we shall see
that nx = S1.
Under the polar coordinate system, we have vr{0, r) = (vx, x) = Xr2  rk+la{6) + o{rk+1), where a{6) is defined by (5.55). By (5.52),
0 < C\ < a(6)
vr(0,rx(0))=O,O<0<2ir}
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Bifurcation Theory and Applications
is homeomorphic to a cycle S1, and all singular points of v near x = 0 are on fl\. It implies that fix C fix Let
9ki= £
^ijX\4,
(5.59)
1 = 1,2.
i+j=k
By Step 3, we know that X2<7fci = X\gk2 Then we infer from (5.49) that 0 < al0 = a2k_n.
(5.60)
For the singular point (£\, 0) € fix of i>,\, we have div vx(xlt0)
= 2A  fca^xt"1  a2k_nxk1
+ o^"1)
= (by aio^i" 1 = A a n d (560)) = (fcl)A + o(A) <0, for any A > 0 sufficiently small. In the same fashion, for any point x 6 £l\, we take an orthogonal system transformation such that x is on the 5iaxis, then we can prove that div v\{x) < 0, which implies that fl\=flx STEP
\/x G fix,
= S1.
5. fl\ contains finite number of singular points. We show that
flx = S\
By the Brouwer degree theory, it follows from (5.49) that deg(i>A,fi,0) = 1,
A > 0
sufficiently small,
in some neighborhood fi C M2 of x = 0. It is known that ind(«A,0) = l,
A^0.
Hence we have
(5.61)
J2 md(vx,Zi)=0.
Let z £ fix be a singular point of via. Without loss of generality, we take the orthogonal coordinate system such that z = (xi, 0). Then by (5.49) and (5.59), the Jacobian matrix of vx at z is given by _, , ,
/(Jfcl)A + o(A)
Dvx(z)=( V
'
U
'
* \i~
\
, i u ) , kUlak0)AJ
2 a
,_... (562)
Dynamic Bifurcation Theory: Finite Dimensional Case
137
where a\0 > 0. Obviously, Dv\(z) has an eigenvalue f3 = — (fc—l)A+o(A) 7^ 0. Hence for any singular point z £ £l\ of v\, the index of v\ at z can only be either 1, —1 or 0. It is easy to see that if the index is 1, then z is a stable node point. Let the index of v\ at z be — 1: index(t>A, z) = —1
(5.63)
When a1_n ^ a\0, z is nondegenerate. Therefore, v\ has a unique unstable manifold at z. When af._n = a\0, divt/A(z) =  ( f c  l ) A + o(A) < 0 .
(5.64)
If the unstable manifold of v\ at z is not unique, then the local structure of v\ at z is topologically equivalent to that as shown in Figure 5.9.
Fig. 5.9
On the other hand, (5.64) means that there is a neighborhood O c R 2 of x = 0, such that div v\(x) < 0,
Vx e O,
which implies that for any open set O C O, \6\ > \6t\,
0
(5.65)
where to > 0 depends on O, Ot = S(t)O, and S(t) is the flow semigroup generated by v\.
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Bifurcation Theory and Applications
However, it is clear that for any open set O C O in domain P as shown in Figure 5.9, the property (5.65) is not true. Therefore, the unstable manifold of v\ at z must be unique We can prove in the same fashion that if the index of v\ at z is 0, then the unstable manifold of v\ at z is also unique. By the PoincareBendixson theorem, all unstable manifolds of the singular points of v\ at z with index —1 and 0 are connected to the singular points with index 1 and 0, as shown in Figure 5.8. Thus by the uniqueness of unstable manifolds for each singular point with either index —1 or index 0, the set of all singular points and unstable manifolds is a cycle S1, and Q x = S1. The proof is complete. • The above proof implies also the following theorem. Theorem 5.11 Ifv\ is an mdimensional C (r > 1) vector field given by (5.48) and satisfies (5.49) for x € K m , then v\ bifurcates from (x, A) = (0,0) to an attractor fi,\. If Q\ contains an (m — 1)dimensional sphere consisting of singular points ofv\, then Cl\ is homeomorphic to 5 m  1 .
5.4
Stability of Dynamic Bifurcation
Let Q. C R" be an open set, x = 0 e £1 and A € / = (a, b) C E 1 . We denote the space of parameterized vector fields by C0M(fi x / , R " ) = { t ; : f i x / ^ R r i  «(0, A) = 0, VA € 1} endowed with the norm
He*,. = sup ( J2 \D>\ + \D*°\ + \^Dxv\ ) . (*,A)enxz ^ H = o
)
Obviously, if v £ CQ'1^ X 7,R n ) (k > 1) then v is fcth differentiate at I E ! 1 and differentiable at A £ / , and v can be expressed as V(;\)=AX + G(;X), where A\ and G{, A) are the same as in (5.16) and (5.17).
(5.66)
Dynamic Bifurcation Theory: Finite Dimensional Case
139
It is known that simple real and complex eigenvalues of A\ are differentiable on A [Kato, 1995], which can be expanded as (3i{\)=ai
+ ai\ + o(\\\), 1
(l
X
Definition 5.4 Let vu v2 6 C Q ' ^ J,R") and A; £ / (i = 1,2) be a bifurcation point of an invariant set Ti{p) of Vi(x,p). We say that both bifurcation points X\ and A2 have the same structure if v\ and v2 are locally topologically equivalent at Ti(pi) and T2(p2) with p\ — \\ = p2 — X2, i.e. there are neighborhoods Ui C R™ of Ti(pi) and a homeomorphism of (f : U\ —> U2 such that
0. Theorem 6.2 Let (6.2)(6.5) and (6.8) hold true. Then we have the following assertions. (1) IfX < Ao, then u = 0 is a locally asymptotically stable equilibrium point of (6.7). (2) If u = 0 is a locally asymptotically stable for (6.7) at A = Ao, then the assertions of Theorem 6.1 hold true for (6.7). Proof.
We proceed in several steps as follows.
STEP l. REDUTCTION TO EQUATIONS WITH FIRSTORDER IN TIME. It
is easy to see that (6.7) is equivalent to ' du — = a.u + v, at < ^=Lxu + a2uav + G(u,X), at (u(0),v(0)) = (<po,ipo),
(69)
Dynamic Bifurcation Theory: Infinite Dimensional Case
155
where <po =
H = #1/2 x H,
H1/2,
equipped respectively with inner products ((ui,v1),(u2,v2))n1 ((U1,V1),(U2,V2))H
Let L\:Hi^H
= (ui,«2)ffi
= (ui,u2)Hl/2
+(VI,V2)H1/2,
+ (v1,v2)H
and G(, A) : Hi > H be two maps defined by Lx = A + B, G(u,v,X) = (0,G(u,\)),
where (u, v) 6 "H\, and
.
A{u
.
fal I\ fu\
>V)={Aal){v)
=
( auv \
{Au + av)>
ZM={a>i°+Bx§Q
= {a>u0+Bxu)
Thus, (6.9) is rewritten as ^=Lxw
(6.10)
+ G(w,X),
where w = (u, v). We infer then from (6.8) that f G : H » W is Cr bounded, { .
(6.11)
STEP 2. PROOF OF ASSERTION (1). Since L\ is symmetric, it has a sequence of eigenvectors {efc(A)} C H\, which constitutes an orthogonal basis of H. Moreover, by taking proper norms for Ha, we can make {efc(A)} common orthogonal basis of Ha for any a £ R. Then it is easy to see that eigenvalues of L\ are given by Pk(X)
= a± ^Jcfi+(3k{\),
fc
= l,2,".
(6.12)
By (6.4) and (6.5), we find t h a t for A < A0) R e p f c ( A ) < 0 , fc = l , 2 ,    .
(6.13)
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Bifurcation Theory and Applications
Furthermore, it is easy to check that the semigroup T\(t) of linear operators generated by L\ is given by
Tx{t) e
{£)<>*#) *!(«) J U J '
^)
(6 14)
'
for any (
*i(t) = coshtCl'2 = l(e~
+ e
$2(i) = sinhtC]!2 = \ ( e < / 2  e  < 2 ) . Namely, for any (
oo
we have
k=l 1
k=N+l
AT
oo
fc=l fc=JV+l
where the positive integer iV is given by 2
a+A
f >0
\
if it < N,
akin.
The direct caclculation shows that there is a constant K > 0 such that for any t > 0, rA(t)=
sup e* t [$ 1 (*) v > + ^ 1 / 2 $ 2 ( t ) V '   i / 2 llvlll/2 + IIV'l0 = l
+ n4 / 2 $ 2 (% + $1(^111/2] (6.15)
KKe*, where by (6.13) /9 = mini Repfcl > 0, k
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Dynamic Bifurcation Theory: Infinite Dimensional Case
and the norm
fc=l
for any a € R l . Thus, by Theorem 3.12, Assertion (1) is proved. 3. PROOF OF ASSERTION (2). It suffices to verify the condition (2.35) in the center manifold theorem, Theorem 2.14. Under the basis {e/j(A)}, the equation (6.10) can be decomposed into the following form STEP
—r^ = —axi + yt at < ^ = Pi(\)xi + a2xi  ayi + G{{u, t) at
if 1 < i < m, if 1 < i < m,
(6.16)
^ = Lxw + PG(u,X), ^ at where m
U = 22xiei
+Wl,
i=l m
i=l e Ex,
w = {wi,w2) Ex = {(w1,w2)
and Cx =
LX\EX
eH
I (wj,ei)H
= O,j = l,2;i = l,
,m},
'• Ex —» ^A is defined by \L*X + a21 alj
\w2j
H = span{e m+1 , e m + 2 , • • •} in H. Hence the eigenvalues of Cx are pk(^) (k = m + 1, m + 2, • • •) given by (6.12), and the semigroup Sx(t) generated by Cx is Sx(t) = Tx(t)\Ex where Tx{t) is defined by (6.14).
: Ex > EX,
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Bifurcation Theory and Applications
As in the proof of (6.15), we infer from (6.5) that
\\Sx{t)\\
W>0,
m i n Rep fe (A), k>m+l
for any A — Ao > 0 small. Hence the conditions of Theorem 2.14 are verified. • The proof is complete. Now we consider the case where m = 1 in (6.4). Let
(G(u,X) = G1(u,X)+o(\\u\\k), \Gi(x«o,A) = W^xGtfaX),
' '
where k > 1, and let
(a > 0),
(6.18)
where e £ £i(Ao) is the eigenvector corresponding to /?i(A) at A = AoTheorem 6.3 Assume m = 1 in (6.4), and assume (6.17) and (6.18). Then there exists a neighborhood U C Hi x #1/2 of (u,ut) = (0,0) such that the system (6.7) bifurcates from (u,ut,X) = (0,0, Ao) exactly two equilibrium points (ui\,0), (it2A)0) € U. In addition, U is decomposed into two open sets U\ and XJ\:
such that
(0,0)edUlndU2x, (Ui,o)eUi i = i,2, and for any (
lim \\ut(t,(p,i>)\\ = 0 ,
t—*oo
where u(^,^,^) w £/ie solution of (6.7).
Dynamic Bifurcation Theory: Infinite Dimensional Case
159
Proof. First it is easy to see from (6.16) that the bifurcation equation of (6.7) is given by
{
dx — = ax + y,
(6.19)
J
J = fa(\)x + a2x  ay+ < G{xex + h{x, A), A), ex >H . By (6.17) and (6.18) we have (G(xei + h(x, A), A), ex)H = oiX\x\kxx
+ o(\x\k).
We make the change of variable x = x, y = y + ax. Then at A = Ao, the bifurcation equation (6.19) becomes
{
dx (6.20)
T
^ = 2aya\x\k1x + o(\x\k), where a > 0 is given by (6.18). By Theorem 5.5, it suffices to prove that (x,y) = (0,0) is locally asymptotically stable for (6.20). To this end, let /i
=y,
h = a\x\k~1x + o(\x\k), 91=0, 32 = 2ay. Then ox ay divg = —2a, /i
160
6.2 6.2.1
Bifurcation Theory and Applications
Bifurcation from Simple Eigenvalues Structure of dynamic bifurcation
Hereafter we always assume conditions (6.2) and (6.3). Let the nonlinear operator G(, A) : Hi —> H in (6.1) have the Taylor expansion near u = 0 as follows G{u, A) = Gk(u, A) + o(ufc),
k > 2 an integer,
(6.21)
where Gk : Hi x • • • x Hi —> H is a k multilinear mapping, and we set Gk(u,X)
=
G1(u,...,u,X).
Let pj(X) G C be the eigenvalues (counting the multiplicity) of L\. Assume that /3i(A) is real, and
{
<0
ifA
t—>oo
= 0, if
(i = 1,2),
where u(t,ip) is the solution of (6.1). (4) The bifurcated singular points vi{\) and U2(A) in the above cases can be expressed in the following form
t/li2(A) = ±\(3i{X)/a \Wk»ex{\)
+ o(/?i/a VC=D).
Theorem 6.5 Assume (6.21) (6.23), k =even, and a ^ 0. T/ien the following assertions hold true. (1) (6.1) bifurcates from (0, Ao) a unique saddle point v(\) with Morse index one on X < Ao, and a unique attractor v(X) £ Hi on Ao < A. (2) If Ao < A, there is an open set U C H of u = 0, and U is divided into two open sets U* and [/£ by the stable manifold T of u = 0 with codimension one in H:
U = u\ + ux2,
u? n C/2A = 0,
r = dU? n dU$,
such that v(X) G J7j\ and lim \\u(t,
Ve
U?,
t—>oo
where u(t,ip) is the solution of (6.1). (3) The bifurcated singular points v(X) of (6.1) can be expressed as v(X) = (/3 1 (A)/a) 1 /('=i) e i + 0 ( / 3 1 / a i/(fei) ) . If we replace the condition (6.23) by Re/3j(A0)>0
i f l < j < n + l,
\Re/3j(A 0 ) < 0
if j>n
+ l,
(6.25)
then the bifurcated singular points of (6.1) are saddle points, which are characterized in the following theorem.
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Bifurcation Theory and Applications
Theorem 6.6 Assume the conditions (6.21), (6.22),(6.25) and a ^ 0 in (6.24). Then the bifurcated singular points of (6.1) from (0, Ao) have Morse index n + 1 on A < Ao, and have Morse index n on Ao < A. Moreover, u = 0 has Morse index n on X < Xo and has Morse index n + 1 on XQ < X. Remark 6.2
In general, if we replace (6.24) by
(G{xex + h(x, Ao), Ao), e\)H = axk + o{\x\k),
ieR,
where h(x, A) is the center manifold function, then for a ^ 0 and k > 1, Theorems 6.46.6 are valid. Remark 6.3 The topological structure of the dynamic bifurcation of (6.1) is schematically shown in the center manifold in Figures 6.16.3.
(a)
(b)
Fig. 6.1 Topological structure of dynamic bifurcation of (6.1) when k = odd and a > 0: (a) A < Ao; (b) A > Ao. Here the horizontal line represents the center manifold.
Proof of Theorems 6.46.6. By the canonical reduction in Chapter 3, the bifurcation equation of (6.1) is as follows: — = A(A)x +
(6.26)
where h is the center manifold function satisfying h(x,X) = o(\x\) V A G K .
(6.27)
By (6.21) and (6.27), (6.26) can be rewritten, near A = Ao, as ^=(31(X)x at
+ axxk+o(\x\k),
(6.28)
163
Dynamic Bifurcation Theory: Infinite Dimensional Case
v
u=0
u=0
l
V
2
i I
1
,
1
(a) Fig. 6.2
—«
,
(b)
Topological structure of dynamic bifurcation of (6.1) when k = odd and a < 0.
» v^
«
—•«
«
u=0
—«
»
u=0 t
(a)
u=0
<— v^
i
(b)
(c)
Fig. 6.3 Topological structure of dynamic bifurcation of (6.1) when k = even and a ^ 0.
where ax = {Gk(e1(X),X),e*1(X))H^a,
if A  Ao.
(6.29)
It is then easy to see that Theorems 6.4 and 6.5 follows from (6.28) and (6.29). The proof of Theorem 6.6 is trivial; we omit the details. The proof is complete. 6.2.2
Saddlenode
bifurcation
Now we consider a class of bifurcations as shown in Figure 4.1, called saddlenode bifurcations. First, we recall a new version of Theorem 4.2 as follows. Theorem 6.7 Assume that the conditions (6.2)(6.5) hold true and the eigenvalues j3j{\) ^ 0 of L\ for all A < Ao. If equation (6.1) bifurcates from (0, AQ) on A < AQ a branch SA of singular points having nonzero
164
Bifurcation Theory and Applications
index near Ao, which is bounded in H x (—00, Ao), then there exists a point (u*, A*) e Hi x K1 with X* < Ao and u* ^ 0 satisfying ind((Lx
+ G),u*) = O,
atX = X",
(6.30)
and there are at least two branches F^ of singular points of (6.1) bifurcated from (u*,\*) on A > A*, i.e. ux>u*, for any ux£T$
(l<j<J,J>
if\>\*+0, 2).
Definition 6.1 We say that (6.1) has a saddlenode bifurcation from (u*,A*) on A > A* (resp. on A < A*) if u* satisfies (6.30), and (6.1) has at least two branches of singular points T* (1 < j < J, J > 2) of singular points of (6.1) bifurcated from (u*, A*) on A > A* (resp. on A < A*). The following theorem is motivated toward to applications in hydrodynamic bifurcation and stability. Theorem 6.8 true, and
Assume that the conditions (6.21)(6.24) with a^O hold Vue#i,AeR\
j
\
1/2,
(6 31)
'
for some Ai < Ao and a constant C > 0, where Ao is as in (6.22). If (6.1) has uniformly bounded attractors for bounded X, and all eigenvalues flj(X) T^O of L\ for A < Ao, then the following assertions hold true. (1) If k =even in (6.21), then (6.1) has a saddlenode bifurcation from (u*,X*) with Ai < A* < Ao, and the connected component C\ containing (u*, A*) of the following set
Tx = {(u,X)eH1 xR1 I Lxu + G(u,X) = 0,u^0} is nonempty at Ao < A < Ao + e for some z > 0. (2) If k =odd in (6.21) and a > 0, then (6.1) has at least one saddlenode bifurcation from (u*,X*) with X\ < A < Ao, and if all singular points on Tx are regular at Ao, then the connected components Cx of Tx containing all singular bifurcation points on X < Ao has at least two singular points of (6.1) at each X with XQ < X < XQ+S for some £ > 0.
165
Dynamic Bifurcation Theory: Infinite Dimensional Case
Proof. Based on (6.30) and (6.31), (6.1) has no nontrivial singular points at A = Ai. By assumption and Theorems 6.4 and 6.5, the bifurcated branches of (6.1) from (0, Ao) on A < Ao is nonempty and bounded in iJxR 1 . Hence, the existence of saddlenode bifurcation follows from Theorem 6.7. It is then routine to prove the rest par t of the theorem by calculatin g the indices of the singular points of (6.1) near Ao. The proof is complete. D 6.3 6.3.1
Bifurcation from Eigenvalues with Multiplicity Two An index formula
In order to investigate dynamic bifurcations of (6.1) from eigenvalues with multiplicity two, it is necessary to discuss the index of the following vector field at x = 0. u =
/anxl + a12xix2 + a 2 2 z \ \bnxl + bi2XiX2+b22X%)'
{
'
'
We assume that the vector field (6.32) is 2nd order nondegenerat e at x = 0, which implies that a^x + b^ ^ 0. Without loss of generality, we assume that on ^ 0. Let A = a\2  4ana 22 , and if A > 0, let au
+ y/E
 a i 2  y/E 2an & = buot? + b12ai + 622,
i = l,2.
The following index theorem will be useful in studying dynamic bifurcation of (6.1) hereafter. Theorem 6.9 Let the vector field (6.32) be 2nd order nondegenerate at i = 0, and a n ^ 0. Then
{
0,
if&<0or
2,
ifauPt > 0 and an/32 < 0,
2,
ifanPi
Pxp2 > 0,
< 0 and anp2 > 0,
(6.33)
166
Bifurcation Theory and Applications
We proceed in several steps as follows.
Proof.
1. When A = a\2 — 4ana 2 2 < 0> t n e following quadratic form is either positively or negatively definite: STEP
anx\ + a12xix2 + a22x\ > 0 (or < 0), V i e K 2 , i ^ 0 , depending on the sign of a n . Hence the following system of equations
f anxj + auxiX2 + a22x\ = £ 2 (or = £2), \ bnx\ + b12xix2 + b22x\ = 0, has no solution for any e ^ 0, which implies that ind(u, 0) = 0, as A < 0. STEP 2. In the case where A > 0, the vector field u given in (6.32) can be rewritten as u =
fan(Xl  aix a )(xi  a 2 a; 2 )\ V 6111? + 612^12:2 + 622^2 )
(6.34)
Since u is 2nd order nondegenerate at x = 0, /3j • /32 ^ 0. By (6.34), u = (0, ±£2)*, with £ ^ 0, is equivalent to Pix\ = ±e2
xx = otix2,
(i = 1,2).
(6.35)
If /?i • /?2 > 0, then one of the systems in (6.35), for either +e 2 or —e2, has no solution, which means that the index of u at x = 0 is zero. STEP 3. When /?i • /?2 < 0, it is easy to see that a i ^ a 2 and A > 0. The vector field u — (ui,u2y given in (6.34) takes the following form:
ui=au(xia1x2)(xia2x2), < u2=
)2 [/?i(si
 a2Z2)2 + /32(xi  aix 2 ) 2
+ "/(xx  axx^ixi  a2x2)}, where 7 = — (2bnaia2 + 6120:1 + 6120:2 + 2622). Let /?i > 0,
/?2 < 0
if a n > 0,
/?i < 0,
/32 > 0
if a n < 0.
(6.36)
167
Dynamic Bifurcation Theory: Infinite Dimensional Case
Then the solutions y = (2/1,2/2) of (6.35) are given by ,
ifan>0, if an < 0,
\otiV2 I a2yf 1/2
)
( ±Pi1/2e
ifa n >0,
t ±/?2" ' e
if an < 0 .
6 37 ( (6.38)  )
Let zi=xi—aix2,
(6.38)
z2 = xi — a2x2.
Then the Jacobian matrix of u is given by
(
dui dui\ /dzi dzi\
~dz~l ~dz~2 I I ~dx[ ~dx~2 \ 9zi 9z 2 / \9xi 9x2 /
It is easy to see that
* fefe <^(!::;)—>»• \9xi 5x2 / Hence we infer from (6.36) and (6.38) that / anZ2 anzi \ detZ?u(x) = (ai  a 2 )det 2(32zi + yz2 20iz2 + jzi . \ (ax  a 2 )
2
(ai  «2) /
On the other hand, by (6.37) we deduce that ' . . f0 z
i = 2/f 
Q
i2/2 = S
(6.39)
2
1/0
I ±(a2a0/321/2£ ± ± f ± (ai  a2)/3r1/2e ± zf = yf  a2yf = 1 [0 Therefore, by (6.39) and (6.40) we arrive at
deti ?tt (^) = {2a"/3l(aiaa)"1(f)a:ba
if an > 0, ifan<0, if an > 0,
(6.40)
if an < 0, ifOll>0
'
(6.41)
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Bifurcation Theory and Applications
By the Brouwer degree theory, we know that ind(u, 0) = deg(u, Br, xo),xo = (0,e2) £ Br
(6.42)
where Br = {x e R2\ \x\ < r}, and r > 0 sufficiently small. It follows from (6.41) and (6.42) that ind(u, 0) =sign det ~Du(y+) + sign det Du(y~) =2, for an/?! > 0 and an/3 2 < 0. We can obtain in the same fashion that ind(u, 0) =  2 , for an/3i < 0 and onft > 0. Thus, the formula (6.33) is proved. The proof of the theorem is complete. • Remark 6.4
If an = 0 and bn ^ 0, we let A = b\2  4&11&12.
If A > 0, we define
51
612  \ M = —26^—'
6ia + \[h &2 =
26 n ' A = ana? + ai2o:i + a22, i = 1,2. Then, the formula (6.33) is written as
{
0
if A < 0 or frfa > 0,
2
if bnPi < 0 and bnJ32 > 0,
 2 if 6u/3i > 0 and bnp2 < 0. Remark 6.5 The index formula (6.33) shows that a two dimensional vector field, which is 2nd order nondegenerate at x = 0, takes only values {0, ±2} as its indices at x = 0. In fact, let u be an mdimensional vector
Dynamic Bifurcation Theory: Infinite Dimensional Case
169
field, which is A;order nondegenerate at x = 0, defined by /
a1
V
«=
xjl
• ••
xjm\
~ji
. . . ™jm
:
E \ji++jm=k
nm a
, X
jijmXl
m
(6.43)
/
then its index at x = 0 is given by
{
0
if m = odd,fc= even,
even
if m = even,fc= even,
odd if k= odd,Vm> 1. Moreover, the index of (6.43) at x = 0 takes values in the following range. 0, ±2, ••• , ± f c m  1 m 1
! ±l,,±k ~
6.3.2
if k = even, m = even, if k = odd,
Main theorems
Under the conditions (6.4) and (6.5), the integers m and r are the algebraic and geometric multiplicities of the eigenvalue /3i(Ao) of LA at A = AO. Here, we assume that m — r — 2, and the operator L\ + G(, A) is secondorder nondegenerate at (u, A) = (0,Ao), i.e. fc = 2 and Gk — G2Let aii(A) = (G 2 (ei(A) iei (A),A) )e *(A))tf, fl22(A)
= (G 2 (e 2 (A),e 2 (A),A),e*(A))H,
a12(A) = (G 2 (e 1 (A),e 2 (A),A)+G 1 (e 2 (A) ) e 1 (A),A),et) i f , 6ii(A) = (G 2 (e 1 (A) ) e 1 (A) ) A),e^(A)) ff , 622(A) = (G 2 (e 2 (A),e 2 (A),A),e^(A)) i/ , 612(A) = (G 2 ( e i (A), e2(A), A) + G 1 (e 2 (A), ei(A), A), e*2(X))H,
where G 2 is given by (6.21), and ej(A),e^(A) (i,j = 1,2) are respectively the eigenvectors of L\ and L*x near Ao: Lxei{\)
= A(A)e,(A), LAe*(A) = /?,(A)e*(A), i,j = 1,2.
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Bifurcation Theory and Applications
Thus, we obtain a vector field
„ m _ fanWxl
+ a12(X)x1x2 + a22(A)a^\
(6.44)
By assumption, u0 is second order nondegenerate at x = 0 near Ao. According to Theorem 4.4, under conditions (6.4) and (6.5) with m = r = 2, if the two vectors (an, ai 2 , o22) and (bu, 6j2,622) are linearly independent near Ao, then there are at most three and at leat one bifurcated singular points of (6.1) on each side of A = Ao. By Theorem 6.9, the index of UQ given by (6.44) at x = 0 is either 0, or 2 or —2. Now we state the main dynamic bifurcations of (6.1) in each situation. We start with the case where ind(uo(Ao),O) = —2. Theorem 6.10 Let the conditions (6.4) and (6.5) with m — r = fc = 2 hold true, L\ + G(,X) be second order nondegenerate at (u, A) = (O,Ao), and ind(uo(X0),0) = —2 for uo defined in (6.44) Then (6.1) bifurcates exactly 3 saddle points with Morse index 1 from (0, Ao) on each side of A = A0. For other two cases, we need to introduce a notation. A set S{6) c R2 is called a sectorial region with angle 6 G [0, 2TT] , if S(9) is enclosed by two curves 71,72 starting with x = 0 and an arc F, and the angle between the two tangent lines L\ and I/2 of 71 and 72 at x = 0 is 0; see Figure 6.4. Let Sr(6) be the sectorial domain with angle 6 and radius r > 0 given by Sr(9) = {x€R2\
\x\
x=0
Fig. 6.4
and x G S{9)}.
171
Dynamic Bifurcation Theory: Infinite Dimensional Case
Theorem 6.11 Assume (6.4) and (6.5) with m = r = 2, and ft(A) = ft(A) near Ao. Let L\ + G(, A) be 2nd order nondegenerate at (0, Ao), and UQ(\) be given by (6.44) Then the following assertions hold true. (1) If ind(uo(Xo),0) = 2, then (6.1) bifurcates an attractor Ax with &imA\ < 1 from (0,Ao) on Ao < A, and A\ attracts a sectorial region Dr{6) in H with angle 9 € (TT, 2TT], and radius r > 0, where Dr{6) = {u = x + v£H\x
= xiei + x2e2 e Sr(9), \\u\\H < r}.
(2) The attractor A\ contains minimal attractors, which are singular points, as shown in Figure 6.5 (a)  (c). (3) If mrf(uo(Ao),O) = 0 and (6.1) bifurcates from (0, Ao) three singular points on Ao < A, then one of them is an attractor, which attracts a sectorial region Dr(9) with 0 < 9 < n, as shown in Figure 6.6 (a) and (b). Remark 6.6 If ft (A) ^ ft. (A) near A = Ao and A ^ Ao, then Theorem 6.11 may not be valid. Consider, for instance, the following equation dx — =Axx + G(x),
(6.45)
where x = {xi,x2)t, ai = 2 + V^,a2 = 2  \/3, and Ax
~ { 0 a2\) '
Q(X\ _ fanxl + ai2xix2 + a22x\\ \bnx\ + b12x1x2 + b22xl)' Assume that G is second order nondegenerate at x = 0 and ind(G, 0) = 2. Let P : M.2 —> K2 be a linear transformation y = P x =
(6.46)
(PII pv) VP21P22J
such that
PG(PIy) = G{y) = M ~ 4 y i 2 / 2 ~ y") . \
2/12/2
)
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Bifurcation Theory and Applications
Here we can choose the coefficients to make G as given here. Then, (6.45) is transformed into
{
£• = 4Ay! + Xy2 + y\
f
4yiy2  y\,
(6.47)
dy , ir 2 = Xy1+y1y2. It is easy to see that y = (0, A) is a unique bifurcated singular point of (6.47) from (0,A0) = (0,0), and (6.47) has no attractor on Ao < A near (y, A) = (0, Ao), which has the topological structure as shown in Figure 6.7.
6.3.3
Proof of main theorems
First order approximation By the center manifold theorem, the dynamic bifurcation of (6.1) is equivalently reduced to that of the following equations
f  ^ = )8i(A)a:i + (G{xiei + x2e2 + h(x, A),e?(A))H,
I f
(6.48)
[  ^ = fo{\)x2 + (G(xiei + x2e2 + h(x, A), e*2(X))H, where h(x, A) is the center manifold function satisfying h(x,X) = o(\x\) for x e R2. Thus, near (a;, A) = (O,Ao), (6.48) can be written as ^ = JxX + F(x,X)+o(\x\2), at
(6.49)
where
JxX
{
/A (A)
0
0 /MA)JW~U(W'
PC \\ _ {anM^i
*
[X A)
\fx1\_((31(X)x1\
' ~\bu(X)xl
+ ai2(A)a;ia;2 + a22(X)x%\
+ bu(X)Xlx2
+ 6 22 (A)^ ) '
and atj,bij are as in (6.44). Since F is 2nd order nondegenerate at (x,X) = (0, Ao), the vector field on the right hand side of (6.49) is a perturbation of J\ + F near x = 0. Hence, by the stability of nondegenerate singular points and attractors (Theorem 5.4), it suffices to prove Theorems 6.10 and 6.11 for the following
Dynamic Bifurcation Theory: Infinite Dimensional Case
173
(a)
\PI
* (]<:
a
•
(b)
(c)
Fig. 6.5 (a) If (6.1) bifurcates one singularity p, the attractor Ax = {p}; (b) If (6.1) bifurcates two singularities pi and p2, then A\ = y U {pi,P2}, where 7 is the orbit connecting pi and p2\ and (c) If (6.1) bifurcates three singularities po, pi and p3, then .AA = 71 U 72 U {PI,P2IP3}, where 7i are the orbits connecting po and pi.
system, which is the first order approximation of (6.48) dx — = Jxx + F(x,\).
(6.50)
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Bifurcation Theory and Applications
(a) Fig. 6.6
(b) (a) A = Ao, (b) Ao < A with {p} being an attractor.
Fig. 6.7
Proof of Theorem 6.10
The proof can be achieved by Theorem 3.10 and the following lemma. Lemma 6.1 Ifind(F{,\0),0) =  2 , then (6.50) bifurcates from (x,X) = (0, Ao) exactly three saddle points with Morse index one on each side of A = A0. Proof. By Theorem 6.9, as ind(.F, 0) = —2, the two vectors (an, ai2,022) and (6n, &12,622) are linearly independent. Therefore it follows from Theorem 4.4 that (6.50) has at most three bifurcated singular points from (0, Ao). We shall prove that (6.50) has just three bifurcated singular points on each side of A = AQ.
Dynamic Bifurcation Theory: Infinite Dimensional Case
175
It is known that ind (Jx +F,0)=
sign [ft(A) • p2(\)} = 1, if A ± Ao,
fc
Y,ind (JA + F, Pi ) + ind (JA + F,0) = ind (F(,Ao),0) = 2, where p» (1 < i < k) are the bifurcated singular points of (6.50) from (0, Ao). Hence, if A ^ Ao, then k
53ind(JA+^Pi) =  3 .
(6.51)
If the number k < 3 in (6.51), then one of the bifurcated singular points, say pi, of (6.50) satisfies that ind(J A + J P,p 1 )>2.
(6.52)
By the Brouwer degree theory, if D(Jx + F)(Pl)^0, we have ind(J A +F,pi) < 1 . Therefore, it follows from (6.52) that the Jacobian matrix of J\ + F at p\ is zero: D{ Jx + F)(Pl) = JX+ ( ^ ^ ] V dxi
= 0.
(6.53)
)
Let pi — (21,^2)1 then we infer from (6.53) that ft + 2 a n z i +C112Z2 = 0 , <
al2zi + 2a22z2 = 0, f32 + 2&22Z2 + &1221 = 0,
(6.54)
bi2z2 + 2bnzi = 0, which, together with J>pi +i ? (p i , A) = 0 , imply that px = 0, a contradiction to pi 7^ 0. Thus, we have shown that k = 3. Prom (6.51) and Theorem 4.4 we have md(Jx + F,Pi) = l, 1 = 1,2,3,
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Bifurcation Theory and Applications
which implies that pt (1 < i < 3) are saddle points with Morse index one. This proof is complete. • Poincare formula In order to prove Theorem 6.11, we need the following lemma, which establishes the Poincare formula; see [Chow and Hale, 1982]. L e m m a 6.2 Let v be a two dimensional Cr(r > 0) vector field with v(0) = 0. Then
(6.55)
ind(v,0) = l + Ueh),
where e is the number of elliptic regions, and h number of hyperbolic regions. Here the elliptic, hyperbolic and parabolic regions E, H and P in a neighborhood U C M.2 of x = 0 are defined as follows; see Figure 6.8: E = {xeU\
S(t)x
and S(t)x
H = {x e U\ S(t)x, S(t)x
+ 0 as t » oo},
$. U for some
P = {x £ U\ either S(t)x > Q(t > oo), S(t)x or S(~t)x
> 0, S{t)x <£ U, or S(t)x, S(t)x
t>to>O}, $ U(t > t0), € U,Vt > 0}.
Fig. 6.8
Proof of Theorem 6.11 The proof is achieved in a few lemmas hereafter. Here, we always assume that (3{X) = /3i(A) = /?2(A) for A near Ao. First, by the homotopy invariance of indices, for A near Ao, md(F(; A), 0) = ind(F(; Ao), 0).
(6.56)
Dynamic Bifurcation Theory: Infinite Dimensional Case
177
Lemma 6.3 Let incLF(, A),0) = 0 or 2. Then for X near Ao, the vector fields F(x,X) have k straight orbit lines with 1 < k < 3: aiXi+piX2=0,
a?+j3?^0,
i = l, ) fc,
(6.57)
where Oi = (Xi/[3i orOi = —fii/cti are the solutions of the following algebraic equation:
{
a22cr3 + (ai2  b22)a2 + {an  h2)a  hi = 0, or
(6.58)
hicr3 + (&12  an)a2 + (b22  o,i2)cr  a2i = 0. Proof. When F(x,X) are secondorder nondegenerate at x = 0 near Ao, °ii + ^li 7^ 0. We assume that an ^ 0. By the homogeneity of F(x, A), a straight line Z2 = ox\ is an orbit line of F(x, A) if and only if FaQc.A) Fi^.A) _ 6na;f + 612^12:2 + ^22^2 anx^ + ai2a;ia;2 + 022^2 _ ^li + bi2
Ifind(F(, A0),0) = 2, then we have
(1) F(x, A) has no hyperbolic regions at x = 0, (2) F(x, A) has exactly two elliptic regions E\ and E2, (3) F(x,X) has no parabolic regions if k = 1, which is the number of solutions of (6.58), and has exactly two parabolic regions Pi and P2, if k>2, (4) the elliptic and parabolic regions E and P are sectorial regions E = S{0i),P = S(92) with 0 < 6i,62 < n,91+e2 = ir, and the edges of Sr(9i) and Sr(62) are the straight orbit lines of F(x,X); see Figure 6.9(a)(c). Proof. Based on Lemma 6.3, we take an orthogonal coordinate transformation y = Ax with a straight orbit line of F(x, A) as the yiaxis. Under
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Bifurcation Theory and Applications
this transformation, the vector field F(x, A) is changed into the following form F(y,X)=(~a^^y)yl
+
(6.59)
^ )
Since ind(.F(, A),0) = 2, b\ j^= 0. Take another coordinate transformation as follows x\ = £>i?/i +£>22/2, x'2 = 2/2Then, by Theorem 6.9 the vector field in (6.59) is transformed into the following form, where for brevity, we omit the primes: F(x, A) = (%) = (a^ ~ «if a)(*i + « 2 x 2 ) \
V
\F2J
bxxx2
)
(6.60)
where a • b > 0, a\, a2 > 0. It is known that a coordinate system transformation preserves the topological structure of F. It is easy to see that (6.60) has the topological structure as shown in Figure 6.9(a)  (c) for o, b > 0 in (6.60). To derive the topological structure in Figure 6.9(a)(c) of (6.60). Let Di, £>2, D3 and D\ be the 4 open quadrants in R2, and the two straight lines x\—ot\X2 = 0,x2 + a^xi = 0 also divide the plane R2 into four regions G R2 I xi  a.\xi > 0,xi +a2x2
Q\ = {(xi,x2)
2
> 0}, > 0},
2
Q3 = {(xi,x2)
G R I xi  axx2 > 0, xi + a2x2 < 0},
Qi = {(xi,x2)
G R2 I xi  axx2 < 0, xi + a2x2 < 0}.
It is easy to see that ' Fi>0
in Qi and Q4,
Fx<0
in Q2 and Q3,
<
F2 > 0 in Dx and D 4 ,
(o.bl)
Fi < 0 in D2 and D 3 . The properties (6.61) ensure that (6.60) has only two elliptic regions E\ and E2, with £1 C R2+ = {(11,12)1x2 > 0} and E2 c R i = {(xi,x2)a;2 < 0};
see Figure 6.10.
Dynamic Bifurcation Theory: Infinite Dimensional Case
X
*2
(a)
179
2
(b)
(c)
Fig. 6.9 Toplogical structure of (6.60): (a) The number of straight orbit lines k = 1, (b) fc = 2, and (c) k = 3.
Thus, by Lemma 6.2, i*1 has no hyperbolic regions, and Assertions (1) and (2) are proved. By (6.58), it follows from (6.60) that the straight orbit lines Li(i = 0,1,2) of F(x, A) are given by LQ : X2 = 0,
L\\
X2 = crixi,
L2 : x2 = (T2X1,
and if o\,
£1,
c*i
(72 =
1
H£2,
a2
for some real numbers 0 < E\ < 1/ai and 0 < e2 < l/a2. Hence we have LiCQiUQ3
(i = 0,12).
(6.62)
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Bifurcation Theory and Applications
. ,  "
• • •  , . .
"
xra2x2
Fig. 6.10
Therefore, Assertions (3) and (4) follow from (6.62) and the radial symmetry of F(x, A). The proof is complete. • Lemma 6.5 / / ind(F, 0) = 2, then (6.50) bifurcates from (x, A) = (0, Ao) an attractor A\ on XQ < A, which attracts a sectorial region Sr(9) with n < 6 < 2TT. Actually, Sr(0) C (Ex U E2 U Pi) n Br, where E\,E2 are the elliptic regions of F, P\ is the parabolic region where all orbits of F reach x = 0, Br = {x S R 2 :r < r}, 6 = 2TT — 6Q, and 9Q the angle of the parabolic region. Proof. We know that under an orthogonal coordinate system transformation, the linear operator
r _ (PM
0 \
is invariant. Therefore, without loss of generality, we take the vector F as given by (6.59). By Theorem 6.9, F(x, A) can be written as F
\F2)
=
{
bx2(Xlax2)
)'
^ 6  63 )
where a • b > 0, a\ + a  ^ 0. We proceed with the case where a, b > 0 and a > 0. The other case can be proved in the same fashion. By Theorem 6.9 we know that cci > a > a2, which implies that thelineszi— o.iX2 = 0(i = 1,2),X2 = 0, andxi— ax2 = 0 are alternatively positioned in M2.
Dynamic Bifurcation Theory: Infinite Dimensional Case
181
Based on the definition of elliptic and parabolic regions, by Lemma 6.4 we obtain that lim Sx(t)x = 0, V x e £ i U £ 2 U Pu
t—>oo
(6.64)
where S\(t) is the operator semigroup generated by F(x,X). On the other hand, we obtain from (6.63) that for any x € E\ U E2 UPi, there is a to(x) > 0 such that Sx(t)x G D = {x G R2 I X! 
(6.65)
It is clear that Pi C D c Ei U E2 U D. Let L>(r) = {x £ D I a; < r}, £>(ri,r 2 ) = {s G D I 0 < n < x < r 2 } . Let T\(£) be the operator semigroup generated by J\ + F(, A). It is known that for A > Ao all orbits of J\x are straight lines emitting outward from x = 0. Therefore, by (6.65) we deduce that Tx(t)x GD,
V t > 0,
x€dD,xj^0.
(6.66)
Now, we shall prove that for any A — Ao > 0 sufficiently small there are ri,r2,r3 > 0 with ri < r 2 < r 3 such that TA(*)x G 0 ( r i , r 2 ) , V i e D(r3),t > tx,
(6.67)
for some tx >0. We know that for A > Ao, the singular point x = 0 of Jx + F has an unstable manifold Mu with dimM" = 2. We take r% > 0 such that the ball Bri C Mu. Then, by (6.66) we obtain that Tx(t)x e D(ri,r2), V xeD(n),t>tx.
(6.68)
If (6.67) is not valid, then by (6.66) and (6.68) there exist An > Ao + 0,t n —> 00 and {a:n} C D(r 3 ) such that I T An («„)!„  > r 2 , V n > l .
(6.69)
Let xn —» xo G D(^3) Then by th4e stability of extended orbits (Lemma 5.1) and (6.69) there is an exgtended orbit 7 of F(, Ao) with starting point XQ € D(r3) which does not reach to x — 0. This is a contradiction to (6.64).
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Bifurcation Theory and Applications
It follows from (6.66) and (6.67) that D{ri,r2) is an absorbing set in a neighborhood U of D(ri, r 2 ). Hence, by the existence theorem of attractors (Lemma 2.2), for A > Ao, the set A\=w(D(n,r2),\), with 0 ^ Ax, is an attractor of (6.50), which attracts D(r3). Applying Lemma 5.1 again we infer from (6.64) that lim max x = 0.
A—>A0
z£Ax
Thus A\ is a bifurcated attractor of (6.50) from (0, Ao). We can deduce from (6.65) that A\ attracts a sectorial region Sr(6) c E1UE2UP1, with 6 = 2TT — #0, where #0 is the angle of the parabolic region P%. The proof is complete. • Lemma 6.6 The attract A\ has dimension dim A\ < 1, and Ax contains minimal attractors consisting of singular points. Proof. It is clear that .4^ contains all singular points of (6.50). We shall prove that Ax does not contain extended orbits homeomorphic to S1. By Lemma 6.3, all singular points of (6.50) must be in the straight orbit lines L of F(x, A), and L are invariant sets of (6.50) which consist of orbits and singular points. Use the method as in the proof of Lemma 6.4, for any straight orbit line L we can take an orthogonal coordinate system transformation with L as its x\—axis. Thus, the vector field F(x, A) take the form of (6.63), and the singular point xo = (x®, x^) of Jx + F on L is given by xo = {x°1,xo2) = (l3(\)/a,O), and the Jacobian matrix of Jx + F at XQ is given by
D +F
^ ^=(~T\ihm)
<67°)
Hence, on each straight orbit line there is only one singular point XQ of Jx + F, and there are two orbits 71 and 72 in L reaching to XQ. Moreover one of 71 and 72 connects from x = 0 to x$. It follows that the attractor Ax containing all singular points has no closed extended orbits. By the PoincareBendixon theorem we obtain that dim.4,\ < 1.
Dynamic Bifurcation Theory: Infinite Dimensional Case
183
When J\+ F has three singular points zt (1 < i < 3), by Theorem 4.4 they are regular, and ind (Jx + F, zi) =  1 , ind (JA + F, z2) = ind (Jx + F, z3) = 1.
(6.71)
It follows from (6.70) and (6.71) that z\ is a saddle point, z2 and z3 are attractors. In this case the attractor Ax has the structure as shown in Figure 6.5(c). When J\ + F has two singular points z\ and Z2, md(Jx + F,Zl)=0,
md(Jx + F,z2) = l,
which implies, by Theorem 4.4 and (6.70), that z2 is an attractor, and z\ has exactly two hyperbolic regions. Thus, Ax has the topological structure as shown in Figure 6.5(b). When Jx + F has only one singular point z, then ind(JA + F,z) = l, which implies by (6.70) that z is an attractor, and Ax = {z} has the topological structure as shown in Figure 6.5(a). The proof is complete. • Note that Lemmas 6.46.6 are still valid for (6.49), Assertions (1) and (2) in Theorem 6.11 follows from Lemmas 6.5 and 6.6. Assertion (3) of Theorem 6.11 is an immediately consequence of the following lemma. Lemma 6.7 Ifind(F,Q) = 0, and (6.50) bifurcates three singular points from (0, Ao) on \Q < A, then one of them is an attractor which attracts a sectorial region Dr(6) with 0 < 6 < IT. Proof. By Theorem 4.4 the three bifurcated singular points pi (1 < i < 3) are nondegenerate, and ind (Jx+F,Pl)
= ind (Jx+F,p2)
=  1 , ind (JA +F,p3) = 1.
Then as in the proof of Lemma 6.6, we can deduce that p3 is an attractor. Since pi and p2 are in the other two straight orbit lines which enclose the parabolic region P, the singular point p3 € P and attracts a domain P n Br = Dr(0) for some r > 0. The proof of the lemma is complete. •
184
Bifurcation Theory and Applications
6.3.4
Case where
k > 3
Now we consider the case where the operator L\ + G(,X) is kthorder nondegenerate at (0, Ao) with k > 3. Let
"o=(U>^fj^;;^\
(6.72)
where a
hJk(X)=
,ejlc,X),e*(X) >H,
» = 1,2,
and Gk : # i x • • • x H\ —» H is thefc—multilinearoperator defined in (6.21). The following theorem is the generalization of Theorem 6.11 to the case where k > 3. Theorem 6.12 Assume (6.4) and (6.5) with m = r = 2, and /?i(A) = /?2(A) near Xo. Let L\ + G(, X) be kth order nondegenerate at (0, Ao), and uo(X) be given by (6.72). If ind(uo(Xo),0) > 1, then (6.1) bifurcates an attractor A\ with dim.4.\ < 1 from (O,Ao) on Ao < A, and A\ attracts a sectorial region Dr(6) in H with angle 9 £ (0,2TT], and radius r > 0. The proof of Theorem 6.12 is based on the fact that if ind(uo(Ao), 0) > 2, by the Poincare formula (Lemma 6.2), there is at least one pair of elliptic regions E\, £7 and a parabolic region P, which may be empty, of «o(A) at x = 0, as shown in Figure 6.11, such that lim S\{t)x = 0,
t—>oo
Vz G Ex U E2 U P,
where S\(t) is the operator semigroup generated by ito(A). Then Theorem 6.12 can be proved in the same fashion as the proofs of Lemmas 6.5 and 6.6; we omit the details. 6.3.5
Bifurcation to periodic solutions
In this subsection, we consider the case where m = 2, r = 1 and k = odd > 3 in (6.4) and (6.21). We also assume that ^(Ao)^0,
V j > m + 1.
(6.73)
Since m = 2 and r = 1, the two eigenvectors vx and u2 of L\ at A = Ao
Dynamic Bifurcation Theory: Infinite Dimensional Case
p
e
185
^ \ \
Fig. 6.11
enjoy the following properties; see Section 3.3.2: LXovi=0, L*Xov2=0,
Lx0v2=av!, LXov{=av*2,
J>0,
i=j,
[ = 0,
i+j.
a + 0,
Let a £ R be the number denned by a = (Gk{vuXo),vZ)H,
(6.74)
where Gk is as in (6.21). Then, we have the following bifurcation theorem of periodic orbits from the real eigenvalues with m = 2 and r = 1. Theorem 6.13 Assume the conditions (6.6) and (6.73) with m = 2,r = 1 and k = odd > 3. Let a be given by (6.74). If a • a < 0, then (6.1) bifurcates from (u, \0) = (0, Ao) a periodic orbit.
Proof. STEP 1. By the center manifold theorem, it suffices to consider the bifurcation of the following equations f ^ \
dx
{~i=
= /31(X)x1 + ax2 + (G(x + h(x, A), A),v\(X))H,
/32(X)x2 + (G(x + h(x,X),\),vl(X))H,
<675)
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Bifurcation Theory and Applications
where x = xiu^A) + x2v2(\), h(x, A) is the center manifold function, Lxv2(X) = f32{\)v2{\) + at;!(A), L*xv*2(\) = (32(\)v*2(\), L*xvl{\)=(31(\)vl+av*2{\), v*(\))H=6ij, {Vi(X),
a / 0,
where Sij is the Kronecker symbol. By (6.21) and (6.74), equation (6.75) at A = Ao reads as
fW(™).
(^6)
where F1(x)=ax2+0(\x1\k,\x2\k), F2{x) = a4 + 0(\Xl\k+1, \x2\k, o: 2  fc Vi, • • • , N • l^l" 1 ). Since a • a < 0 and k = odd > 3, we have ind(F,0) = l.
(6.77)
STEP 2. We now prove that the number of elliptic regions of F at x — 0 is zero, i.e. e = 0. Assume otherwise, then there exist an orbit 7 of (6.76) connected to x = 0, i.e.
lim S(t)x = 0, Vz G 7,
t—*oo
where S'(i) is the operator semigroup generated by (6.76). Let 7 can be expressed near x = 0 as ^2 = /(zi), (Zi,^) £7
It follows from (6.76) that for any ( i i , ^ ) S 7 ax2 _ axk + Odxtl^1, l n H / ^ ) ! *  1 , \xl\2\f{x1)\k'i, • • • , ^ l ^ 1 ! / ] ) dxi o/CiO + OdnlM^xi)!*) Thus we obtain af{xl)f'{x1) + 0(\Xl\k, / f c )/' = mf + 0(xi*+1, I x i l ^ i / H ,
(678)
187
Dynamic Bifurcation Theory: Infinite Dimensional Case
which implies that f{x) = f3xm + o{\x\m),
2 < m = ^~
(6.79)
Therefore, from (6.78) and (6.79) we get a = am/32, where a • a < 0, m > 2 and /? ^ 0. It is a contradiction. Hence e = 0. STEP 3. By (6.77) and the Poincare formula (6.55), h = 0. Therefore i = 0 must be a degenerate singular point, and is either (a) a stable focus, or (b) an unstable focus or (c) a singular point having infinite periodic orbits in its neighborhood. The case (c) implies a bifurcation to periodic orbits for (6.75). For the case (a), x = 0 is an asymptotically stable singular point of (6.76). Then by Theorem 5.2, the equation (6.75) bifurcates from (x, A) = (0, Ao) an i^attractor SA on A > AoFor the case (b), x = 0 is an asymptotically stable singular point of the vector field —F(x), therefore the vector field
_(ft(A)
0
\x_F(x)
bifurcates from (0, Ao) an 5>1attractor EA on A > Ao, which implies that (6.75) bifurcates from (0,Ao) on A < Ao an S1 repelor EA, which is an invariant set. STEP 4. Now, we need to prove that the S^invariant set EA contains no singular points. Consider the following equations /?i(A)x 1 +aa: 2 +0(a:iM a :2*)=0,
(31(\)x2+axk1+O(\x1\k+1,\x2\ki\x1\i)=O. Hence
X2 = aaxi'1
/3i(\)a1x1+0(\x1\k,\p1\k),  ft(A)/%(A) + 0(\Xl\k, *ifc/?i) = 0.
(6.80)
By aa < 0 and /?i(A)/?2(A) > 0, (6.80) has no solution near (x, A) = (0, Ao). Hence, there is no singular points in SA, which means EA must contain a periodic orbit. The proof is complete. •
188
6.4
Bifurcation Theory and Applications
Stability for Perturbed Systems
In this section, we shall present two theorems on the bifurcated attractor s of systems with perturbation , which will be used in studying the bifurcation phenomena for the Taylor problem; see Chapter 10. One theorem is on the general case and the other on the case with simple eigenvalues.
6.4.1
General case
Consider the perturbe d equation of (6.1) given by 
= (L>+Si)v + G(v,\),
(6.81)
where L\ and G ar e as in (6.1), and Sf^ € Ba is a perturbatio n operato r of L\, depending continuously on A € R, such that (6.82)
\\Sio\\B,<e.
Here Ba = C(Ha, H) is the space of all linear bounded operator s from Ha to H, i.e. Ba = £(Ha, H) = {B : Ea  H linear bounded}. We always assume that 0 < a < 1. By the spectra l theory of the linear complete continuous fields [Kato, 1995], under conditions (6.5) and (6.6) as s > 0 sufficiently small, there exists a paramete r AQ approximating to Ao such that the eigenvalues {/S^(A)} of L\ + S^ satisfy
{
< 0
for A < A£ (6.82)
= 0
for A = A^
>0
for A > Ag
ifl
i f j > m i + l, >Rej3?(A§)<0 where 1 < mi < m, and m is as in (6.4). The main theorem in this section is the following attracto r stability theorem for the perturbe d equation (6.81). Theorem 6.14 Let the conditions in Theorem 6.1 hold true. Then there aree > 0 and 5 > 0, such that as 5  e Ba satisfies (6.82) and 0 < A — AQ < 5, the following assertions hold true.
Dynamic Bifurcation Theory: Infinite Dimensional Case
189
(1) There exists a neighborhood U C H of u = 0 such that the equation (6,81) has an attractor S  C U such that dimS^<m,
0 i S,
and Ti\ attracts an open and dense set UQ of U. (2) Each element u\ e E^ can be expressed as v\ € Eo, < lim vx = 0,
(6.84)
A>A 0
lim K/K=0, A—*AQ £—.0
where Eo is as in Theorem 6.1. (3) If u = 0 is globally asymptotically stable for the unperturbed equation (6.1), and (6.81) has a global attractor for any A € M., then S  attracts any bounded open set of H. Proof. We shall apply Theorem 5.4 to prove this theorem, and remark that Theorem 5.4 is still valid if the domain of each vector field depends on the field. We know that the eigenvalues {/3f.(\)} of L\+Sl depend continuously on the operator 5  ; see Ch. IV  3.5 of [Kato, 1995]. Therefore, by condition (6.5) there are numbers £ > 0 and 6 > 0, as S^ e Ba satisfies (6.82) and  A — Ao < S, the eigenvalues of L\ + S^ satisfy RePj(\) <  7 ,
V j > m + 1 for some 7 > 0.
(6.85)
Let L\ = Lx + S{. By (6.4), (6.5) and (6.85), the spaces Hx and H can be decomposed respectively into
JJi = ££©££, and Hi = [email protected]^, H = Et®E$£,
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Bifurcation Theory and Applications
such that dim Ei = dimi?*£ = m, and for i = 1,2,
£$ = L\Er.E?+E*, The operators £^ and £Ae have eigenvalues {/?i(A), ,/?m+i(A)} and {/3f (A), • • • ,/?m+i(A)}, while the operators £A and £A£ have eigenvalues {#j(A)  j > m + 2} and {/3?(A)  j > m + 2} respectively. Thus, by the center manifold theorems in Section 2.2.4, there are C 1 functions h : O —» ^(fl), ^ : OE — JB2A£(^), with O c E± and C7£ C ^ ^ being neighborhoods of a; = 0, such that the attractors of (6.1) and (6.81) in the locally invariant manifolds
Mx = {(x,h(x))\xeOcE?}, £ M x = {(x,h£(x))\xeOecE?e}, are homeomorphic to the attractors of the following ordinary differential equations respectively dx — = £$x + PG{x + h(x), A),
XGOC
£j\
(6.86)
OJXJ
^
= C\ex£ + PeG(x + he(x), A),
x £ e O £ C Et,
(6.87)
where P : Hi —» E± and P£ : H\ —> ^ £ are the canonical projections. It is known that
•hm ££ = £*, , lim£^=^,
(6.88)
lim h£ — h.
In addition, by Theorem 6.1, (6.1) bifurcates from (u, A) = (0,Ao) an attractor EA C MA for A > Ao. Hence it is easy to see that as Ao — 5 < A < Ao the system (6.86) has a unique trivial attractor x = 0 in O, and as Ao < A < Ao + S, (6.86) bifurcates from (x, A) = (0, Ao) an attractor, which is denoted by SA C O, and SA attracts O \ {0}. By the attractor stability theorem, Theorem 5.4, we infer from (6.88) that as £ > 0 is sufficiently small, (6.87) has an attractor EA in Oe for all
Dynamic Bifurcation Theory: Infinite Dimensional Case
191
 A — Ao  < S, such that E^ —» Ex
a s e  t 0.
Obviously d i m £  < m. By (6.83) we deduce that 0 £ S  for all A > Ao. Thus, Assertion (1) is verified. By Assertion (3) of Theorem 6.1, Assertion (2) follows from (6.88) and (6.89). Finally, Assertion (3) follows D from Theorem 5.4. The proof is complete. Remark 6.7 In the above proof, the center mainfold theorem is used by considering perturbation of (6.1) with 5  ; see [Temam, 1997] and [Henry, 1981]. 6.4.2
Perturbation at simple eigenvalues
We now consider the case where m = 1 in (6.4), i.e. the first eigenvalue /?i(A) of L\ is simple. Let VQ £ Hi be the eigenvector of L\ at A = Ao: LXov0 = 0,
VD
= 1.
We assume that L\ is symmetric, m = 1 in (6.4) and (6.5), G(u) is bilinear, (G(u,v),v)=0, \/U,VGHO.
(6.89)
Here as G is bilinear, one can write G as G(, •), which is linear for each argument. We remark here that the condition (6.89) is necessary for the uniqueness of a pair of equilibrium points in the attractor S^ in the floowing theorem; otherwise, E  may contain more than two points. Theorem 6.15 Let the conditions in Theorem 6.14 and the conditions (6.89) hold true. Then we have the following assertions. (1) The attractor Yj\ of (6.81) obtained in Theorem 6.14 consists of exactly two equilibrium points of (6.81), i.e. T,\ = {w^u^} such that
{
u^ = ai{\,e)vQ+wi{\,e), "2 = a2(A,£)uo + w2(X,e), Wi{\,e) = o(ai(A,e)) G Hi,Oi € E, i = 1,2,
(6.90)
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Bifurcation Theory and Applications
where at > 0, and a(X, e) —> 0 as e —» 0, A —> Ao. (2) There is a neighborhood U C H of u — 0, and U can be decomposed into two open sets U± and U^:
u = u\ + ux2,
t/A n t/2A = 0, o £ du? ndu£
with uA £Uf (i = 1,2), such that
lim t>(£,v?)uA = 0 ,
as(peU?
(i = 1,2),
t—»oo
where v(t,(p) is the solution of (6.81) with v(0,ip) = tp. Furthermore, if (6.81) has a global attractor for each X eR1, then U can be taken as any bounded open set of H. Proof. It suffices to prove that there are exactly two nonzero singular points of the operator U\ + G in H\ near u = 0 as e > 0 and A — Ao > 0 sufficiently small. We shall proceed by using the LyapunovSchmidt method. The following equation L\u + G(u, A) = Lxu + Sexu + G(u, A) = 0
(6.91)
can be reduced to the following onedimensional equation f3f(\)x + P?G(xv£0 + y£(x, A), A) = 0,
x e R,
which can be rewritten as (3£1(\)x
+ P 1 G ( x v 0 + y ( x , \ ) , \ ) + g ( x , e ) = 0,
x £ l
(6.92)
where
jg(x,e)=o(\x\),
(6.93)
and y(x, A) e E% is the solution of the equation £$y + P2G(xv0+y,\) = 0.
(6.94)
Since the linear operator L\ is symmetric, the eigenvalues {/?fc(A)  k = 1,2,... } of L\ are real and complete, and the eigenvectors {vk} (v\ = VQ)
193
Dynamic Bifurcation Theory: Infinite Dimensional Case
constitute an orthogonal basis of H. Then we have y = ^2fL2yiv3> an( ^ (6.94) becomes oo
0j(X)yj
+ {G(xvo
+ J2ykvk,X),vj)H
= O,
(6.95)
j = 2,3,.
fc=2
By (6.89) we find OO
OO
OO
2
(G(xv0 + ^2 VkVk, X),VJ)H = <*jX + ^2ckxyk fc=2
fc=2
+^
(6.96)
cikykyk
i,k=2
where a^, c^, c^ are constants, and atj = { G ( V O , \ ) , V J ) H ,
j = 2,3,.
It follows from (6.5), (6.95) and (6.96) that yJ = \l3j\1aja?+o{xt),
j = 2,3,.
(6.97)
On the other hand, we see that OO
PiG(xvo + y,X) {G(xvQ +
^2ykvk),v0)H fc=2
=(by (6.89)) OO
=
^2aixyj 3=2
OO
+ Yl c^ykyk
(6.98)
i,k=2
By (6.97) and (6.98), (6.92) reads P\{\)x  ax3 + g(x,e) + o{\x\3) = 0,
(6.99)
where
a = f> J r 1 a J 2 >0, 3=2
as G(VQ, X) ^ 0. Because G is analytic, so is g(x,e). By (6.93) we find g(x, e) = b(e)x2 + o{x2), where b(e} —> 0 as e —> 0.
(6.100)
194
Bifurcation Theory and Applications
Then it follows from (6.83), (6.99) and (6.100) that there are only two nonzero solutions of (6.99) near x = 0 as A > Ao and e > 0 sufficiently small; they are given by
K.)^>l.) +W W ,
'
(6.101)
2(7
Thus, the proof of the theorem is complete.
•
Remark 6.8 The coefficients at{\, e) (i = 1,2) in (6.90) are the numbers given by (6.101); namely <*1 =
^ ACT
~,
"1 =
^
lf b
^
= KS) > 0>
2,(7
ai =
^ , a2 = \ if b = b(e) < 0. la la It is clear that if b ^ 0, e.g. b > 0, then the attractor «£ = ~~a2Vo + W2 is bifurcated from (0,A0), and the other one, u\ — OL\VQ +W\ is not, i.e. lim u\ =  wo + o   )
T^
0.
In fact, the number b(e) in (6.101) can be expressed as &(e)=
6.5
Notes Theorem 6.1 is first proved in [Ma and Wang, 2004e], and Theorems 6.2 and 6.3 are new here. Sections 6.2 and 6.3 are based on [Ma and Wang, 2004a; Ma and Wang, 2004b], and Section 6.4 is taken from [Ma and Wang, 2005d].
Dynamic Bifurcation Theory: Infinite Dimensional Case
:'
Fig. 6.12 cation.
^ ^
195
\
When 6(e) ^ 0, the attractors (6.90) are originated from a saddlenode bifur
Chapter 7
Bifurcations for Nonlinear Elliptic Equations 7.1 7.1.1
Preliminaries Sobolev spaces
Let Cl C M.n be an open set, and Ck(Q) (k > 1) the space of all fcth differentiable functions on $7, endowed with the norm uCfc = V^ sup\Dau\, a
where a — (ai, • • • ,an), at > 0 (1 < i < n) are integers, \a\ = J3"=i a«i and
Du=— dxi'
D uy
92u
dxidxj'
Let
c£(n) = {«e cfe(n)  suppu c n}, where supp u is compact support set of u defined by suppu = closure of {x G Q  u(x) ^ 0}. Let Lp(£l), 1 < p < oo (or L°°(fi)) be the space of real functions denned on Q with the pth power integrable (or essentially bounded real functions), endowed with the norm
\HLP = \J \u\pdx\ , 197
198
Bifurcation Theory and Applications
(or, for p = oo, II U L°°
= ess.sup a
W(X)).
For 1 < p < oo and nonnegative integer k, the Sobolev spaces are defined as follows Wk'p(Q) = {ue L p (fi)  Dau G L p (ft), a < fc}, with the norm
\\Dau\\Lr.
\\u\\w,.P = £
\<x\
As p = 2, we write HkQ.) = Wfc'2(fi) which are Hilbert spaces with the scalar product
(u,v)Hh = V
/
DauDavdx.
Jn
\a\
We define WofclP(ft) = the closure of C£°(fi) in Wrfc'p(fi),
For integers /c > 0 and 0 < a < 1, we define the spaces = {u G Cfc(fi)  [Z)"«]a < oo, /? = fc}
Ck'a(n) with the norm
ucfc.° = \\u\\ck + Yl [DPu]a, /3=fc
Ha= sup
K*)y,
Embedding Theorems. Lei fi C R n 6e a bounded domain and 1 < p < oo, then
Theorem 7.1
^'P((1)CL«(S1)
VI < ? < — ^ ,
W0fe'p(O) C Lq(Q)
VI < g < oo,
P
1 01
W£' (Q.) C C" ' ^) \/kp>n,
n>fcp,
n = kp,
m+a =
k.
199
Bifurcations for Nonlinear Elliptic Equations
Moreover, the inclusions are continuous, namely TIT)
\\u\\Li < C\\u\\wkr
ifl
wc'm,a < Cu^fc,P
ifm + a = k
IX —
T, Kj) 77.
,
kp > n.
where C = C(n, ft, q). Theorem 7.2 have
Let Cl C M.n be Lipschitz, not necessarily bounded, then we
Wk'p(Q.) c L"(Q)
q =  ~  , n > kp, n — kp m+a = k, kp > n,
Wk'p(n) C Cm'a(Q) and the inclusions are continuous. Compactness Theorems.
Theorem 7.3 Let Q c M.n be a bounded domain, then the following embeddings are compact W^(n)^L"(Q),
Q<^~,
W^p(n)^L"(n),
q
W*'p(£l) ^ C°
n>p, n=p,
a < 1 ,
p > n.
Interpolation Theorems. Theorem 7.4 follows
For p < q
where e > 0 is an arbitrary real number. Theorem 7.5 Let Q C K" be a Lipschitz domain and e0 > 0 be given, then, for any e (0 < e < EQ) we have
H > < C[e\u\m,p + e»\\u\\LP},
\ukP= E \l \Dau\pdXP, a=fc
Un
J
200
Bifurcation Theory and Applications
where C = C(m,p,£ 0 ,ft), 0 < j <ml,
n= ^ j .
Trace Theorems. Theorem 7.6 Let Q, C Kn be a domain with class Cm+1. u € W£'p(n), p > 1, we have
Then for any
Dau an= 0, a.e., Va < m  1. 7.1.2
Regularity
estimates
We consider the linear elliptic equation given by
{
n
n
(7.1)  ^2 a^DijU + ^2 biDiU + cu = f(x), x E Q, u\da = V,
(7.2)
where 0 C Mn is a bounded domain, and n
^2 aijtej > m2 (A > 0), Var e U, £ € M". (7.2) The following theorems provide the Schauder global estimates and the Lp~estimates for the equation (7.1). Global Schauder Estimates. Theorem 7.7 Let Q C Rn be of class C2'a, and aijr bit c, f G C°>a(Q.),
1 such that for 1 < p < p0, the solution of (7.35) is unique; (3) for e > 0 sufficiently small, the solution of (7.35) near p = 1 can be expressed by
< C[/ L , + \\v\\W2,v + \\U\\LP] ,
where C is a constant depending on n, p, Q, A and the L°° modular of a,ij, bi and c(x).
Bifurcations for Nonlinear Elliptic Equations
201
The following is the Agmon's theorem. We consider the linear elliptic equation as follows n
Au =  Y
Di(aij(x)Dju) = f(x),
i£(l
(7.3)
where a^ satisfy (7.2). The conjugate operator of A is given by A*u =  Y Theorem 7.9
Dj{aijDiu).
Let Q, be of class C2 and a{j £ C^Q). Let u € Lq{Q) and
f G L"(fi), p,q>l.
If for any v G C 2 (f2) 0 W o l l P (ft),
/ u • A*vdx = / / • iicfa:, then u e W2'P(Q) n Wd'p(O) is a sironp solution of (7.3), and IUW*.
where C = C(n,p,Q,, A).
7.1.3
Maximum
principle
We denote n
n
Lu = — 2_j QijDijU + Y, biDiii + cu,
x £ £1,
where aijy b,, c G C°(Q), and Cl is C 2 . Theorem 7.10 Let L 6e on elliptic operator, i.e. a,ij satisfy (7.2) and c = 0. IfueC2(Q.) satisfies Lu > 0 (< 0)
in Cl,
then u reaches its minimum (its maximum) at boundary dQ,, providing u is not a constant. The following is the Hopf maximum principle. Theorem 7.11 Let c(x) > 0 in fi and u € C2(Q) satisfy Lu > 0. / / io € dQ, u(x0) < 0 and u(x) > u(x0) Vx e 0, then du(x0) an
202
Bifurcation Theory and Applications
where n is the outward normal at XQ £ dQ,. 7.2 7.2.1
Bifurcation of Semilinear Elliptic Equations Transcritical
bifurcations
In this section, we consider the following system of elliptic equations: r
£
Aui + Aui +gik(ui,u2)+
Y2 9iP(u) = 0, p=k+l
Au2 + \u2 + g2k(ui,u2) + ] T g2P(u) = 0,
(7.4)
. w a n = 0, where u = (ui,U2), ans giP(ui,U2) (k ^ 2,i = 1,2) are pmultilinear functions such that r
Gi(u1,u2) = gik(ui,u2) + ^2
9iP(u),
p=fe+i r
G2{ui,U2)=g2k(ui,U2)+
^2
52p("),
p=fc+l
are C°° functions. We remark that the above system can be considered as the steady state equations of the reactiondiffusion equations (8.10) for the case where c = 0. Let gik(ui,u2) = ^2
aiju\u{,
i+j=k
. . hjulu^.
92k{ui,u2)=
*jT
(7.5)
i+j=k
Let po > 0 be an eigenvalue of —A with the Dirichlet boundary condition, having multiplicity m, and {ei, • • • , e m } c Hi be the eigenvectors corresponding to po,  Ae* = poei, ,
e
ilan
= 0
'
/ aejdfl = Sij. v JQ
(i = l , . . . , m ) (7.6)
203
Bifurcations for Nonlinear Elliptic Equations
Notice that p0 is not necessarily the first eigenvalue. Assume that (xi, • • • ,xm,yi, • • • ,ym) = 0 is an isolated zero point of the following algebraic equations 5Z
S
Yl
Y,
a'riuhljXri • ••*riyi1 ' • • Vlj = 0, Pl1...ril1.:lixriXriVl1'yii=O,
(7.7)
for 1 < s < m, where K^nhii
= aH / e n • • • e^e*! • • • e(jes dx,
Prtnhlj
= bij / e n • • • e ^ e * i • • • ehes J Cl
dx

We see that the eigenvalue of L\ at A = po has multiplicity 2m. By Theorems 4.1 and 4.4, we can obtain the following theorem. Theorem 7.12 Under the assumption (T.I) with k = even, the equations (7.4) have at least one bifurcated branch on each side of A = po. Moreover, if m = l,k = 2 in (7.7), and the two vectors (a20,oii,a02), (&20,^115^02) are linearly independent, then the following assertions hold true. (1) There are at most three bifurcated branches of (7.4) on each side of A = p 0. (2) If the bifurcated branches on A > po (resp. on A < po) are regular, then the number of branches is either 1 or 3. (3) If the number of branches on a given side is 3, then all the 3 branches must be regular. (4) If the number of branches on a given side is 2, then one of them is regular. Proof. Let Hi = H2(n,R2) n H^{Q,M2), H = L 2 (fi,R 2 ), and the operators Lx : Hi > H and G : Hi > H be defined by
L\u = (Aui + Xui,Au2 + \u2), G(u) = (G1(u1,u2),Ga(u1,u2)). It is easy to see that the linear operator Lx has eigenvalues j3j (A) given by 0iW = Pi + \
204
Bifurcation Theory and Applications
where pj is the jth. eigenvalue of —A. Then we have
{
< 0
if A < po,
= 0
if A = po,
> 0
if A > po,
/3j(po)^O,
(pjo = po)
j^j0.
The eigenvectors corresponding to (3j0 (A) at A = po are iV = ( e r , 0 ) ,
wi = (Q,ei),
l ^ r , l ^ m .
Thus, for Gk = {gik,92k) we have (Gk(vri,(Gk(vri,
,vri,wh,wli),va) • • • , vri, wh,
• • • wtj),
=a'li...trih...i., ws)
= (3
S
T1...
^...ij•
In view of Theorems 4.1 and 4.4, we obtain this theorem. The proof is complete. • When m = 1 in (7.7), i.e. po is a simple eigenvalue of (7.6), the condition (7.7) amounts to saying that (x, y) = 0 is an isolated singular point of the following equations
^2 on^y3 = 0, 
i+j=k
£
i+j=k
bijxW = 0,
(7.8)
and that / ek+1 dQ ^ 0,
(7.9)
Jn where e is the eigenvector of (7.6). When p0 is the first eigenvalue of —A, (7.9) holds true. Let m = 1 and k = 2 in (7.7). If the coefficients in (7.5) satisfy that a2o/3i < 0, a2o/32 > 0
(or fc2o/3i > 0,62o/32 < 0),
(7.10)
Bifurcations for Nonlinear Elliptic Equations
205
where fa = &20
i = l,2,
—an ± )L vafi—4002^20 ~^ . Z<220
a
2 ,. ^ n n ~ 4«2oaO2 > 0,
(or (3i = a2oOif + anoii + aO2,i = 1,2, "1,2 =
 6 1 1 ± y/^11  46 2 o&o2 ,2 . . . .n x ^ ^ > H i  4&20&02 > 0 ) , ^020
then, from Theorems 6.9 and 6.10, we obtain the following result. Theorem 7.13 Let the condition (7.10) hold true. Then the equations (7.4) bifurcate from (O,po) exactly three branches on each side of X = po. When k = odd, by using Corollary 4.1, we obtain the following result. Assume that the operator in (7.5) is a potential operator, i. e. the coefficients ciy and bij satisfy (j + l ) a y + 1 = (t + l ) 6 i + l i l
O^i,j
(7.11)
We also assume the algebraic equations ' (\po)x+ J2 OijxV = 0, % k . . (A  po)y + £ bijxY  0
(712)
have finite number of solutions near A = po. Theorem 7.14 Let po > 0 be a simple eigenvalue of the scalar Laplacian with the Dirichlet boundary condition, and k = odd in (7.5). If (7.11) and (7.12) hold true, then the equations (7.4) bifurcate from (O,po) at least four branches on both sides of X = po. Remark 7.1 The condition (7.12) with A ^ po is equivalent to the following condition: The algebraic equation Y, {aij  b^lj+1)zj+1
~bko=0
(7.13)
i+j=k
has only finite number of zero points z £ I 1 , i. e. there is at least a nonzero coefficient in (7.13): 2 J \aij — bi~ij+\\ + \bko\ ^ 0. i+j=k
206
Bifurcation Theory and Applications
Indeed, when A ^ p0, the solutions of (7.12) must be on the straight line a2y = a\x with a2,a\ € R1 satisfying
z=
;r = £ 6 ^ Z 7 12 aiizj> 2
i+j=k
i+j=k
which leads to (7.13). Now, we address the following equations with c^O: r
{
Ait! + \ui + cu2 + gik(ui,u2) + ^2 9iP(u) = 0, P=k+i
Au2 + Xu2 + g2k(u1,u2) + ^2 g2p(u) = 0,
(7.14)
P=fc+i
u
lan = °
We consider only the bifurcation of (7.14) from a simple eigenvalue po of (7.6). Thus, the eigenvalue of LA at A = po has the algebraic multiplicity m = 2 and geometric multiplicity r = 1. The eigenvectors of L\ at A = po are given by vl = (e,0),
vZ = (0,e),
where e satisfies (7.6), and the eigenvectors of L*x at A = p0 are «I(e,0),
t£ = (O,e).
They satisfy LPovi = 0 , i>2=0,
LPov2 =cvi, L;ovl=cv*2.
Let a = (Gfc(t;i),U2> = / g2k(e,0)dx = bko, then based on Theorem 4.3, we have the following theorem. Theorem 7.15 Let po > 0 be a simple eigenvalue of (7.6). If the number a = bko 7^ 0, then the following assertions hold true. (1) As k = even, there exists a unique bifurcated branch of (7.14) on side of X = po
eac
h
Bifurcations for Nonlinear Elliptic Equations
207
(2) As k = odd, if a • c = b^o • c > 0 then there are exactly two bifurcated branches of (7.14) on each side of A — po(3) As k = odd , ifac = bkoc < 0 then equations (7.14) have no bifurcated branches from (u,X) = (0,po). (4) Each branch of (7.14) bifurcated from (O,po) is regular, and the solutions u\ of (7.14) in the bifurcated branch F(A) C H\ can be expressed as ux = ±\a\XPof\1/{k1)e 7.2.2
+
o(\Xp0\^k^.
Saddlenode bifurcation
Let po > 0 be the first eigenvalue of (7.6). We assume that 9ir(xi,x2)x1 +g2r(x1,xa)x2
^ a(zaf + 1 + \x2\r+1)
(7.15)
for some constant a > 0. Theorem 7.16
Under condition (7.15) we have the following assertions.
(1) If c ^ 0 and bfo ^ 0, then for k = even and k = odd with cbko > 0 the equations (7.14) have a saddlenode bifurcation point (UQ,XQ) with Ao < po(2) If c = Q,k = even,and (gik{x),g2k(x)) is kth order nondegenerate at x = (x\,X2) — 0, then the equations (7.4) have at least one saddlenode bifurcation point {UQ, AO) with Ao < poProof. We shall apply Theorem 6.7 to prove this theorem. By Theorems 7.12 and 7.15, there exist bifurcated branches from (u,X) = (0,po) on A < po. Let EA be a bifurcated branch from (0,po). By Theoremm 6.7 it suffices to prove that SA is bounded in H x (co, po). We infer from (7.14) and(7.15) that r f Vu 2 u 2 I [\   ^ \ \ ~ cuiu2  ^2(giP(u)ui P=k
J Q
+ g2P(u)u2)} dx
> I [l Vw ! 2  A M 2  ^1^2 + a ( K  r + 1 + w 2  r+1 ) in
 52(ui5i p + u2g2P)} dx. P=k
(7.16)
208
Bifurcation Theory and Applications
By (7.15), r > 3 is an odd number. We take .
*
=
2(r + 1 
ri
TO)
P=
'
r 1
7+T^>
9=
r 1
3
^2'
^m^r'
then
JO"**
\Jy H"P[/j"(""1'"
<  / M2 da:+
Pin
for any e > 0 and v e Hi.
/  w  p+1 dx
9 in
(7.17)
^
It follows from (7.16) and (7.17) that /
Vu2  Au2 cuiu2
 Y^(uigip + u2g2P) dx
>2[iv«i a +H a +(iu 1 r +i +i« a r +i )]dx > 0, for any A < —N, and u £ Hi with u ^ O , where JV is a an integer. On the other hand, by
/ \u\m dx^ce'^+s
[ \u\r+1 dx,
(7.18)
m^r,
for some constant c > 0, we infer that / [V«2 + a(\Ul\r+1 + \u2\r+1)  A \u\2  culU2 r
~ ^2(uigip + u2g2P)] dx P=k
> I Vu2 dx  M, (7.19) Jn for A < —A^ and some M > 0. Prom (7.16), (7.18) and (7.19) we deduce that EA is bounded in H x (—oo,po). The proof is complete. • Remark 7.2 Under the hypotheses of Theorem 7.16, the equations (7.4) and (7.14) have nonzero solutions at A = p0 the first eigenvalue of —A.
209
Bifurcations for Nonlinear Elliptic Equations
7.3 7.3.1
Bifurcation from Homogenous Terms Superlinear case
We consider the equations given by f (~l)mAmu \Dau\dn
 AM*" 1 * + G(x, u,...,
D2mu) = 0,
= Q, \a\ < m  l ,
(7.20)
where p > 1, m > 1, and f2 C K" is bounded. Assume that G G C 0 0 and for A near 0 G{x,\z,.,\Q=o(\\\),
AeR 1 .
(7.21)
Let X = Hlm{Sl) n F o m (ft), Y = L 2 (fi). It is known that the equation
C
"
'
\ Dau\9Q
= 0,
'
Va < m  1
(7.22)
V
7
has a nontrivial solution uo G X for
{
2n
—
if n > 2m,
n2m
if 1m > n.
oo
The following theorem is a direct consequence of Theorem 4.10. Theorem 7.17 Let u0 G X (u0 =£ 0) be a solution of (7.22). Under the condition (7.21) if the following linear equation ((l)mAmvp\uor1v a
\D v\Bn
= 0,
= 0,
Va < m  l ,
has no nonzero solution, then the equation (7.20) has a bifurcation from (u,X) = (0, oo), and the bifurcated branch (u(x,X),X) of (7.20) can be expressed as u{x, A) = XVbVuoix) lim v(x, A) = 0. A—»+oo
+ A1/(P1)v(rc, A) (A > 0),
210
Bifurcation Theory and Applications
7.3.2
Sublinear case
By using alternative bifurcation theorems, we shall investigate global bifurcations of the following problems: ( (_!)"• Amu pu + AM*"1** + G(x,«,.., [Dau\dn
= 0,
Dm~lu) = 0,
\a\<ml,
(7.23)
where 0 < 6 < 1, p > 0, Q C R" is bounded, and G satisfies (7.21). The eigenvalue equation of (7.23) is given by I
\Dau\dn
= 0,
\a\<ml.
(7.24)
Assume that (A\) The real number p > 0 in (7.23) is not an eigenvalue of (7.24) and the sum of multiplicities of eigenvalues of (7.24) in (0, p) is an odd number. (A2) There is a constant C > 0 such that
G(*,oi
(JO.
1
ifn >2,
if 2 > n.
We denote E = {(u, A) G flj*(n) x (0,00) I (u, A) satisfy (7.23), u / 0}, So = the connected component of £ containing (u, A) = (0,0). Theorem 7.18 Let (Ai) and (A2) hold. Then Ao = 0 is a unique bifurcation point of (7.23) in [0,00), and there is a bifurcated branch on A > 0. Moreover, at least one of the following two assertions holds true. (1) Eo is unbounded in H^(Q) x (0,oo). (2) So contains (uo,0), where UQ^O satisfies j (l)mAmu pu + G{x, «,•, Dm~lu) = 0, \Dau\9n = 0, \a\<ml.
211
Bifurcations for Nonlinear Elliptic Equations
Proof. We shall apply Theorems 4.6 and 4.7 to prove this theorem. Let X = H!P(Q). We define mappings L, h, G : X > X by the following inner products, respectively {Lu, v) = — I pu • vdx,
Jn
(hu,v) = / u ~1uvdx, Jn {Gu, v } = f G{x, « , • • • , D "  1 ^ • v d x . Jn Thus, equation (7.23) is equivalently written as u + Lu + Xhu + Gu = 0,
(725)
u £ X.
It is clear that L, h, G : X —> X are compact operators. We shall show that the following problem has no bifurcation point in A€(0,l]. f (  l ) m A m «  XP(u  \u\*xu) + +G(x,«,.., a
\D u\dn
= 0,
Dm^u)
= 0,
\a\ < m  l .
(7.26)
Denote by ^ i = {x € fi  u(a;) < 1} and fi2 = ^ \ fii Then we have /
Jni
u2da; < / u1+5da;, Jn
I u2dx < / w4da;.
Jn2
Jn
It follows from (7.26) that
j [ A m / 2 u  2 + \ p { \ u \ 5 + l  u2) + G(x, « , .  • , D 1 " " ^ ) ^ dx > I [  A m / 2 u  2  Xpu4 + G(x, « , • • • , Z?" 1  1 ^)^] rfr > [ \Am'2u\2dx Jn
+ o(\\u\\%m),
which, by the Poincare inequality, implies that (7.26) has no bifurcation point in A £ (0,1], and (7.26) has no bifurcated branches on A > 0 if A = 0 is its bifurcation point. Hence, by Theorem 4.6, the equation (7.23) has at least a bifurcation point in A € [0, p], and if A = 0 is a bifurcation point of (7.23) then there is a bifurcated branch of (7.23) on A > 0.
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Bifurcation Theory and Applications
Thus by Theorem 4.7, it suffices to prove that there is a nondecreasing function r\ of A > 0 such that (7.23) has no nontrivial solution in Brx. In fact, for any A > 0 there is an £\ (0 < e\ < 1) such that / u2dx < Xp1 f \u\1+5dx, Jn,x Jn
Vu ^ 0,
(7.27)
where Qex = {x £ Q,  \u(x)\ < £\}. Obviously, £\ is a nondecreasing function of A > 0. On the other hand, we have / u2dx < e^2 / u4dx, Jn\n.x Jn
Wu + 0.
(7.28)
Furthermore, for any £\ there is a r\ > 0 (r\ is a nondecreasing function of £\, therefore r\ is also a nondecreasing function of A > 0) such that for anyue£rx{0} I [/\ m/2 u\ 2 + G(x, u, • • • , Dmlu)v^
dx
>C\\u\2Hm+o(\\u\\2Hm)
(7.29)
> pe^2 I uAdx. Jn It follows from (7.27)(7.29) that for any u e Brx{0}
f [  A m / 2 u  2  p u 2 + Au 1+ * + G(a:,«, ,£> m ~ 1 u)u] dx >p [ [Xp'luf+t+efu^u^dx Jn >0,
which implies that (7.23) has no nontrivial solution in Brx. This theorem is proved. D
213
Bifurcations for Nonlinear Elliptic Equations
7.4
Bifurcation of Positive Solutions of Second Order Elliptic Equations
In this section, we shall investigate bifurcations of the following problem with respect to the two parameters p and A,
{
Au = Xup + f(x,u,Vu,V2u),
leftcR",
u\dn = 0,
(7.30)
u > 0 in ft, where A > 0, ft is C2
{

u, Vu, V2u)DijU = Xup + F(x, u, Vu, V 2 u),
aij(x,
u\dn = 0,
(7.32)
u > 0 in 0. Actually, equation (7.32) can be written as  aij(x)Diju = \up + G(x, u, Vu, V2u),
{
u\an = 0,
(7.33)
u > 0 in ft, where (aij{x)=aij(x,0,0,0),
\G(x,^<:)=F(x^,0+aij(x^,0aij(x). It is clear that (7.33) is essentially the same as (7.30). Meanwhile, we shall also consider the global bifurcation for the following equation J Lu = Xup + g(x,u,Vu), \ u\9n = 0,
a; eft CM",
u > 0 in ft,
where 0 < p < 1, and Lu = —a,ij(x,u,'Vu)DijU + bi{x,u,Vu)DiU + c(x, u, Vu)w.
214
Bifurcation Theory and Applications
7.4.1
Bifurcation in exponent parameter
We consider the problem
{
 An = Xup, u\dn = 0,
xeCl, (7.34)
u > 0 in Q. Let Ai be the first eigenvalue of —A, and u\ > 0 in $7 with jitic?2 = 1 be the eigenfunction corresponding to Ai. We remark that if u(x,p) satisfies
fA« = w , { u\dn = 0,
*en,
P*i,
(7.35)
u > 0 in fi,
then u(x,\,p) = (\\;1)1/VP)u(x,p) satisfies (7.34). Hence to investigate (7.34) it suffices for us to consider the problem (7.35). It is known that for 0 < p < 1, the solution of (7.34) is unique. Let S be the set of all solutions of (7.35) in C2(Q) x [0, 2±). We have the following result. Theorem 7.19 In S, there is a connected component S in C2(Q) x [0, 2±) such that [C2(ft) x {p}] ± 0 ; (1) for anyO
u(x,l ±e) = ^ ( e ) ^ ! ±su±(a;,e), where a ± (e), v±(x,s)
are C1 on e > 0 sufficiently small, and
lim tr^e) = a0,
aQ = eL/'n"? 1 "''!^/^ »!*«],
£—»0
lim v1*1 (a;, e) = VQ(X),
215
Bifurcations for Nonlinear Elliptic Equations
and vo satisfies
{
— Av = \iv + AiaoWiln(aoUi), u\an = 0 ,
f
x € fi,
/ uivdx = 0. Jn
Remark 7.3 Let id  \\Tp : C 2 (0) —> C 2 (n) be the completely continuous field corresponding to (7.35), and
Tp(v)= [ G(x,y)\v(y)\*dy, Jn where G(x, y) is the Green function, we shall see later that (ind(idX1Tp,u(x,p)) = 1, 0 < p < 1, < n+2 I deg(^A 1 T p ,5 p ,0) =  l , K p <  ^  ,
(7.36)
where B p = {u£ C2(ft)  0 < e < wC2 < Rp} with BpnS = En[C 2 (fi) x {P}]Proof of Theorem 7.19. By (7.77) later, we see that for 0 < p < 1, there exists a constant /3 > 0 independent of p such that the solution of (7.35) satisfies (/JAOVdp) <
IKS.JOHCO.
(7.37)
First, we shall verify that there is a constant C > 0 independent of p such that for all 0 < p < 1 we have \\u(x,p)\\Ci
(7.38)
and for any p0 > 0 there is a constant Ci > 0 independent of p with Po < P < 1 such that W^PJHC^CL
By (7.35) we obtain / Vw(a;,p)2d:r = Ai / w 1+p (a:,p)^
Jn
Jn
in view of the Poincare inequality / \Vu\2dx > Ax / \u)2dx
Jn
Jn
(7.39)
216
Bifurcation Theory and Applications
one deduce that / \u(x,p)\2dx<
fi,
0
(7.40)
where fi is the volume of fi. By the Agmon's theorem (Theorem 7.9) and the Sobolev imbedding theorem (Theorem 7.1), (7.38) follows from (7.40). By the Schauder estimates, for (7.35) we have \\u(x,p)\\c, < C[\\u\\Co + AilKIM
(7.41)
Taking a = p0, from (7.38) and (7.41) we can obtain (7.39). Prom (7.37) and (7.39) it follows that lim u(x,p) = uo(x)
p>0
and uo(x) satisfies  A u = Ai, ! u\dn — 0,
x e fi u > 0 in fi.
It is easy to see that ind(idAiTo,u o ) = 1
(7.42)
By the uniqueness of (7.35) for 0 < p < 1 and the homotopy invariance of degree, from (7.38) and (7.42) one can deduce md(id\1Tp,u(x,p)) = l,
V0
Next, we shall show that for any 1 < p\ < ^ § , there is a constant C > 0 depending only on p\ and fi such that for all solutions u(x,p) of (7.35) with 1 < p < pi, we have Hx,p)\\C2 < C,
l
(7.43)
By the results in [de Figueiredo et al., 1982], one can see that for any e > 0 sufficiently small, the set {u(x,p)} with 1 + e < p < pi is uniformly bounded in C2(Cl). Hence we only need to estimate u(x,p) near p = 1. Multiplying both sides of (7.35) by ui and integrating them, we have / u\ • u(x,p)dx = / ui • up(x,p)dx.
217
Bifurcations for Nonlinear Elliptic Equations
By the Holder inequalities
f
\r
V~'\ f
Jo,
Un
J
/ u\ • u(x,p)dx < / uicfcc
/
Ua
l1/p
uiup(x,p)dx\
J
we get that / u1up(x,p)dx < / mdx.
Jo.
Jo,
(744)
Let e(x) > 0 in Q, satisfy J  Ae = 1, i 6 ( ! \ ean = 0. Multiplying both sides of (7.35) by e(x) and integrating them, we get / u{x,p)dx = Ai f e{x)up(x)dx. (7.45) Jn ./n Because u\(x) > 0 in 0, and by the Hopf s maximum principle (Theorem 7.11), we have  ^ > 0 for x S dn, arx where rx is the inward normal vector at x € dCl. Hence there is a constant C > 0 such that ui(rx) > C\rx\,
for 0 < \rx\ < e, Vx € dQ..
Hence there is a constant K > 0 such that e{x)
VxeQ.
It follows from (7.44) and (7.45) that / u(x,p)dx < K\! / ui • up(x,p)dx < KXi f mdx. JQ Jn Ju By (7.35) we see that / \Vu(x,p)\2dx < C [ up+\x,p)dx. Jo. Jn
(7.46)
(7.47)
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Bifurcation Theory and Applications
In view of (7.46) and the Holder inequality / up+1(x,p)dx = f u1'" • up+Vdx Jn Jn
< I / u(a:,p)ds:j * \J upq'+1dx\
(7.48)
i1/'9'
r /•
U1+PI'(x,p)dx\
Hence, from the Sobolev's inequalities and (7.47), (7.48) it follows that r r
/ \u\2n^n~2Ux\
in2/n
f
\Vu\2dx
r
i 1/q'
f
u1+Pi'dx\
Taking ,
2n
1
n2
then we have p
Jn
Vl
(7.49)
Then, the bound (7.43) follows from (7.47)(7.49). Now, we prove the assertion (2). Let wp{x) = \\u(x,p)\\]j\u(x,p). By (7.35), wp(x) (1 < p) satisfies
f Aw = \1upI{x,p)w, 1
n
(p>l),
[ w an = 0.
(7.5U)
By (7.43), for any convergent subsequence of {u(x,p)} with p —+ 1 and p > 1, without loss of generality we still denote it by {u(x,p)}, we have 0 < lim vP1^)
= g(x) < 1 in C°(Cl).
It is obvious that g{x) ^ 0, otherwise by (7.50) we deduce that limp^j wp(x) = 0 in HQ(Q,), and reach a contradiction with u;p#i = 1. Let wp *• wo in HQ(Q), then u;p converges to w0 almost everywhere in Q,
Bifurcations for Nonlinear Elliptic Equations
219
hence w0 > 0 in fl. We can see that w0 ^ 0, otherwise by (7.50) one deduce that lim p _i wp(x) = 0 in HQ(£1). Then, u;0 satisfies
{
 Aw0 = Ai5(x)iy0,
xetl,
wo\dn  0.
(7.51)
By the maximum principle, we have that WQ > 0 in f2. Thus Ai is the first eigenvalue of (7.51). Hence we have Ai=
inf
.
, \ _. .
(7.52)
On the other hand, by assumption, Ai is also the first eigenvalue of —A, which implies that Ai=
inf
. \, .
(7.53)
By 0 < g(x) < 1, it follows from (7.52) and (7.53) that g(x) = 1 in fi. It is clear that for (7.35) with p > 1, the operator id — \\TP is differentiable at u(x,p) £ C2(O), and the equation corresponding to id — \xf3Tp(u(x,p)) is as follows [ &v = p\1pup1(x,p)v,
i£(l,
[ v an = 0.
(7.54)
It is easy to see that (3 — p~l is the first eigenvalue of (7.54) which corresponds to the eigenfunction u(x,p) > 0 in fi. On the other hand, because lim pup~1(x,p) = g(x) = 1 in Cl,
p—>i
we can see that there is a real number £ > 0 sufficiently small such that for any p (1 < p < 1 +e) and the solution u(x,p) of (7.35), the equation (7.54) has only one eigenvalue j3 = p~l in [0,1]. Since /? = p~l (/? < 1) is a simple eigenvalue of (7.54), for any solution u(x,p) of (7.35) with 1 < p < 1 + e, we have ind(idAiT p ,u(x,p)) =  1 . We know that for any p (1 < p < ~^), deg(idX1Tp,Rr,0)
= l>
220
Bifurcation Theory and Applications
where ^ = {116 C2(ft)  0 < 5 < \\u\\C2 < R}, and S small enough, R > 0 great enough such that all solutions u(x,p) of (7.35) are in i? r . Hence the assertion (2) holds true. Assertion (1) follows from (7.36) and the homotopy invariance of topological degree. Finally, we verify Assertion (3). Consider the case fA
= W*,
( u\dQ = 0,
" " •
(7.55)
u > 0 in Q,
where s > 0 sufficiently small. We assume that the solutions of (7.55) are of the form ' u(x, ±e) = a±(e)ui ±£v±(x,e), u1v±(x,e)dx = 0,
<
(7.56)
Jo. 0
±
n
\<bl)
[v ±\an = 0, here /i±(e) = £1Ai(a±wi ± ev*)1^ ±
±
e^X^Ui ±
 (±X1)v  (±Ai)o; u1ln(a ui). Because near £ = 0 we have (o^ui ± ev*)1^
= (a ± u 1 ± ew±)(a±ui ± eu ± ) ±£ = ( ^ u i ± eu111)!! ± e l ^ ^ u i ± eu*) + o(e)].
By assumption, 0 < Ci < a ± (e) < C2 < 00 for £ > 0 sufficiently small, it is easy to see that limK±(e)=0
in C°(O).
e+0
Denote by Cj(ft) = {v € C°(fi)  / n « • M s = 0} and P : C°(fi) »
221
Bifurcations for Nonlinear Elliptic Equations
C°(fi) the projection defined by
{
Pv = v — 7U1, j =
v • u\dx/
Jn
I u{dx.
Jn
Thus, the equation (7.57) can be decomposed into the following system r  Av± = A i t; ± + P[±\1a±u1 l 4. f 4^ Jn a ± l n a ± [ u\dx + a* [ ullnuidx Jn Jn
l n ( a ± u i ) + K ± (e)], ( 7  58 )
+ / K±u1dx Jn
= 0.
(7.59)
It is clear that if (7.58) and (7.59) have solutions (u ± (£),o; ± (£)) for e > 0 sufficiently small with 0 < C\ < ^(e) < Ci < oo, then (7.56) are the solutions of (7.55). Let id  XiL  g± : C°(Q.) x R ' x l 1 ^ C°(Q) be the operator corresponding to (7.58), and
Lv=
Jn
g±(s,a,v)
G(x,y)v(y)dy, = P / G(x,y)[±\1aui\n(aux) Jn
+
K±(e,a,v)]dy,
where G(x, y) is the Green function. It is easy to see that (e, a, v) = (0,a o ,vo) satisfies (7.58) and (7.59), here (ao,vo) is as in Assertion (3). We see that
D^O.ao.flb) = P f G(x,y)[DvK±{0,ao,vo)]dy Jn
= 0.
Hence, we have Dv[id  XiL  3±](o,ao,l;o) =id
AiL : C°(O) » Cj(fi)
and it is invertible. By the implicit function theorem, for each e, a) with 0 < e and Q — a0 \ sufficiently small, there exists a unique solution v(x, e, a) of (7.58) which is C 1 on (e, a) near (s, a) = (0, a0) in the norm topology of C°(fi). Inserting v(x,e,a) in (7.59), thus the existence of (7.58) and (7.59) near (0, a0, v0) is equivalent to the existence of the algebraic equation (7.59)
222
Bifurcation Theory and Applications
near (0, a0). We know that
[
f
/
— / u\\nuidx/
Jn
"
I
I u\dx\
Jn
is a solution of the following algebraic equation
J
f(a) = a In a / u\dx + a / u\\nuidx = 0 Jci Jn and f'(a0) = f u\dx ^ 0. Jn Hence, (7.59) has a solution a(e), and a(e) is C 1 for e > 0 sufficiently small. Thus, Assertion (3) is proved. The proof of this theorem is complete. 7.4.2
Local bifurcation
In this subsection, we consider the local bifurcation and uniqueness of the following problem
f Au = g(x,u,Vu,V2u)+X(up + f(x,u,Wu,V2u)),
<
[ wan = 0,
u > 0 in f2,
(7.D(J)
where 0
0 sufficiently large. Proof of Theorem 9.1. We shall prove this theorem by using Theorems 3.16, 6.1 and Remark 6.1, together with the LyapunovSchmidt reduction procedure. It is easy to see that the eigenvalues and eigenfunctions of the linear operator L\ : H\ —» H defined by (9.3) are given by (3k(X) = Afc2 
(g(x,\z,\t,K) = o(\\\), \f(x,Xz,\^XC)=o(\\n
(7.61)
uniformly for ( i , z , ^ ( ) 6 f l x / x / n x / " n2+1 . Let Uj satisfy
f  A « = «>, x e n , p ^ l , [ wan = 0, ix > 0 in Q.
(?62)
Bifurcations for Nonlinear Elliptic Equations
223
We denote by ri(A)=
_max g(x,X,X£,X£), ien,?=i,lCI=i /(x,A,A£,AC), r2(A) = _ max x6n,ci=i,KI=i r(A) =max{A 1 r 1 (A),A p r 2 (A)}. By (7.61), it is clear that lim*—o r(X) = 0. If p = 1, from Theorem 1.11, we immediately obtain the following theorem. Theorem 7.20 Under the condition (7.61) with p = 1, near (u, A) = (0, Aj) there is a unique branch of solutions of (7.60), and for A — Aij > 0 sufficiently small, the solution {u\,\) in the branch can be expressed as
{
u\ = ±tu\ + tv(x, t), v(x, 0) = 0, A  Ai = (j,(t), and fi(0) = 0.
For p ^ 1, we have the following results. Theorem 7.21 Under the assumption (7.61), if 0 < p < 1 near (u\,\) = (0,0), there is a unique branch of solution of (7.60) and for A > 0 sufficiently small the solutions {u\,\) in the branch satisfy ux(x) = XWriuKx)
+
\1/{1p)v(x,X),
t,(x ) A) c2 , P =O(r(A 1 /( 1 rt)) moreover, v(x, A) is C1 in A for A > 0 sufficiently small in the norm topology
ofC2(ty.
Theorem 7.22 true:
Under the assumption (7.61), the below assertions hold
i) If 1 < p < ^  , and the equation
{
— Av = pu[p~1v, v\dCl = 0
has no nonzero solution, then (7.60) has a bifurcation solution (u\,X) from (u,X) = (0,+oo),u\ e C2'p(Cl), and for X > 0 sufficiently large, the solutions satisfy ux(x)  XV^riuKx) + A 1 /( 1 P) U (X,A),
224
Bifurcation Theory and Applications
Hz,A) c 2 =O(r(A 1 /( 1 ")))
forX> +00
1
moreover, v(x, A) is C in A for A > 0 sufficiently great in the norm topology ofC2(£l). ii) There is a number pa > 1 such that for any 1 < p < po, near (u\, A) = (0, +00), there is a unique branch of solutions of (7.60). Proof of Theorem 7.21. Assume that (ux,X) satisfy (7.60), and liniA+o«A = 0 in C2'p(fl), we want to show that in the norm topology of C2'P(ty, \im\1^1riux=u*1. Denote by ux(x) = \l/^^u\
+ h(X)v{x, A),
v(x,\) = K(x)  A 1 ^ 1  ^ ] • K  Ava^u^, h(X) =
\\uxX1/^'^u*1\\C2,P.
Plugging this expression in (7.60), we deduce that r  Av = h1(X)g(x, ux, Vux, V2ux) + A/ l  1 (A)(A 1 /( 1  p '< + h(X)v)ai A 1 /( 1 ri/ l  1 (AK" + Ar 1 (A)/(x, U A,Vu A ,V 2 U A), I v\dQ = 0.
(7.63) Since {v(x, A)} is bounded in C ' (f2), for any subsequence of {v(x, A)}, there is a convergent subsequence in C2(f2). If h(X) ^ C^A1^1"^) for A —> 0, then there is a sequence {A/t} with limA>oo A^ = 0 such that limfc.oo /i1(Afc)A^/(1~p) = 0. Then by (7.61) in C°>P(n) we have 2p
lim
hllg{x,uXk,VuXk,V2uXk)
= lim
h^gixMiK'Xy^ul+vJMih^X^^Vul+Vv)),
k—>oo
lim Xkh^1 k—*oo
f{x,uXk,VuXk,V2uXk)
= lim (X1k/{1p)h^)1phpf(x,hk(h^Xl/ilp)ul k—*oo
+vx),
hk{KlX)J{1p)Vu\ + V«A), hk(hklXl^p) V2u\ + V2vA)) = 0, lim Xkh^iXl'^^ul k—too
+ hkvxy = 0,
Bifurcations for Nonlinear Elliptic Equations
225
where hk = /i(A&). If limfc^oo v(x, Afc) = vo(x) ^ 0 in C2(fi), then by (7.63) and the equalities above, we can deduce that vo satisfies
j  Av = 0,
xsfl,
\ v\aa = 0. This is a contradiction with VQ ^ 0. If lim^oo v(x, Xk) — 0 in C2(fi), then by Schauder estimates, we have Hx,Xk)\\cp
< C[\\v(x,Xk)\\co + \\F{x,Xk)\\Cp},
where F(x, A) = h1(X)g(x,ux, Vux, V2ux) + A/I 1 (A)(A 1 /( 1 P) U * + h(X)v)? X1^1^h1(X)u*1p
+
Xh1(X)f(x,uxyux,W2ux).
But by the assumptions and the equalities above, we have lim Kx,Afc) C o=0 )
K—»OO
lim
k—>oo
\\F(x,Xk)\\CP=0.
And one obtain a contradiction with \\v(x, A)c2,P = 1. Hence, we can assume that u\ = A 1 /^ 1 "*'^ + X1^1~p^v(x, A), and {v(x, A)} is bounded in C2'p(ft). Recall that v(x, A) satisfies
{
 Av = K + «]P ~ % P + )Cmip)g(x,ux,
Vux, V2uA)
+ AT^/(X,UA,VUA,V2UA),
wan = 0,
(7.64)
Uj +v > 0 in fi.
If there is a subsequence {A^} with lim^oo A^ = 0 such that lim v(x,Xk) = vQ(x) ^ 0 in C2(U) k—*oo
then by (7.61), and (7.64) we deduce that VQp satisfies p f  Av = [u{ + v}  u{ , xeQ, J l < , [ v\9n =0, u*1+v>0 in Q.
(7.65)
By the uniqueness of (7.62), it follows that (7.65) has only a solution v = 0, a contradiction. Thus, we have that lim,\>o v(x, A) = 0 in C2(Q), and by
226
Bifurcation Theory and Applications
(7.64) and the Schauder estimates, this implies that lim^o v(x, A) = 0 in C2'p(fi). Now, we shall verify that near (u, A) = (0,0), there is a unique branch of solutions of (7.60). The statement above tells us that we only need to show that the equation (7.64) has a unique solution branch near (u, A) = (0,0). In fact when fc 2 sufficiently small, we have that u\ + v > 0 in fi. Denote by idTF\ : Co'7(f2) —• Co'7(n) the operators corresponding to the equations (7.64), where 0 < 7 < p < 1 and Co' 7 (0) = {«£ C2^(Tl) v\d(i = 0}, and
G{x,y)[\ul+v\ru\p)dy,
Tv= [ Jn
Fxv = [G(x,y)[\1H1rig(y,\1K1ri(v.t A 1/(1 P ) (V 2 U * +
A i/dp))( V u *
V
2w))+
+v),X1^1'P\Vu*1 + Vv),
xl/{lp)f^
A l/(1P) (u * +
v^
+ Vv), A 1 ^ 1  ^ ^ 2 ^ + V2v))]dy,
where G(x, y) is the Green function. By the condition (7.61), it is easy to check that the operators F\ are continuous and differentiable, and .F((0)A=O = 0. Since 0 < 7 < p, the operator T : C Q ' 7 ( 0 ) —» CQl7(fi) is compact. We want to show that the operator T is differentiable in a neighborhood of v = 0 in C 0 ' 7 (fi). We only need to check the state for v = 0. We know that there is a constant C > 0 such that u\{rx) > C\rx\,
for all 0 < \rx\ < e,
where rx is the inner normal vector at x £ dQ, and £ > 0 sufficiently small. Because for any v € C0'7(f2), there is a constant C\ > 0 such that v(rx)\
for all 0 < r x  < e.
Thus we deduce that for a function v £ C0'7(f2) there is a constant C > 0 such that — < C on fj. u* Moreover, we shall show that v • u*1~^1~ £ CP(Q). In fact for x, y € Q, if
227
Bifurcations for Nonlinear Elliptic Equations
i( ) ^ u i(y)i then we have
u x
[«I («) tt j (x) ul{x)  ul(y) a:I/
l W
U y) p
^ ut(y)\\Xy\P v(x) , p >«(a:)ui(i/)i>(l/)"iO»0 «!(*) * «I(a:)«i(y)a:/I>
\ul(y)  U ;(X)P Ky)u;(y)  ut(x)\^ \xy\P ul(x)ul(y)
< J^L  V U .  P + f iM. "\ 2p_ '"'I VWP + 2 j^L iv.tr < 3CV«^P + 2C1PVvp,
(by uj(a:) > uj(y)), (7.66)
where C = maxjT«(a;)m(a;)1. Hence for v £ Co'7(fi),
f Jn
G(x,y)vu\'(1p)dyeC^p(U).
By the inequality (7.66), we can see that if Vk —> 0 (A; —> oo) in C*o'7(fi), then «i ~ • w*:cp —> 0 (fc —» 0), by the Schauder estimates, one can deduce that lim fvk = lim / G(x,y)ur (1 ~ p) ^fedy  0
fc—»oo
fc—+oo
in
C^lty.
JQ
Thus, the operator T is continuously differentiable at v = 0, and T'(0) = f. Hence we have id  T'(0)  F\(0)\\=o = id  f, and the equation corresponding to id — XT is as follows f  At; =  ^  i / , < p < ( v\dn = 0.
x € SI
(7.67)
By the KreinRutman theorem, the first eigenvalue Ai of (7.67) is simple and corresponds to a positive eigenfunction, and there are no positive eigenvalues which correspond to the eigenvalues A^ (fc = 2,3, • • •) of (7.67). Because A = p~l is the eigenvalue of (7.67) which corresponds to the positive eigenfunction u*(x), (7.67) has no eigenvalue in (O,;?"1), which implies
228
Bifurcation Theory and Applications
that the operator id _ r'(0)  F A (0) A=0 = id  f : C^(U) > C^(Ti) is invertible. By the implicit function theorem, we can see that near (v, A) = (0,0) there is a unique branch of solution of (7.64), and for A > 0 small enough, the solution of (7.64) in the branch v(x, A) is C 1 in A in the norm topology of Co'7(H). Finally, we check that t;(z, A)C2,P = 0(r(A1/(1P>)). Let v(x,X) = h(X)w(x,A), h(X) = \\v(x,\)\\c.r, w(x,X) = \\v{x,A)^p x v(x,X). By the conclusion above, we know that limA_>0 h(X) = 0. By (7.64) w(x, X) satisfies f  Aw = X1^1P)h1(X)g(x, ux,Vux, V2uA) + h^iX)^ l
l ft^AJuJ"+ \*hh (\)f(x, { w\en = 0.
+ v]'
2
ux,Vux, W ux),
(7.68) If h(X) ^ OiriXWri)), then by (7.61) and the definition of r(A), there is a subsequence of {A}, that we still denote it by {A}, such that in CP(Q) lim [X1^1^h1(X)g(x,ux, VuA, V2^A) ° + X'Thh\X)f(x, ux, Vux, V2ux)) = 0.
A
(7.69)
We need to show that in CP(Q,) lim [JT^AXuJ + v(x, A))p  /i~1(A)u*p  pu^wix,
A)] = 0.
(7.70)
A—•()
In fact, we only need to verify that in C°(Q)
(7.71) where d(x) = dist(x,dCl). Because ^ and ^ are uniformly bounded in C°(f2), noting that limA_,o MA) = 0> w e have
lim (^Y r(l + ^ A ( x ) r  l  ^ ( A ) z A ( x ) 1 = A»o \ d J [ h(X) J where zx(x) = Hd^l. This means (7.71) holds. Since u>Ac2.p = 1; {w\} n a s a convergent subsequence in C2(fi), that we still denote it by {wx}. If wx > iy0 ^ 0 in C2(fi), by (7.69) and
229
Bifurcations for Nonlinear Elliptic Equations
(7.70), from (7.68) one deduces that w0 satisfies (7.67). And we read a contradiction. If w\ —> 0 in C2(Q), then by the inequality (7.66), we have that
lim wx(x) • wp (1 ~ p) (x) = 0 in Cp(ty and by (7.70), it implies that lim A^AJKu! + vx)p  u{p\ = 0 in C P (Q). A—y\j
And from (7.69) and the Schauder estimates, it follows that
U>AC2>P
—*
2
0 (A —» 0), it is contrary to IUAC P — 1
Hence h(X) = \\v(x, A) ca , P = O(r(A1/(1~P))). The proof is complete.
Proof of Theorem 7.22. Prom Theorem 4.10, by using the same methods as in Theorem 7.21, one can derives the claim i). And we only need to verify the claim ii). As the proof of Theorem 7.21, we can show that if mru»ou.\ = 0 in 2 7 C ' (fi), 0 < 7 < 1, and u* satisfies (7.60) with 1 < p, then u\ must be the form expressed in the claim i). By theorem 7.19, there is a real number p0 > 1 such that for any 1 < p < p0, the problem (7.62) has only one solution u\, and the following problem ( Aw ^pu^'w,
xeil,
\ Han =0
(7.72)
has no nontrivial solution. Putting u\ in (7.60), v(x,X) satisfies (7.64) with p > 1. Making the transform of A = t~l in (7.64), we have /  Av = K + v)p ~ u*/ + F(x, t.v, Vv, V2v), \ v\dn = 0,
(7 73)

where F{x,t.v,Wv,V2v) = t~p^g(x, t ^ r (u* + v ), t?^r (Vui + Vu), t^ (V2itj + V2v))
+ tp**f(x, t& («I + v), t& (V«I + Vv), t^ (V2ul + V2v)).
230
Bifurcation Theory and Applications
In view of (7.61) we can see that Fj(x>0,0>0) = i^(i,0,0 I 0) = i^(x,0,0,0) = 0. Denote by id — T — Ft : C2'7(fi) —> C2'7(f2) the operators corresponding to the equation (7.73) and Tv= I G(x,y)[\u\+v\>ul]dy, Jn Ftv = / G(x, y)F(y, t, v, Vv, V2v)dy Jn where 0 < 7 < p — 1. It is clear that id — T — Ft are continuous and differentiable, and id  T'(0)  F(=o(0) = idT: fv=
C2'7(fi) + C2^(Ti),
f G(x,y)pu*1p1vdy. Jn
Since 7 < p — 1, by the Schauder estimates, we deduce that T : C2'7(Q) —> C2'7(J7) is a compact operator. On the other hand, id — T corresponds to the equation (7.72) which has no nontrivial solutions for 0 < p < po. Hence id — T : C2i7(fi) —> C2'7(f2) is invertible. By the implicit function theorem, we get the claim ii). The proof is complete. Now, for the problem ( Au = g{u) + X(up + /(u)), \ u\an =0,
xen,
u > 0 in Q,
(7.74)
with the condition: (A1) \g(z)\,\f(z)\
Bifurcations for Nonlinear Elliptic Equations
Proof.
231
In fact, we only need to check that for the problem
I  Au = up + F(t,u), \ u\aa = 0, u > 0 in Q, and under the conditions (7.61) and (^4i)(^43), there is a priori bounds of solutions uniformly for 0 < t < 1, where F(t,u) = t~^g(t^u) +t~p^f(t^u) And the uniformly priori bounds of solutions can be deduced from the results in [de Figueiredo et al., 1982]. • Remark 7.4 In (7.50), if the functions f,g are independent of V2u, namely /,g £ Cl(Q, x I x / n ) , then by Remark 7.3, one can get the conclusions that for any p (1 < p < ^±2) there is a solution u\ of (7.52) such that near (u, A) = (0, +oo) there exists at least a branch of solutions of (7.50), and for A > 0 large enough, the solutions (u\, A) in the branch satisfy
J ux(x) = A^rtuifc) + xWrivfa A), 1
lim t;(x > A) ca =0.
^ At+oo
Remark 7.5 It is well known by the Pohezaev's inequality that if Cl c Rn is a star shaped domain, then the below problem
\ u\an = 0 ,
u > 0 in fi
has no solution in C2'7(fi) (0 < 7 < 1). Therefore for the star shaped domains fi C Rn, the critical index p\ = ^~ in (7.50) is essential. But if the domains Q satisfy Hq{Q.,z) / 0 for some q > 0, then the restriction P < S^l i n (750) ma Y b e relaxed. 7.4.3
Global bifurcation from the sublinear terms
Give a quasilinear elliptic operator Lu = —aij(x,u,Vu)DijU + bi(x,u,Wu)DiU + c(i,u,Vu)«.
232
Bifurcation Theory and Applications
We consider the global bifurcation of the below problem
(Lu = g(x, u, Vu) + X(ua + f(x, u, V«)), \ wan = 0, u > 0 in 0,
(7.75)
where 0 < a < 1, and a^, 6j, c, f, g £ C 7 ^ x I x In), 0 < •y < a. Suppose that (Ai) a,ij(x,z,£) — a,ji(x,z,£), and x, we have
where (3 > 0 is a constant. (A2) For any 2 > 0 , ( e I " and x 6 0, we have [f{x,z,£), g{x,z,(), c(x,z,£)>0, \/(a: l 0,0)=0 1 5(x,AzlA^)=O(A). Theorem 7.24 Under the conditions (Ai) and (A2), Ao = 0 is a unique bifurcation point of (7.75). Moreover, if the following problem has no solu
tion in C2'T(0)
{
Lu = yV q(x,u, Vu), 7
(7.76)
u\dn = 0 , u > 0 in 0
and E = {(u,A) e C 2 ' 7 (Q) x i?+ I (u,A) satisfies (7.75)} iften i/ie connected component C of E which contains (u, A) = (0,0) is unbounded in C 2i7 (fi) x i? + , and for any (u\,X) GT, we have the estimates C\T±=
<\\u\\co
(7.77)
where C > 0 is a constant independent of A.
Proof. By the theory of linear elliptic equation, it is known that for h G C 7 (n x l x l " ) and any v £ C 1 ' 7 ^ ) , the following problem has a unique solution u G C2'7(fi)
{
 aij(x,v,Vv)DijU + bi(x,v,Vv)DiU + c+(x,v,Vv)u
= h(x,v,Wv),
u\an = 0 (7.78)
233
Bifurcations for Nonlinear Elliptic Equations
where +i
( c (z,z,£), for z > z 0 , [ 0, for z < z0
^
and c(x, zo,£) = 0. We define Th : C 1  7 ^) > C 1  7 ^) by T7i(t>) = u. By Theorem 10.4 in [Gilbarg and Trudinger, 1983], Th is a compact mapping. It is easy to see that for h\,h2 € C7(J7 x l x R 2 ), we have T(\iht + X2h2)(v) = XtThily) + \2Th2{v). On the other hand, for each v e C 1>7 (0), there is a Green function Gv{x, y) such that the solution u of (7.78) can be expressed as follows u= I Ja
Gv(x,y)h(y,v,'Vv)dy
where Gv(x,y) is a nonnegative symmetric function [Krasnoselskii and Zabreiko, 1984]. By the uniqueness of (7.78), we have Th{v)=
f Jn
Gv{x,y)h(x,v,Vv)dy.
Thus the existence of (7.75) is equivalent to the existence of nontrivial solution of the following equation in C 1 ' 7 ^ ) , u = Tg+(u) + \Tf1(u) + XTf2(u),
(7.79)
where
7i(z,uJvu) = Ma> \ t t I
v? ^
*+/
n^
j f{x,u,Vu), ^0,
u>0, u < 0.
By Theorem 4.9, we only need to prove that there is a nondecreasing function r\ > 0 of A > 0 such that for any t > 0 the following equation has no solution in Brk, u = XTfi(u) + Tp(u) + tTK,
(7.80)
and when t = 0, (7.80) has no nonzero solution in Brx, where K e C(Ti) and P G C7(f2 x E x R n ) are respectively arbitrary positive function and
234
Bifurcation Theory and Applications
nonnegative function, and 5 + = {u G Cl>i(p)  u C i., < r 7 , u > 0 in Q}. Suppose that uA € i ? ^ (uA 7^ 0) satisfies (7.80), then from TP(ux) > 0 and TK > 0 we can derive that ux(x) >X f Gux (x, y)u$dy, Jn On the other hand, (7.80) also implies that
x £ Q.
/ L«A = A< + P{x, ux, Vux) + K(x) > 0,
{ ux\au = 0.
(7.81)
(7.82)
By the strong maximum principle, from (7.82) we get ux(x) > 0 in O.
(7.83)
From (7.81), for any open subset fi CC fi, we have
I ux{x)dx >X [ «X(x) [ / Gux(x,y)dy] dx h Jn L/n J
(784)
>Ainf / G u , ( x ) 2 / ) ^ • / uldx. yen u n J Jn Let uA co = £. We want to probe that there is a j3e > 0 such that inf [_G^{x,y)dy>pe.
(7.85)
yznJn
Taking e G C^°(fi) such that 0 < e(x) < 1, e ^ 0 and supp e C ( i , then the solution of the below equation
{
 a,ij(x, ux, Vux)DijUe + bi(x, ux, Vux)DiUe + c(x, ux, Vux)ue = e(x), we an = 0
can be expressed by ue{x)=
I Gux(x,y)e{y)dy Jn
= I Jh
GUx(x,y)e(y)dy.
Hence we have
f Gux(x,y)dy > I Gux(x,y)e(y)dy = ue(x).
Jn
Jn
(7.86)
Bifurcations for Nonlinear Elliptic Equations
235
By the strong maximum principle, from ft CC 0, it follows that there exists a constant j3e > 0 such that infue(s)>/?e.
(7.87)
(7.86) and (7.87) imply that (7.85) holds. On the other hand, let ux > 0 (A > 0) in C ^ f i ) . Erom the proof of Theorem 10.4 in [Gilbarg and Trudinger, 1983], we see that ue = Te(ux) > Te(0) (A > 0) in C^ity
(7.88)
where wo = T"e(0) is a solution of the below equation f  ai:, (a;, 0,0)D ijU + b^x, 0,0)Aw + c(x, Q, 0)w = e(x), \ wan = 0. It is easy to see that inf / G0(x,y)dy > inf uo(x) > /30 > 0. By (7.85) and (7.88), we know that there is a constant j3 > 0 independent of A such that
inf JGUx{x,y)dy > 0 > 0. Prom (7.83) and (7.84) it follows that (ux{x)dx Jn
> A/3 • [_v%(x)dx> \p\\ux\\co1 [ux(x)dx. Ja Jn
By (7.83) / ux{x)dx > 0. Ja Thus we obtain AA^<Kc°
(A/?1^)
(7.89)
Therefore (7.89) is a necessary condition for the existence of (7.80) and (7.79). Let rx — ^ A 1 ^ (0 < fo < Pi) Then rx > 0 is the nondecreasing function in Theorem 4.9. This theorem is proved. •
236
Bifurcation Theory and Applications
Corollary 7.1 7/fi C Rn (n > 3) is a star shaped domain, then Ao = 0 is a unique bifurcation point of the below equation r  Au = u" + X(ua + f(x, u, Vu)), \ u\dn = 0,
u > 0 in f2,
(7.90)
where p>n+^—2, 0 < a < 1. Moreover, the connected component C of T, containing (it, A) = (0,0) is unbounded in C 2 ' 7 (fi) x M+, where E = {(tt,A) € C2)7(fi) x M+  (u,A) safe/ies (7.90)}. Proof.
By the Pohezaev inequality, the equation
{
2
 A n = up, p>n+
u\dn = 0,
2,
n
u > 0 in il
has no solution, this corollary follows from Theorem 7.24. 7.4.4
Global bifurcation from the linear
•
terms.
Now we consider the global bifurcation of the below problem  Lu = g(x, u, Vu) + A(w + f(x, u, Vu)), \ u\dn = 0,
it > 0 in £1.
(7.91)
Let Ai be the first eigenvalue of the following problem
{
 aij(x)DijU + bi(x)DiU + c(x)u — Au, uan = 0,
where aij(x) = a,ji(x), c(x) > 0, a^, bu c £ C(Ct), 0 < 7 < 1, and there is a constant /? > 0 such that
Lemma 7.1
Ai satisfies the following inequality 0 < Ai <
. .
where G(x, y) is the Green function.
1
.
„.
rr
(7.92)
Bifurcations for Nonlinear Elliptic Equations
237
Proof. It is well known that G(x,y) is a symmetric positive function and the eigenvector ui(x) corresponding to Ai is positive which satisfies that ui(x) = Xi
Jn
G(x,y)ui(y)dy.
For any f!0 C fl we have .
/no "!(*)<**
= x
SnM )
f
1—' [fno ( >y)dy\dx G x
Because ui(x) > 0 for x G Q, we derive that
f Ui x
1 G x
d
dx
Jno ( }[fno ( >y) y\ <
F^
j , vnoca
infxefio [/n0 G(x, y)dy\ Hence the inequality (7.92) holds. The proof is complete.
•
For the problem (7.91) we assume that (B\) a,ij(x,z,£) satisfy the condition (Ai) in Theorem 7.24 ; (B2) dij, bu c, f, g£ C^(Q x R x R"), 0 < 7 < 1, and
0 < f{x, z, ft, g(x, z,0, c(x, z,0 Vz > 0, £ G R" /(ar > 0,0)= 5 ( a :,0,0) = 0f (B3) there is a constant a > 0 such that lim
2>o,c—0
fl(».^0=a. z
Denote by Ei = {(«,A) G C2T(fi) x R I («,A) satisfy (7.91)} Let X\(v) be the first eigenvalue of the following problem
i
 aij(x,v,Vv)DijU + bi(x,v,Vv)Diu + c+(x,v,'Vv)u = Xu, u\dn = 0.
Ai(0) is the first eigenvalue of (7.93) with v = 0.
(7 93)
'
238
Bifurcation Theory and Applications
Now we are in a position to state and prove our main theorems in this subsection. Theorem 7.25 Under the assumptions (B{)(Bz), if X\(0) > a, then there is a real number Ai > 0 such that the problem (7.91) has at least a bifurcation point in [0,Ai]. Moreover, there is a bifurcation point A* £ [0, Ao] such that the connected component C of Si which contains (u, A) = (0,A*) satisfies at least one of the following i) C is unbounded in C2'7(fi) x l ; and ii) C contains (uo,O) where UQ satisfies
{
Lu = g(x,u,'Vu) u\en — 0,
u > 0 in Q,.
Theorem 7.26 Under the hypotheses of Theorem 7.25, if f(x,\z,\£) = o(A), then the bifurcation point A* £ R of (7.91) is unique, and A* = Ai(0)a. Proof of Theorem 7.25. For a function h e C~*(Q. x R x R") we define the mapping Th : C1''y(fi) + C 1 ' 7 ^ ) as in Theorem 7.24. We consider the following equation u = Tg+(u) + XTu+ + XTf+(u)
(7.94)
where Tu+ = f Gu(x,y)u+{y)dy. Jn We shall apply Theorem 4.9 to prove this theorem. It is suffices to check the conditions (A\) and (A2) in Theorem 4.9. Take Ao = 0. By (B3), g+(x,u,J\/u) — au+ — O(UCI.Y), and a < Ai(0), hence there is an e > 0 such that for any 0 < t < 1, the equation u = tTg+(u) has no nontrivial solution in B£ = {v £ C1'7(fi)  fci.y < £} therefore (A2) is verified. Next, we need to check the condition (Ai). It is known that for any A > Ai(t;) and q(x) > 0, x £ Q, the following problem has no solution in C2'7(fi)
{
— aij(x,v,'Vv)DijU + bi(x,v,Vv)DiU + c+(x, v, Vv)u — \u = q, «an = 0,
u > 0 in Q.
Bifurcations for Nonlinear Elliptic Equations
239
Let Ai = [infxen faGo(x,y)dy] , fi CC £1 an open subset. We shall show that Ai (0) < Ai < +oo . By Lemma 7.1 we can obtain that Ai(0) < Ai. Similar to the proof of (7.85), one can derive that Ai < +oo . Define the mapping A(v)=
[Gv(x,y)dy. Jo.
If we can prove that A(v) is continuous, by Lemma 7.1, it is easy to get that for any A > Ai there exists r\ > 0 such that when v G C1>7(f2), IMIc1^ < r7i w e have
and obviously r\ > 0 is an increasing function of A > Ai. Using the method as in Theorem 10.4 in [Gilbar g and Trudinger, 1983], one can prove that when vn(x) —> vo(x) in Cln(Q), / GVn(x,y)dy > / GVo(x,y)dy Jn Jn uniformly for x 6 ft. Hence A : C 1'7(f2) —» C(il) is continuous. By the proof above, we see that for any A > Ao. the following equation has no solution in BTx u = Tg+(u) + XTu+ + XTf+(u) + q{x) where q G C7(i7), q > 0 in Q is arbitrary . The condition (yl2) is verified, therefore this theorem is proved. The proof of Theorem 7.1 is obvious, here we omit the details. Example 7.1
{
We consider the following problem  Au = up + Au + AVuQ, u\an = 0, u > 0 in Q,.
0
1 < p,
(7.95
By Theorem 7.25, we can get that the problem (7.95) has a bifurcation point Ao > 0, and if p > ^  (n > 2), 0 is a star shaped domain, then there exists a bifurcation point A* > 0 such that the branc h of (7.95) connected to (it, A) = (0, A*) is unbounded in C27(Q) x R. Let Ai be the first eigenvalue of —A, we have
240
Bifurcation Theory and Applications
Corollary 7.2 problem
A* = X\ — 1 is a unique bifurcation point of the following
{
 Au = Xu + ueu2, u\9n = 0,
x e £2 c R"
u > 0 in fl
(7.96)
and the branch of (7.96) from (u,X) — (0,A*) either is unbounded, or for any X € (—1,A*) the problem (7.96) has a solution. Corollary 7.3 IfCt C M.n (n > 3) is a star shaped domain, then X = X% is a unique bifurcation point of the following problem Au <
= Xu + up,
u\on = 0, u > 0 in fl
p>—, n
( 7  97 )
and the branch of (7.97) from (u, A) = (0, Ai) is unbounded in C2'"i(?L) x R.
Remark 7.6 In Theorem 7.25, the coefficient f(x, z, £) is not required to be the higher order function on z and £ near (z, £) = (0,0). Hence the global bifurcation in Theorem 7.25 can not be obtained from the Rabinowitz's global bifurcation theorem. 7.5
Notes
7.1 The material in this section can be found in many basic books on Sobolev spaces and partial differential equations; see, among many others, [Adams, 1975; Lions, 1969; Mazja, 1985; Gilbarg and Trudinger, 1983; Agmon, 1959; Agmon et al., 1959; Agmon et al., 1964; Aubin, 1982]. 7.2 This section is based on the authors' recent work on bifurcations from higher order terms regardless of the multiplicity of the eigenvalues. The results presented here are new. 7.3 Theorem 7.13 was proved by the first author's Ph D thesis [Ma, 1990]. 7.4 This section is based on Ma's Ph D thesis [Ma, 1991a; Ma, 1991b], under the supervision of P. L. Lions. He would like to use this opportunity to express his deep appreciation to Professor Lions for his valuable guidance.
Chapter 8
React ionDiffusion Equations
8.1 8.1.1
Introduction Equations and their mathematical
setting
Reactiondiffusion equations are basic equations in many problems in science and engineering. The general form of reactiondiffusion equations is given by CJ1J
— =AAu + Bxu + D\Vu + G(x,u,Vu),
x e fl,
OX/
' u\m=0
for £ \
u(x, 0) =
( 8 D
=0),
On an
J
Here fi C M"(n ^ 1) is an open set, and u = (ui, • • • ,um) (m ^ 1) is a vector function. Other notations are given as follows. (1) A is a positive diagonal matrix of diffusion coefficients given by
o ••. 0 h
A=
\ 0
• ••
(82)
fim)
where /ij > 0 (1 < i ^ m). (2) B\ is an m x m parameterized matrix
(
611(1,A) ••• 6 l m (i,A) \ bml(x,X) 241
•••
bmm(x,X)/
242
Bifurcation Theory and Applications
(3) The operator D\ • V is defined by
(
m
n
„
m
n
a
\
ES>«£EE*«£ •
where faj(x, A) and dkij(x, A) depend continuously on (a;, A) G O x R1. (4) G = (Gi, • • • , G m ) is a continuous vector function defined on J7 x Rm x M mx " satisfying
G(*,£,C)=o(£,C)We set
r
(
i7 = L 2 (O,R m ),
ff^^n.R^n^cn.R"1), for^1 = (ueF2(fi,Rm)  ^ =o}V
. V
[
on en
(83)
)J
It is known that the linear operator A : Hi —> i? defined by >lw = —J4AM is a sectorial operator satisfying (3.27), and Ha = D{Aa) = W2a'2(Q.)(0 < a < 1). Obviously the linear operators BA,D : Hi > ff defined by SAu = 5 A u, Vu = D\ Vu are bounded. Therefore, by Theorem 2.6, the linear operator L\ : Hi —> H defined by L\u = AAu + B\u + DxVu is a sectorial operator, and for any a € R1, Ha = D(L"). For 0 ^ a < 1, Ha enjoys the following properties (HacWk'q(n)
{
[HacC°'5fl)
if
fcn/<7«C2a^,
n
(84)
if 0 < < K 2 a   .
Lemma 8.1 Let fi C R" 6e Lipschitz, and L : WmP(Q) > Z,P(ft) a sectorial operator for some m > 1 and 1 ^ p < oo. Then for 0 ^ a ^ 1 £/ie spaces X a = D(La) satisfy Xa c Wk' q{0) X°cCM(fl)
if k n/q < ma  n/p, q^p, if Q^k + 5<man/p,
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ReactionDiffusion Equations
and the inclusions are continuous. 8.1.2
Examples from Physics, Chemistry and Biology
FLOW OF ELECTRONS AND HOLES IN SEMICONDUCTORS: Let V be the
electrical potential, u and v represent the concentrations of electrons and holes respectively. The equations describing the flow of electrons and holes in a semiconductor are written in the following form: ' AV = a(uv A), du « — = jiiAu  ftdMu • VIO  Fi(«.«). dv — = /x2Av + (32div(v • W ) + F2(u, v),
(8.5)
V OX
where A, a, fx^, /i 2 , /3i, @2 are positive constants, and F\,F2 are the recombination rates, i^(0, 0) = 0 (i = 1,2). Equations (8.5) are supplemented with the Dirichlet boundary condition V\an = 0,
u\9n = 0,
v\9n = 0.
(8.6)
Let G(x, y) be the Green function of  A . The problem (8.5) with (8.6) is rewritten as ' du — = /iiAu + p\u — u\u{u — v) — criVu • VF — Fi(u, v), i
dv
— = /J 2 AU  p2v + cr2v(u — v) + cr2Vu • VV + F2(u, v), . u\m = 0,
V\OQ
= 0,
where pt = Aa/3j > 0, <7j = afc > 0 (i = 1,2), and
V= [ G(x,y)(X + vu)dy. Jn If Fi and F2 are Cr(r > 1), then
{
^1(21, ^2) = anz! + a12z2 + gi(zi,z2), F2(zi,z2) = a2izi +a22z2 + g2{zi,z2), gi(z1,z2) = o(\z\), t = l,2.
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Bifurcation Theory and Applications
CHEMICAL REACTIONS IN CATALYST: Let Si,ical species involved in k independent reactions,
, S m be the m chem
m
J^aijSi^Oil^j^k), i=l
Vi the concentration of Si and T the temperature. The reaction equations read k d ^ = HiAvi + ^TaijFjivi,
(1 < i < m),
,vm,T),
— = KAT J2Y1 "oA^^i, • • • , I'm, T), j=li=l
where «,//»,ay are constants, Fj(vi, • • • ,vm,T) is the rate of the j — t/i reaction, Fj(0,••• ,0,T) = 0, and /3j the partial molar enthalpy of the i — th species, which is a constant. The boundary conditions are as follows
JT\dQ = O (or an^ = 0 ) , < [wisn=0
an (1 < z < m).
EQUATIONS FROM CHEMICAL REACTOR THEORY: The equations are
given by
{
du
d2u
.
.
~dt = / U l 9 x 2 +a^u + a^v + 9i(u,v), dv d 2v . . where /xi, /i2 > 0, a,j are constants, 0 < a; < 1, and gi(u,v) are C°° functions satisfying 5i ( w , w )
= o(M 3 + M3).
The boundary conditions are given by u — v = 0 at x = 0, a; = 1.
245
ReactionDiffusion Equations
THE HODGKINHUXLEY EQUATIONS: The equations describe the nerve impulse transmission, and are given as follows.
<9V ox* d2u2
(dUl at du2
. .,, , .
>
^=M2^+/l(«l)(^K)« 2 ).
(
du3 ~oT =
9u4 ~dt
d2u3 M3 1^2~
(g7)
^("O^altii)  «3),
2
d ui "ax2"
= M4
~ W4)>
f^vv1^1)
where 0 < x < L, fi(z) > 0, 1 > /i»(z) > 0, i = 1,2,3, and 9i — 71^2^3(^1  *i) + liu\{ui + 52) + 73(ui  S3), H >0,5i> 0. In this model, u\ is the electric potential in the nerve, and u?,113,114 are chemical concentrations. The boundary condition is Ui(0) = Ui(L) = 0 ,
1 < i < 4.
(8.8)
Let problem (8.7) and (8.8) can be equivalently rewritten in the following form
1 u(0) = u{L) = 0, where u = (ui,u2,u3,u4), B = (bij(x)), ,
hk=z~ uu
k
bjk =
u = v
^—[fj(u1)(hj(u1)uk)]u=v,
<>Uk
for 2 < j < 4, and G(u) = o(u). GINZBURGLANDAU EQUATIONS: The following equations arise in the study of superconductivity of liquids. They read
(
2
du
(8.9) =AAu
+
u\u\
u,
xen,
( g 9 )
u\dn = 0, where Q C K" is a bounded open set, u = [u\, • • • , un), and A is as in (8.2).
246
Bifurcation Theory and Applications
8.2
Bifurcation of ReactionDiffusion Systems
In order to show how the abstract bifurcation theorems in Ch.4 and Ch.6 work, in this section, we consider only the bifurcation problem for the following simple equations —j = Aui +Aui +cii 2 + Gi(ui,u 2 ), < QT = Au2 + \u2 + G2(m,u2),
(8.10)
(wi,«2) an =0,
(u1,u2){x,0)
= (cpi,
n
where $7 C M (l ^ n < 3) is a C°° bounded open set, c is a constant, A 6 R1 is a parameter, and Gi{u\,u2)(i = 1,2) are C°° function, which can be expressed as '
r
Gi(ui,u 2 ) =
^2giP(ui,u2), p k
: G2(ui,u2) = ^2g2P(ui,u2),
(8.H) (2 < k < r ^ oo),
v=k
and giP(ui,U2) (i = 1,2) are pmultilinear functions. Let Hi and H be as in (8.3) with m = 2. The equations (8.10) can be rewritten in the following abstract form
{
du
,
_,, .
(8.12)
Tt=L^ + G ^ (8.12) u(0) = if, where the operators L\ : R\ —> H and G : H\ —* H are defined by L\u — (Awi + Aui + cu2, Au2 + Au2), G(u) = (Gi(ui,U2),G2(«i,ti2)).
In Section 8.1, we know that L\ is a sectorial operator, and by (8.4), it is easy to see that G : Hy > H (7 > 3/4) is C°°. 8.2.1
Periodic solutions
When /9o > 0 is a simple eigenvalue of (7.6), by Theorem 7.15, if a • c = bko • c < 0, the equations (8.10) have no steady state solutions bifurcated
247
ReactionDiffusion Equations
from (0, po). However, as complement of Theorem 7.15, from Theorem 6.13, we obtain the following theorem, showing that (8.10) bifurcate from (0, po) a periodic solution provided a • c < 0. Theorem 8.1 Let po > 0 be a simple eigenvalue of (7.6). If k ^ 3 is an odd number and a • c = b^o • c < 0, then the equations (8.10) bifurcate from (u, A) = (0, po) to a periodic solution. It is interesting to investigate the stability of periodic solutions for (8.10). The following theorem provides a sufficient condition for the bifurcated periodic orbit being an attractor. Theorem 8.2 Let po > 0 be the first eigenvalue of (7.6), and k — odd, bko • c < 0. / / dlvGk(x1,x2)
=^
+^
OX\
<0
forx^O
small,
(8.13)
OX2
then the bifurcated periodic solution of (8.10) from (0, po) is an attractor. Proof.
By the attractor bifurcation theorem (Theorem 6.1), it suffices to
show that u = 0 is locally asymptotically stable for (8.10) at A = po By reducing to the center manifold, we only need to prove that x — (x\, £2) = 0 is locally asymptotically stable for the following equations —
f dx
=cx2+gi{x1,x2),
( 8  14 )
. . 2 £j = 92(xi,x2), where 9i(xi,x2) = agik(xi, x2) + o(\x\k), g2{x1,x2) = ag2k(x1,x2) + o(\x\k), a=
f efe+1(a;)da:, Jn
and e(x) > 0 in fi is the first eigenvector of (7.6). By (8.13), x = 0 is an isolated zero point of divG^. Because divGfc is a (k — l)order homogeneous function, there is a constant 5 > 0 such that 8rk~l < \dwGk(rxurx2)\,
V x\ + x\ = 1, r > 0.
Hence Tp + rp < 0,
V x\ + x\ < £,
x =£ 0,
£ > 0 small.
(8.15)
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Bifurcation Theory and Applications
From the proof of Theorem 6.13, we know that x = 0 is a degenerate singular point of (8.14), which is either (a) a stable focus, or (b) an unstable focus, or (c) a singular point having infinite number of periodic orbits in its neighborhood. In view of (8.15), both cases (b) and (c) can not occur. Therefore x — 0 must be locally asymptotically stable for (8.14). The proof is complete. D
8.2.2
Attractor
bifurcation
Let po > 0 be the first eigenvalue of (7.6), c = 0 in (8.10) and k = 2 in (8.11). For the coefficients a^ and btj in (7.5), we assume that a2Of3i > 0,
a 20 /? 2 < 0
(or
62o/?i < 0,
&20/32 > 0)
(8.16)
where Pi = b2Oa? + bncti + b02,
* = 1, 2
—±1 2a 20 (or fa = CL20o% +ancti + a 02 ,
"i > 2 =
"1,2 =
)L
fen ± \/b\x — 462O&o2 fell ^T1
(an  4a 2 0 a 0 2 > 0) i = 1,2, ,2
,, , n\  4fe2Ofe02 > 0).
By Theorems 6.9 and 6.11, we obtain immediately the following theorem, which shows that (8.10) bifurcate from (0,po) to an attractor. Theorem 8.3 hold true.
Let the condition (8.16) hold. The following
assertions
(1) Equations (8.10) bifurcate from (0, po) on A > po an attractor A\ with dimA\ < 1, and A\ attracts a sectorial domain Dr(8) C H with some angle 9 (n < 9 sj 2TT) and radius r > 0. (2) The attractor A\ contains minimal attractors which are singular points, as shown in Figure 6.5(a)(c). In Section 8.1.2, we see that many models arising in physical, chemical and biological problems require the unknown functions u»(l < i < m) are nonnegative. Hence, it is interesting to consider the bifurcated attractors of (8.10) which contain nonnegative functions. To this end, we assume that
Van, z 2 > 0
(* = 1.2).
(817)
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ReactionDiffusion Equations
Theorem 8.4 Under the hypotheses of Theorem 8.3, if (8.17) holds true, and the following two algebraic equations a02z3 + (an  bO2)z2 + ("20  bn)z  b20 = 0, 3
&202 + (&n  a20)z
2
+ (b02  an)z  a02 = 0,
(8.18) (8.19)
have only positive solutions z > 0, then the functions (ui,U2) in the attractor A\ of (8.10) bifurcated from (O,po) are positive, i. e. u\ > 0, u2 > 0 in
n.
Proof. form
The bifurcation equations of (8.10) are reduced to the following £• = (A  po)xi + agu(xi,x2) <^ k
= {XPo)x2+ag22(x1,x2)
a=
Jn
+ o(\x\2), + o{\x\2),
(8.20)
e3(x)dx > 0
The functions (1*1,1/2) i n attractor A\ of (8.10) can be expressed as Ui = Xie(x) + o(\x\),
i = l,2,
{xi,X2) —»0 as A —»/9o, where(a;i,a;2) are in the attractor ,4A of (8.20) bifurcated from (x, A) = (0, po) Hence, we only need to prove that xi > 0,
x2 > 0,
V (zi, a;2) G A
(8.21)
Because (512,522) is 2order nondegenerate at a; = 0, it suffices to consider the equations
{
—A = ( A  p o ) ^ ! + a5i 2 (a:i, x2), dx
(8 22)
'
 ^ = (^  Po)x2 + ag22(xi,x2). By (8.16) and Theorem 6.9, the vector field (512,522) has at least two straight line orbits and at most six straight line orbits, connecting to x = 0, and half of which reach to x = 0, the other half depart from a; = 0. We denote the straight orbits reaching to x = 0 by L t , the straight orbits departing from x = 0 by Lj. The domains P+ and P~ enclosed by L~l, Li[ and Lj~, L~ (i = 1 or 3) respectively are the parabolic domains of (512,522) at x = 0, and all orbits in P+ reach to a; = 0, in P~ depart from
250
Bifurcation Theory and Applications
x = 0. The both domains Ex and E2 (Ei U E2 = M2 \ (P+ + P~)) are elliptic domains. For the straight lines Lj +LJ : ajiXi = aj2x2, the number Zj = ctji/aj2,aj2
^ 0 ( resp. Zj = a^/a^ijOji ^ 0) satisfies
the equation _ X2 _ g22(xi,x2) _ b02z2 + bnz + b20 xi 9\2{x\,x2) a02z2 + anz + a20 ( xi gi2 a20z2 + auz + aO2\ \
resp. z = — = — = r— — , x2 P22 o20z2 + bnz + b02 J
which implies that Zj = a.j\/a.j2 ( resp. Zj = ctj2/ctji) satisfies (8.18)(resp. (8.19)). Hence, from (8.17) and the condition that (8.18) and (8.19) have only positive solutions it follows that all straight orbits L+ of (5112,922) are in the domain M+ = {(£1,2:2)1 X\ > 0,x2 > 0}. Thus, when A > po all singular points of (8.22) must be in Lj~, and the attractor A of (8.22) bifurcated from (0, p0) satisfies that A C P+ C R+, which implies (8.21). • The proof is complete. Remark 8.1 The conditions that a.ij,bij < 0(i + j = 2) are equivalent to (8.17), and the conditions b2o, ao2 < 0, a n — b02 ^ 0, bn — a2o ^ 0, a02, b20 < 0 imply that (8.18) and (8.19) have only positive solutions. We now consider the case where the index of (<7i2> 522) at x = 0 is zero. Assume that a2n  4a2OaO2 < 0,
or ft • /32 > 0,
(8.23)
where ft and ft are as in (8.16). Theorem 8.5 Under the condition (8.23), if (8.18) (or (8.19)) has three real solutions, then the following assertions hold true. (1) Equations (8.10) bifurcate from (O,po) on X > po an attractor A consisting of a single singular point, which attracts a sectorial region Dr{6) for some 6(0 < 6 < n) and r > 0.
ReactionDiffusion Equations
251
(2) If (8.17) holds and (8.18) (or (8.19)) has only nonnegative solutions, then the singular point (ui,U2) («4 = {(1x1,112)}) is positive, i. e. u\ > 0,u2 > 0 in ft. Assertion (1) of Theorem 8.5 is derived from Theorems 6.9 and 6.11, and Assertion (2) can be proved in the same fashion as the proof of Theorem 8.4. 8.3
Singularity Sphere in 5""Attractors
In this section we study the set of singular points in a bifurcated attractor for the following equations £ = Alt + Xu  \u\2 u,
16SI,
(8.24)
where u = (u\,• • • ,um) is vector function, fi C M n (l ^ n ^ 5) a bounded open set, and A G M1 a parameter. Equations (8.24) are supplement with either the Dirichlet boundary condition: u\da = 0, or t h e Periodic
boundary
(8.25)
condition:
u(x + 2kn) = u(x),
k = (fci, • • • , kn), ki£Z
(8.26)
Equations (8.24) appear in some physical and fluid mechanical problems. When m = n, (8.24) are the equations given by (8.9), and when m = 2, (8.24) are referred to the GinzburgLandau equation. We can see that (8.24) are invariant under an action of orthogonal group O(m), i.e. under an orthogonal transformation u = Bu,
B G O(m) a unit orthogonal matrix,
the form of equation (8.24) is invariant. This property implies that the steady states of (8.24) occupy an (m — l)dimensional sphere. 8.3.1
Dirichlet boundary condition
We consider the problem (8.24) and (8.25). Let {pk\ 1 < k < 00} and {efe 1 ^ k < 00} be all eigenvalues (counting multiplicities) and eigen
252
Bifurcation Theory and Applications
functions of —A:  Aek = Pkek, 0 < pi < p 2 < • • • ,
'
e
fclan = °»
/ e^ej da; = <5y.
\ Jn Theorem 8.6 ing assertions.
For the Dirichlet boundary condition, we have the follow
(1) If X < p\, then u = 0 is globally asymptotically stable for (8.24) and (8.25). (2) If pi < X, the problem (8.24) and (8.25) bifurcates from (it, A) = (0,pi) to an attractor T,\ homeomorphic to an (m — 1)dimensional sphere S"™"1, and SA attracts H \T, where T C H is a manifold with codimension m. (3) S A consists of singular points of (8.24) and (8.25). As a corollary, when m = 2 corresponding to the complex GinzburgLandau equation, the bifurcated attractor SA = S1, consisting of only steady states. Proof. obtain
(1). Multiplying both sides of (8.24) by it and integrating it we
/ \u\2 dx =  [ [VM2  A \u\2 + u4l da:
~
2 at Jn JQI By the Poincare inequality and the Holder inequality / Vit dx ^ pi / u da;
Jn
Jn
/ H 2 da; < Q 1/2 • / \uf dx\ it follows from (8.27) that Au22^_filH42 which implies that
Urn «i a = 0,
forA
J
(8.27)
253
ReactionDiffusion Equations
for any u satisfying (8.24) and (8.25) with u(0) = u0. (2). We shall apply Theorems 6.1 and 5.11 to prove Assertion (2). It is known that the eigenfunctions of L\ = A + XI corresponding to f3\ (A) = A — p\ are given by Vi = (Snei,
1 < i < m.
,Simei),
By the canonical reduction to the center manifold, the bifurcation equations of (8.24) are written as g. = (\Pl)zizi dt
/ iTzjVjfeldx
+ oQzf),
•/« U.
(8.28)
l^i^m.
By Assertion (1) we know that z — (zi, • • • , zm) — 0 is globally asymptotically stable for (8.28) at A = px. Hence, by Theorem 6.1 the equation (8.28) bifurcate from (z, A) = (0, pi) to an attractor SA which attracts Mm \ {0}. We need to prove that SA = S"71"1. The vector field in (8.28) can be written as v = fj,zG(z),
G(z)=G3(z)
/i = A  / 9 i > 0 ,
z=
(zi,
,zm),
3
+ o(\z\ ),
G3(z) = (azi\z\2
,••• ,azm\z\2),
a =
Jn
efdx.
Obviously {G3(z),z) = a\z\4. By using Theorem 5.11 and Assertion (3), one can show that the bifurcated attractor Y,\ of (8.28) from {0,pi) on A > p\ is homeomorphic to an (m — l)dimensional sphere Sm~1. Assertion (2) is proved. (3). Let
{
u = (ui,
,um),
oo
Ui =
^2xikek.
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Bifurcation Theory and Applications
Then, the stationary equations of (8.24) can be equivalently expressed in the following form oo
(A  Pl)xu  ] T ftXik = 0 1 °° —^ Yl fixV
xik =
for 1 < i < m,
(8.29)
for 1 < i < m, k > 2,
(8.30)
where fk~
Jn
\u\2ejekdx.
For k > 2, let
Then for k > 2, (8.30) can be expressed as 1 °° Xik = Xa [Xifcli  — T ^ / ^ « . It is easy to see that for k > 2 and x = (xn, • • • , £ m i) € IRm,
M1
=
o(x2),
f;/^ = o(N4). 3=2
By induction, we define 1 ^ 
°° A
J=2
Then, we have r T ..i
_
V
^—^
f^ firi ...
ti
[xifcl^Odrrl21"). (8.31)
255
ReactionDiffusion Equations
Therefore, we infer from (8.30) that oo
(8.32)
Xik = xu J2 [xik]r . r=l
Putting (8.32) in (8.29) we obtain oo
oo
(A  pi)xa  xnfl  xu ] T YJ f* [Xik\r = °>
l
m

k=2r=l
Hence, (8.29) are referred to oo
oo
(Api)/ 1 X!EA f c [^fc]r = 0, K U m . 1
(8.33)
From (8.31) we see that [xik]r = [xjk]r , V 1 < i, j < m,
r > 1.
Hence, the equations (8.33) are the same, which can be represented by one equation. We note that m
fl=a(£xl)
+ o(\x)2),
E E / f M r = o(x4).
fc=2r=l
Thus, from (8.33) we get the steady state bifurcation equation of (8.24) near A = pi as follows m
^x?1+o(x2)=a1(Ap1),
a=
ejdx.
(8.34)
Obviously, as A — p\ > 0 sufficiently small, the set of solutions of (8.34) is an (TO — l)dimensional sphere. Therefore the set S\ of singular points of (8.24) and (8.25) is also a sphere S\ = Sm~x. Because S\ c SA, by Assertion (2) we have that S\ = SA = S m . The proof is complete. •
256
Bifurcation Theory and Applications
8.3.2
Periodic boundary condition
Theorem 8.7 assertions.
For the periodic boundary condition, we have the following
(1) If X ^ 0 , then u = 0 is globally asymptotically stable for (8.24) and (8.26). (2) IfX>0, then this problem bifurcates from (u, A) = (0,0) to an attractor SA = Sm~l consisting of singular points, andY,\ attracts H\Y, where F is the stable manifold ofu = 0 with codimension m. (3) If 1 < A, then the problem (8.24) and (8.26) bifurcates from (u,X) = (0,1) to an invariant set Y,\, which is a (2nm — 1)dimensional homological sphere S2™'1. (4) The invariant set T,\ contains at least Ck ( k
1
1
k
manifolds T = T x S™' , T = S x • • • x 5 of singular points of (8.24) and (8.26).
—— 1
)
singularity
the ktorus, i.e. T consists
Proof. The proof of Assertions (l)(3) is the same as the proof of Theorem 8.6. We only have to prove Assertion (4). We know that the eigenvalue problem
{'Aek
= Pk&k
(8.35)
'
y ek(x + 2kn) = ek(x),
has eigenvalues given by p k = k2 +  + k l ^ l ,
h = 0,l,
,
l < i < n ,
and the eigenfunctions corresponding to pk are given by sin(fciX! H
+ knxn),
cos(fcxXi +
h knxn).
It is clear that the first eigenvalue pi = 1 has eigenfunctions sinxj,
COSXJ
(1 < j < n).
Let L\ + G : H\ —» H be the operator denned by L\u + Gu = Aw + Xu — \u\ u. It is clear that the space Hj = < u e Hi  u = ^ i / f c sinkxj,y k = (yu, • • • , ymk) e R m > { k=l )
257
ReactionDiffusion Equations
is an invariant subspace of L\ + G. Then, the LyapunovSchmidt reduction equations of L\u + Gu = 0 on Hj are given by oo
(A  l)ya  J2 fiVik = 0 ,
(1 < i < m),
(8.36)
(8.37) where J.2TT
fj. = (27T)""1 / Jo
\u\2
smrxjsinkxjdxj,
oo
u = ^ Vk sin kxj. k=i
In the same fashion as used in (8.29) and (8.30), we can derive that the solutions of (8.36) and (8.4) constitute an (m — l)dimensional sphere
^=r£aismxj+o(\a\)\\af
= ^=(X2)1/H
We note that the problem (8.24) and (8.26) is invariant for the translation of solutions u(x,t)^u(x
+ 9,t),
9=(91,.
,0n)€R".
Hence the functions in the set $o = {u(x + 6)  u G $} . are singular points of (8.24) and (8.26). Therefore the set r = U06T"0 is a manifold homeomorphic to S 1 x Sm~l, which consists of singular points of (8.24), (8.26). Likewise, for a multiple index (j\, • • • ,jk), 1 ^ j r ^ n, j r ^ jk as r ^ fc, we take e = sin Xjx + • • • + sin Xjk
258
Bifurcation Theory and Applications
Obviously, the space
He = lu€H1 u = JTykek\ I
fc=i
J
is invariant for L\ + G. In the same fashion as above one can get that L\ + G has a singularity manifold in Tie: r = Tfcx5m1,
Tk = S1 X    X 5 1 . Th • '
*'(TI
The number of indices (ji, • • • , jk) for k < n is C% = proof is complete.
8.3.3
• • A* 4 1 ^
—
. The D
Invariant homological spheres
It is not difficult to prove that as A crosses any eigenvalue pk with multiplicity r ^ 1 of —A, then the equations (8.24) will bifurcate form (u, A) = (0, pk) on A > pk to an invariant rm — 1 dimensional homological sphere, which contains singularity sphere. Theorem 8.8 Let pk > 0 be an eigenvalue of —A with the Dirichlet boundary condition or the periodic boundary condition, and py. have multiplicity r ^ 1. Then the following assertions hold true. (1) Equations (8.24) with (8.25) or (8.26) bifurcate from (0,pk) to a unique rm — 1 dimensional invariant homological sphere E^. (2) For the Dirichlet boundary condition, the invariant homological sphere E\ contains at least Y^=i Cn singularity sphere Sm~x. (3) For the periodic boundary condition, Y,\ conditions at least C^ (r = 2s, k ^ s) singularity manifolds T = Tnk x 5 m  1 for some rik (1 ^ nk ^ n ) . Proof. Let {ej, , e r } be the eigenfunctions of —A corresponding to pk The reduction of (8.24) to the center manifold is given by ^
= (A 
Pk)v
 P{\v\2v) + o(\\u\\%
(8.38)
where P : H —> Ek is the canonical projection, Ek the eigenspace of pk, and v = YH=I Viei
Prom (8.38) we get \jt\M%
= ^  Pk)\\v\\%  \\v\\4 + o(\\v\\4).
(8.39)
ReactionDiffusion Equations
259
It is easy from (8.39) to see that v = 0 is asymptotically stable for (8.38) at A = pk As in the proof of Theorem 8.6, we can achieve that (8.38) bifurcates from (0,Pk) on A > pk an attractor H\. Therefore Y,\ is an invariant manifold. Assertions (2) and (3) can be proved in the same manner as in the proof of Theorem 8.7. The proof is complete. • Remark 8.2 It is known that the equations (8.24) have a global attractor for any A £ K1. It is reasonable to conjecture that the global attractor of (8.24) may be a sphere Skm~1, where k is the sum of multiplicities of all eigenvalues pj < A. 8.4
BelousovZhabotinsky Reaction Equations
8.4.1
Setup
In this section, we study the dynamic bifurcation of the following BelousovZhabotinsky reaction equations in chemical dynamics dui 2 —— = fiiAui + a(ui + u2  uiu2  pwi) du2 1 —— = p,2Au2 + — (7«3 u2 uiu2) a at ' du3 — = /i 3 Au 3 + o(ui  u3) u\an = 0, u(z,0) = <> /
(840)
Here fi C R n (l ^ n < 3) is a bounded and smooth domain, u = (u\,u2, u3) chemical concentrations satisfying u, ^ 0 (i = 1,2,3), and the constants Hi, H2, A«3, a, P, 7 , 5 > 0.
It is known [Smoller, 1983] that D = {(uhu2,u3)\
0 ^ U l < a,0 < u2 < b,0 < u3 < c}
(8.41)
is invariant for the BelousovZhabotinsky reaction equations (8.40), provided that a > max{l,/3 1 },
c> a, b > jc.
260
Bifurcation Theory and Applications
Let R+ ={(*!,• •• ,xm)eRm\
Xi>0,
l^i^m},
* = (Mi»At2,A*3,a,7,(J) e K + . Let #1 = # 2 (f2,R 3 )n.ff 0 1 (ft,R 3 ), H = L2(Q,M.3). The operators LA = — A\ + B\ and G{,\) : Hi —* H are defined by ^ A « = (—/xiAui, JJ,2AU2,
1 < 5^u = (aui + m i 2 ,
M3AU3),
7 "2 + — "3, Su3 + Sui),
G(u, A) = (—CXU1U2 — a/?Wj,
(8.42)
U1W2,0).
Then the BelousovZhabotinsky reaction equations (8.40) can be rewritten as the following operator equation in H: (%=Lxu + G(utX),
( 8 4 3 )
I "(0) = V. where A = (fj,i,(i2,H3,a,J,S). We shall address the bifurcation and stability for the BelousovZhabotinsky reaction equations in the invariant region D given by (8.41). 8.4.2
Bifurcated attractor
Let pk and ek be the fcth eigenvalue and eigenfunction of the Laplace operator —A with the Dirichlet boundary condition. We know that the first eigenfunction is positive ei(x) > 0 V i e O .
(8.44)
Let A(A) be a function of A = (^1,^2,^3, a, 7,8) defined by A(A) = ajS  {nipi  a)(fi2api + l)(/i3Pi + <*), and Ao = (/x?,^,^,a o ,7 O ,<5 0 ) satisfy that A(A0) = 0,
(8.45)
261
ReactionDiffusion Equations
where pi > 0 is the first eigenvalue of —A with the Dirichlet boundary condition. Let r = {Ao e R6+  Ao satisfy (8.45)}, a matrix E(X) be defined by E(X)=[
/(l*ipia) a 0 (/iaPi + O
V
<*
0
0 aJ7
\ ,
(fx3Pl+8)J
(8.46)
and two real positive number a, 6 > 0 by • a = [a o 5 o (l + M °aV)(M3Pi + <5°) +
1+A V
+
*° + M^iJ/n
X
'
(847)
2
' 6=5o(^i + 0(«°) [^3Pi + 0 ( ^ V + l) \
a°l+$aopi\ Jn
Then, we have the following bifurcation theorem for the BelousovZhabotinsky reaction equations. T h e o r e m 8.9 For any Ao = ( / i ; , ^ , ^ , a o , 7 ° , < 5 0 ) (/ii,/i2,M3j a >7>^) G K^_ satisfy that A ^ Ao and
e T, if X =
A(A) > A(A0) = 0, i/ien the following assertions hold true. (1) The problem (8.40) bifurcates from (u, A) = (0, Ao) a unique singular point u\(x) S Hi with u\(x) > 0 in Q, which is an attractor. (2) The singular point u\(x) can be expressed as
( ux(x) = frWab^voix) + o(\^(X)\),
!»*(*) = ^(6°+^^,
i
f ^ e ! , ^ ,
(8 48)
'
where a,b > 0 are gwen by (8.41), and /?i(A) t/ie eigenvalue of (8.^6) with(5i(X) > 0 . (3) there exists an open set O C H with u = 0 € O such that u\ attracts O n D, where D is the invariant region (8.4I), i. e. for any <j> =
262
Bifurcation Theory and Applications
(
0.
Proof. The proof is achieved by applyinging Theorem 6.4 to (8.43), and will be divide into several steps as follows. STEP 1. We know that L\ is a sectorial operator and the nonlinear operator G defined by (8.42) satisfies that
G(, A) :He>H
is C°° for some 6 < 1.
Now we verify conditions (6.22) and (6.23). All eigenvalues /3,fc(A) (j = 1,2,3, k = 1,2, • • •) of L\ satisfy det(/?jfc(A)J£fc(A)) = O,
(8.49)
where Ek(X) (fc = 1,2, • • •) are the matrices as follows
(
(mpka)
0 5
0
a
(wAfc + O 0
\
a^l , {n3Pk + 5))
(8.50) (8.50)
where pk is the kth eigenvalue of —A with the Dirichlet boundary condition. It is easy to see that the function A(A) is given by A(A) = adetEi(A).
(8.51)
Hence we have det^(A) = 0 if and only if A e I\ Furthermore, we see that „..,. f dA dA dA dA dA 0A\ . VAA = — , — , — ,   ,  U 0 \dfii
op,2 9^3 da 07 do J
onT,
and F can be expressed as a realvalued function 7=
(M1P1 ~ l)(M2Q!Pi + 1)(M3/Ji + S) ~5 . Hi,fJ2,H3,ot,6 > 0 .
It follows that F is a 5dimensional manifold in Rj., which divides R^. into two disjoint,connected components R6+=^u^,
DinD 2 = 0,
dDindD2 = T,
263
ReactionDiffusion Equations
such that det£;1(A) = /3 1 1 (A)/3 2 1 (A)/33i(A)^'
^
^
'
(852)
On other hand, we have 3
^Re/^A^traceofi^A) j=i
(8.53) 1
=  [(MiPi  a) + {mPi + a" ) + (/^/>i + <*)] and there exists Ai = (/i^/x^/ij.a 1 ^ 1 ,^ 1 ) € R+ such that A(Ai) ^ 0, n\Pl  a1 > 0.
(8.54)
It follows from (8.51), (8.53) and (8.54) that 3
J]Re/3 jl (A 1 )<0,
A!GD2.
(8.55)
Then we infer from (8.52), (7.87) and (8.55) that /?u(A)>0, /3U(A) = 0,
Re/? 2 1 (A)<0, Re/?21(A) < 0,
Re/?3i(A)<0, A G Z?2,
(8.56)
Re/?3i(A) < 0, A € T.
(8.57)
We deduce from (8.52) and (8.57) that /?n(A)<0,
Re/3 2 1 (A)<0,
Re/33i(A) < 0, A e A ,
(8.58)
Hence condition (6.22) follows from (7.17) (8.58). Now we need to show that the eigenvalues Re/?ifc(Ao) < 0, V Ao G T, k > 2.
(8.59)
Let Ao = (/x?,^,/u^a o ,7 0 ,(5 0 ) G T. Then by (8.45) we have nlpr  a0 > 0
(8.60)
Thus we infer from (8.60) that detEfc(Ao) = i  [ao7o5o  (tiolPk  ao)($a°pk
+ l)(/x°pfc + 8°)]
< ^ h,7o<*o  (/i?Pi  a°)(/i§a 0 P l + lj^gpx + <50)]
264
Bifurcation Theory and Applications
for pk > p\, k > 2. Namely detEfe(Ao) <detE 1 (Ao) = 0,
V k > 2, A0 G I\
(8.61)
In addition, it is easy to see that
p*(Ao) = £i(A), \A = ( / x?p^r i , M Vpr 1 ^Wr 1 ^ 0 ,7 0 ,<5°).
(8.62)
Hence we have ft*(Ao)=fti(A),
V f c > l , j = 1,2,3.
(8.63)
We obtain then from (8.61) and (8.62) that A e £>i, and (8.59) follows from (8.58) and (8.63). By Theorem 6.4, it suffices to prove (8.48). The eigenvector t>o(A) of L\o corresponding to /?n(A0) = 0 is given by vo(x) = (xiei,a;2ei,X3ei),
e\ satisfies (8.44),
and (a;i,a;2i £3) is the eigenvector of the matrix i?i(Ao): a0
/(jtlKcP)
0
0 0
1
(fiPl + (a ) )
0
7°(a )
\ 1
/Xl\
U
2
= 0,
which has a unique solution
Hence the eigenvector VQ(X) is as given in (8.48). The eigenvector 1*0(2;) = (xiei,X2e2ix3e3) °^ ^A 0 c o r r e s P° n ding to /?n(Ao) = 0 satisfies that /_
(/i0pi_a0)
0
a 0
V
0
6
0
0
v . j v
1
 ( ^ P l + (a ) ) 0 \[x*2 = 0, 0 1 0 °(a )(/igpi+J )/ W 7
which implies
«g(x) = U(l + AVi)ei^°(«°)2ei, T ^ X  e i ) •
ReactionDiffusion Equations
265
Finally the constants a, b in (8.47) are obtained by a= (vo,Vo)H = / vo(x) • v£(x) dx Jci b=
(G(VQ,XO),VO)H
= / [ao(x1X2+/3x21)x*1+xix2xya0]eldx Jo, Hence (8.48) follows by direct computation, and u\ is in invariant region D of (8.41). The proof is complete. • Remark 8.3 Only nonnegative solutions of the BelousovZhabotinsky reaction equations are meaningful in chemistry. 8.5
Notes
8.1 The model for the flow of electrons and holes in semiconductors is due to [Van Roosbroeck, 1950]; see also [Henry, 1981]. The equations for chemical reactions in catalyst were introduced in [Gavalas, 1968; Aris, 1969]. The equations from chemical reactor theory are given in [Verma and Amundson, 1970]; see also [Cohen, 1973]. The HodgkinHuxley equations describing the nerve impulse transmission were proposed in [Hudgkin and Huxley, 1952]. In the original model, n? = M3 = A*4 = 0. Here we follow the slight modification by [Smoller, 1983]. For the complex GinzburgLandau equations, see [Temam, 1997]. 8.2 The results in this section are proved using the bifurcation theory developed by the authors in [Ma and Wang, 2004a; Ma and Wang, 2004b], also given in Sections 4.1, 6.2 and 6.3. 8.4 The equations (8.56) serve as a model for the BelosouvZhabotinsky reactions in chemical dynamics; see [Hastings and Murray, 1975; Kopell and Howard, 1973]. The attractor bifurcation theorem for the BelousovZhabotinsky reaction equations is new here.
Chapter 9
Pattern Formation and Wave Equations 9.1
KuramotoSivashinsky Equation
9.1.1
Setup
The KuramotoSivashinsky equation in onedimensionai space with periodic boundary condition is given by ( du
dAu
< diu
d2u
du
n
diu
no
(91)
. u(x, 0) =
/
(9.2)
u(x,t)dx = Q,
J — IT
where fi > 0 is a constant. Equation (9.1) is derived by differentiating the following original KuramotoSivashinsky equation in the spatial variable x, and letting u = dv/dx: dv
d4v
d2v
l(dv\2
_
We shall discuss the attractor bifurcation of (9.1) and (9.2) in the following two cases: (1)
CASE WITH ODD SOLUTIONS.
In this case, we look for solutions of (9.1) 267
268
Bifurcation Theory and Applications
and (9.2), which are odd functions with respect to x. Hence, we set
H1 = {ve H*er(Tr,ir)  v(x) = v(x),
f
v dx = 0},
J — IT
H={v£ L2(ir, n)  v(x) = v(x), /
J—v
v dx = 0}.
Here Hper(—ir,ir) is the Sobolev space with periodic boundary conditions as given in (9.1). (2) GENERAL CASE. In this case, we look for solutions without the oddness assumption. Let
H1 = lv e H*er(Tr,n)  J \ dx = o  , H = Iv e L2(—7r,Tr)  /
vdx = o\.
In both cases, we define the operators L\ — —A + \B and G(,A) by
(9.3) Thus, the problem (9.1) and (9.2) can be rewritten into the following abstract form in H (resp. H): (d^
= Lxu + G(u,X),
(Q4)
[ «(o) = VHere A — fi"1,
9.1.2
T
— [it.
Symmetric case
In this subsection, we give an attractor bifurcation theorem for the KuramotoSivashinsky equation in the space H consisting of odd periodic functions. Theorem 9.1 For the KuramotoSivashinsky equation (9.1) with (9.2) defined in H, the following assertions hold true. (1) The equation (9.1) with (9.2) bifurcates from the trivial steady state solution u = 0 to exactly two steady state solutions u\ and ui £ H,
Pattern Formation and Wave Equations
269
when A = /x" 1 crosses the first critical value \Q — 1. Both bifurcated solutions can be written as
{
ui = a(X) sin x + o(\a\),
u2 = a(X) sinx + o(a),
(9.5)
a(X) = V ( 4  A ) ( A  1 ) / A .
(2) There exists a number 1 < manifold F C H of u = 0 space H into two open sets of U\ and u
e < 4 such that if 1 < A < e, t/ie stafc/e twYA codimension one separates the phase U\ and U%, which are basins of attraction i. e.
UlnuZ = V>,
du\ n du2x = r, UiGUl
i = l,2,
and lima\\u(t,
= O,
(9.6)
for any (p G U{ (i = 1,2), where u(t,ip) is the solution of (9.1) and (9.2). Remark 9.1 The equalities (9.5) and (9.6) imply that for any initial value ip £ H \F, the solution u(t,
k\
efc(x) = sinfcx, for k=l,2, • • •, and satisfy the conditions (6.4) and (6.5) with m = 1 at A0 = l. Obviously, the operator G(,X) : Hi —> H denned by (9.3) satisfies the orthogonal condition in Theorem 3.16, and at Ao = 1 the equation (9.4)
270
Bifurcation Theory and Applications
has no invariant sets in J?o=span{sinz}. In fact, Ana2 G(a sinx,Xo) = Ana2 sin x cos x = ——sin2x ^ EQ. Moreover, equation (9.4) has a global attractor for any A > 0 in H [Nicolaenko et al., 1985]. Thus, by Theorems 6.1, 3.16 and Remark 6.1, we only have to prove (9.5). We verify (9.5) by the LyapunovSchmidt method. Let u G H and u = Yl'k>=ixksiiikx. Then the steady state bifurcation equation (9.1) can be expressed as TT(1 — u)xj — / JK
— irf3nxn— /
77T
u— sinx dx — 0,
(9.7)
dx
u—sinnxdx dx
= 0,
Vn ^ 2,
(9.8)
where pn = n2(n2/il),
fi
= \1.
We know that u— =  ^2 kxkxj (sm(j + k)x + sin(j  k)x) .
(9.9)
It follows from (9.7)(9.9) that ^ oo
(1  n)Xl + T I
Xixi+1
= 0,
"nl
xn = X3 V kxkxnk ^ « U=i
(9.10) oo
I
 V nxkxn+k , Vn > 2. *=i J
(9.11)
By induction, we infer from (9.11) that
U^+^P),
(912)
(x n = a(nK + oflnl"), where a(n) is a constant. We derive then from (9.10) and (9.12) the following bifurcation equation (lfi)xx^xl
+ o(\x1\3)=0.
(9.13)
Pattern Formation and Wave Equations
271
From (9.13) we obtain the bifurcation solutions oo
^(x) = y^ xf sin kx, fc=i
if = ±4v/(4/xl)(lM),
xt=o(\xf\),
where /x = A"1 and k ^ 2. Thus, (9.5) is verified. The pr.oof is complete. 9.1.3
General case
We now consider the attractor bifurcation of the KuramotoSivashinsky equation in the general case without the oddness assumption. Namely, we study the dynamics in the phase space H, consisting of periodic functions. Theorem 9.2 In the space H, the KuramotoSivashinsky equation (9.1) with (9.2) bifurcates from (u, A) = (0,1) on A > 1 to an attractor SA = 5 1 , which attracts all bounded sets in H \T, where T is the stable manifold of u = 0 with codimension two in H. Moreover, T,\ consists of only steady states of (9.1) and (9.2). Proof. The eigenvalues /3k(\) and eigenfunctions ek(x) of L\ : H\ —> H are given by ftnl(A)  ftn(A) = n\X  n2), e2ni(x) = sinn2, e 2n = cosnx,
(914) (9.15)
for any n > 1. These eigenvalues satisfy the conditions (6.4) and (6.5) with m = 2 at Ao = 1. Furthermore, it is easy to see that the conditions of Theorems 6.1 and 3.16 are satisfied by (9.4), the equations (9.1) with (9.2) bifurcate from (u, A) = (0,1) to an attractor T,\ with 1 ^ dimSA < 2, which attracts H\T, where F is the stable manifold of u = 0 with codimensional two in H. Next, we shall prove that T,\ = S1. The reduced equations of (9.4) to the center manifold on H are as follows
{
^ T = (A  l)xi + i [" at
IT
J
G(u,\)smxdx,
—• = (A  l)x2 +  I G{u,X) cos xdx,
(9.16)
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Bifurcation Theory and Applications
where u = zisinx + a^cosz + $ (x 1,0:2), and $ is the center manifold function. We know that G(u, A) is 2—multilinear with respect to u, and the terms of the first order approximation of (9.16) as in (3.55) are zero: (G(xiei + x2e2, A), e*)H = 0, 3 = 1,2, where e\ = e\ = sinx, e% = e\ = cos a;. Therefore we have to consider the terms of the second order approximation of (9.16). By the formula of the second approximation (3.62) and (3.63), (9.16) can be rewritten as •
,
2
— = (A  l)xi + 5Z al^XiXjXi+odxl3), (9.17)
A
_ ^ = (Al)x 2 + J2
a x x x
li i J '+o(\x\3)'
i,3, ' = 1
where
^' = S(2/3 1  A ,)<e B l e>) g < G ^' e " A ) ^ ) g x (G(ei, e n , A) + G(e n , ei, A), e*k)H, where (3n and e n are given by (9.14) and (9.15), and en = en. By direct computation and e~ekdx, we obtain that A2
1
aill
~
a
=
222
a
4(2A/? 3 )' "i a
112+ 121+ a 211 =°>
al22 + a J 1 2 + a J 2 1 =  4 ( 2 ^ _ A ) ,
Pattern Formation and Wave Equations
273
and
A2
°' 22 = " 4 ( 2 0 !  f t ) ' a
"2
+ a ? 2 1 + O 11 =
2
a
A2 ~4(2/Jift)'
a
°122 + 212 + 221 = ° • Thus, the equations (9.17) are in the form as follows
j ^ = ^ 1 >;pr^ + °> + *i s >.
(9l8)
Here 2ft  ft = 4(4  A) + 2A  2 > 0 for A near Ao = 1. By Theorem 5.10, it follows from (9.18) that the attractor of (9.18) bifurcated from (x, A) = (0,1) on A > 1 is homeomorphic to S 1 . Hence STEP 3. Finally we verify that SA consists of singular points of (9.1) and (9.2). By Theorem 9.1, the problem (9.1) and (9.2) has a steady state solution u\ = a(A)sina; + h(x), h(x) = o(\a\). Because of the invariance of (9.1) and (9.2) for the translation
u{x,t) —> u(x + 0,t), the functions Ux{x
+ 6) = a{\) s\n{x + 6) + h(x + 6), 9e R1
are steady state solutions of (9.1) and (9.2), and the set $ = {a(A)sin(a; + 9) + h{x + 6) \ oo < 6 < +00} is a circle S1 in H. Therefore E^ = $. The proof is complete. 9.1.4
S1—invariant
•
sets
As A crosses n—th critical value n 2 , the KuramotoSivashinsky equation bifurcates to an invariant set LA in H homeomorphic to S1, which consists
274
Bifurcation Theory and Applications
of steady state solutions of (9.1) and (9.2), i.e. we have the following theorem. Theorem 9.3 In the space H, the problem (9.1) with (9.2) bifurcates from (u,X) = (0,n 2 ) on X > n2(X = /i" 1 ) to an invariant set Yl\ = S1, which consists of steady state solutions of (9.1) and (9.2). Proof.
Function u G H can be expressed as oo
ek as given by (9.15).
u = ^2xkek, fe=i
Then the reduction equation of (9.4) to the center manifold near A = n 2 is in the following form
{
X2
Z~l = P2n(X)x2ni + 
G(u,X)sinnxdx,
ir
dZ
(9 19)

—TT = /32n(A)z2n +  / G(u, X) cos nx dx. at IT J_n As in (9.17), (9.19) can be rewritten as ^
i
= /32nX2n_! +
J2 a
^ f = frnx2n +
b^X.XjX,
» 1 <«*< 3 »
J2
+ O(\X\3),
3
^kxiXjxk + o(\x\ ),
(9.20)
where x = (x2ni,X2n), and
K
^
S
m^2nl,2n
(2A B /3j)<e m> e m >H
We find that (G(ej,ek, X),em)H = 0,
for In — 1 ^ j , k < 2n, m ^ An — l,4n.
Namely b k=
* W2nMn[{G{ei>ek>X)>einl)H x (G(ei,e4n_i,
A) + G ( e 4 n _ i , e u A),e p ) H
+ {G(ej,ek, A), ein)H(G{ei,
e4n, A) + G(e 4 n , eit A), ep) ],
275
Pattern Formation and Wave Equations
for 2n — 1 ^ i,j, k,p ^ In. By direct computation, (9.20) ar e as follows
{
^T— = Ajn^nl  ——
— r ^ l n l + ^ n  l ^ L) + o(a:3),
«* 4(2/32n  Pin)  ^ = /?2nZ2n ~ T7^3 a~^(X2n + ^ n ^ n  l ) + o(o;3), «E 4(/P2n — P4nJ
for X =
(x2nl,X2n)
The remaining par t of the proof is the same as in the proof of TheoD rem 9.2, and the theorem is complete. 9.2
CahnHillard Equation
9.2.1
Setup
The CahnHillar d equation, which involves a fourthorder elliptic operator , read s
JW= A K «'
(S.21)
[u(x,0)=
where x G fi C M. , 1 ^ n ^ 3,
(
K{u) =  a Au  \u + f(u), ^
(922)
t
f(u) = ^ afc«fc, fc=2
a > 0, afc(2 ^ A; ^ 2p + 1, p ^ 1) ar e constants, A € R1 is a parameter , u = u(x,t) is a scalar function, and fi C Mn is bounded and sufficiently smooth domain. The equation (9.21) is supplemented with either THE NEUMANN BOUNDARY CONDITION:
£_£!_„
onan,
where n is the unit outward normal on dfl, or
(9.24)
(
THE PERIODIC BOUNDARY CONDITION:
u(x + 2kir) = u(x) Vfc = (&!,••• ,kn), where Q = [0, 2TT]™, k = (k1: • • • , kn), an d k{ £ Z.
(9.24)
276
Bifurcation Theory and Applications
In this section, we always consider the case where u has vanishing average / u(x, t) dx = 0, Vi > 0. Let
H=lu£L2(n)

fudx = o\.
For the Neumann condition (9.23), we define
H1 = {ueH*m I / n «^ = 0,^ f l 0 =^an=o}, and for the periodic boundary condition (9.24), we define
#i = j u e # 2 ( f l )

f udx = 0,u(x + 2kn) = u(x), Vfc e Zn\ .
Then we define the operators L\ = —A + B\ and G : Hi —> H by
{
Au — aA2u, (9.25)
Bxu = XAu,
G(u) = A/(«), where f(u) as in (9.22). Then the CahnHillard equation (9.21) is equivalent to the following operator equation
{
du
,
Tt=L^U
_,. .
+ G
^
(9.26)
«(0) = ip.
9.2.2 Neumann boundary condition We first consider the case where Q. C R" is a general bounded domain. Let Pk and ek be the eigenvalues and eigenfunctions of the following eigenvalue
Pattern Formation and Wave Equations
277
problem  Ae f c = <
dn
9Q
pkek,
~ U'
(9.27)
/ efc dx = 0.
v JO,
It is known that the eigenfunctions of (9.27) satisfy 0 < P\ ^ P2 ^ • • • , pk > oo (fc > oo), and the eigenfunctions {e^} of (9.27) constitute an orthogonal basis of H. Especially, the eigenvalues of (9.27) satisfy ^
dn
a n
0 °'
Jfcf 1e2 1 2 . . . '' •
Hence, {e*} is also an orthogonal basis of Hi under the following equivalent norm
r /• i1/2 2 2 H i = / A u rfx . Theorem 9.4 For the CahnHillard equation (9.21) with the Neumann boundary condition (9.23), assume that ai = 0 and as ^ 0 in (9.22), then the following assertions hold true. (1) If 03 > 0 and the first eigenvalue p\ of (9.27) has multiplicity m ^ 1, then the problem (9.21) with (9.23) bifurcates from (u,\) = (0,ap\) on A > api to an attractor Y,\ homologic to Sm~1. Furthermore, Y,\ attracts O\T for some neighborhood O C H of u = 0, where T is the stable manifold of u = 0 with codimension m. In addition, if m = 1, T,\ consists of exactly two steady state solutions, and if m = 2, then S,\ = S1. (2) Let the k—th eigenvalue pk of (9.27) have multiplicity mk, then the problem (9.21) with (9.23) bifurcates to an invariant set, homologic to the mfc — 1 dimensional sphere, from (u, A) = (0,apk) on X > apk if a^ > 0, and on X < apk if a$ < 0.
278
Bifurcation Theory and Applications
Proof. Obviously, the eigenfunctions {e^} of (9.27) are also eigenfunctions of the linear operator L\ = —A + B\ defined by (9.25) and the eigenvalues of L\ are given by fc
0k(\)=Pk(\apk),
= l,2,.
The reduced equations of (9.21) and (9.23) to the center manifold near A = apk are given by dx f £ = PkWxi  Pk / f(u)eki dx VI < i ^ mk, m Jn
(9.28)
where {e^i, ,ekmk} are eigenfunctions of (9.27) corresponding to pk, f(u) is given by (9.22), and u = ^2xieki + h(x), h{x) = o(\x\) the center manifold function. Since a2 = 0 and 03 > 0, (9.28) can be rewritten as follows —± = 0k(\)xi  a3pk / {Y^X:jekj)3ekidx a t
•*&•
Let
j=i
Or mk
+ o(\x\3), 1 < i < mk. (9.29)
n
\
f v3ekidx,
, / v3ekmk
Jn
dx
J
,
Then we have mk
(g(x),x) = ^2xi in > Ca;4,

v3ekidx
(9.30)
for some C > 0. By (9.30), a; = 0 is locally asymptotically stable for (9.29), then this theorem follows from Theorems 5.2 and 5.10. •
279
Pattern Formation and Wave Equations
Now we consider the case where the domain f2 is an n—dimensional cube, i.e. 0,= [0,7r]n. The first eigenvalue of (9.27) is p\ — 1, and the first eigenfunctions are given by e\ = cosrri, • • • , en = cosrcn. 26 Theorem 9.5 For Q = [0, vr]n, if the constants aas > —a^ in (9.22), then the following assertions hold true. (1) The problem (9.21) with (9.23) bifurcates from (u,X) = (0,a) on A > a to an attractor Y,\ homologic to Sn~1 (if n = 2, EA = S1). (2) The attractor Y,\ contains exactly 3 n —1 steady state solutions of (9.21) with (9.23), which are regular. Furthermore, when n = 2, EA = S1, and has the structure as shown in Figure 9.1.
*•
•+
~7\
*
1
"*—
t
Fig. 9.1 When n = 2, the attractor E,\ = 5 1 contains eight singular points z; (1 < i < 8), with Z2ki being saddle points and Z2k being minimal attractors (1 < fc < 4).
Proof. To prove this theorem, we need to consider both the center manifold and the LyapunovSchmidt reductions. The proof is divided into a few steps in the following.
280
Bifurcation Theory and Applications
STEP 1. We need to consider the second order approximation for the center manifold reduction. The eigenvalues and eigenfunctions of L\ = —A + B\ are as follows
•pK(\) = <
\K\2(\a\K\2),
K =
(ku...,kn),
h£N
(1 < i < n),
2
lK\ = k> + ... + kl, and e
K{x)
= COS k\Xi • • • COS knXn.
By (3.62) and (3.63), the vector field in the second order approximation of the center manifold reduction for (9.21) is given by
Mv) = y f t W ! / + G2(y) + G3(y) + G23(y),
(9.31)
where y = (j/i, • • • ,yn) £ Kn,/3i(A) = A  a, and G2(y) = a.2 I / « 2 e i dx, • • • , / v2en dx I ,
\Jn f
O n V = ^2viei> i=l
a
Jn f
u 3 eidx, ,
) \
v3endx , Jo. J
(9 32) '
v
e
i = COSXi,
and n \ G23(y)=\ J2 ^iikViVjyk, , J2 a?jkViViVk I , J \i,j, *=i i,i,fc=i / (
I
,
^
2\K\2a22
f
,
f
(933)
,
281
Pattern Formation and Wave Equations
Direct computation implies that / v2eidx=
/ (yi cosxi+
in
Jn
•••+
cosxn)2cosXidx
yn
= 0,
J'a v eidx— Jn/ (yi cos x\ + • • • + y cos x ) 3
E
n
7T n a 2 I
i
°^W^ =— T
4
n
3
cos Xi dx
o
4
V^ 2 I
4(A4a)ft y « + 2(A2a)/? 1 yi g y *l
Thus, the following center manifold reduction equation of (9.21) (934)
^=/31(X)y + v1(y)+o(\yn
2 with vi = — (G2 +G3+ G 23 ), can be rewritten, by (9.31)(9.33), in the form d
^ = {Xa)yi1
aiy3 + a2yiJ2yl
L
kiti
+ <>(\y\3), 1 < t < n,
(9.35)
where
hl«+(AtoUw* l".=3«3+ ( A _ 2 o ) _ f t ( A ) «l
(936)
STEP 2. At the critical value \ = a, the numbers in (9.36) are 3 1 2 ^lO2) = o° 3 ~ ^~a2> ^2(a) = 3a3 «2From (9.34) and (9.35) we see that (vi(y),y) = (^i E y f + 2a2 J > M ) .
282
Bifurcation Theory and Applications
Therefore if <j\ (a) and a2 (a) satisfy for cr2K(a) > 0 , ' for<7 2 (a)<0,
{0 a1U(a)> I ' \ 2
/ (9.37); V
then we have n
( 9  38 )
(vi(y),y)^cJ2yf'
for some C > 0 and A — a > 0 small. It is clear that the condition (9.37) is equivalent to the assumption aa.3 > 7=0%, By using Theorems 5.2 and 5.9, Assertion (1) follows from (9.38). STEP 3. LYAPUNOVSCHMIDT REDUCTION EQUATION. The singular points of (9.35) are also associated to the steady state solutions of (9.21) and (9.23); here we use the LyapunovSchmidt reduction to illustrate a method for rinding the implicit function. The steady equation of (9.21) can be expressed as
^/3i(A) yi + / A/(u) • cosxi dx = 0, 1 Jo, yK
= R <\M\ 112 /
A
1 ^ i ^ n,
/ M e * dx, \K\ > 2.
(9.39)
(9.40)
where u = YL\K\^\VK&K, Vi = yK=(&n, An)We find that
/ Af(u)eKdx = \K\2 f f(u)eKdx, Then (9.39) can be rewritten as —f3i(\)yi  / (a2u2 + a3u3) cosxi dx + o(\u\3) = 0. Jn ^ For K = (ki, • • • , kn), we denote Vij = VK Vij=yK
*£ kj = ^ ifki = kj = l
an
d ki = 0 for i ^ j , and ki = 0 for I j= i, j .
(9.41)
283
Pattern Formation and Wave Equations
We infer from (9.40) that
^
=
2 x 22a2
/", v ^ ( T n 
4(A4a) y g i ^
= Q_4^W»^2 / Vi
i
=
^2 o j )0082ar
4 x 2a2
o7\—TT~^Viyi
cos2
cos2x
^
f
^
J
dx+
°(l^ 2 )'
2
o
/i i9%
I cos^zicos'zjdx + oflj/l"),
where y = (j/i, • • • , yn) G E n . Namely ' ^ (
=
2(A^)^
+ O(y2)
'
^  ( A ^ ) ^ + °(y2)' .y K =c(H 2 ), VJRT>4.
(9 42)
'
Based on (9.42) we have
u2cosxidxJ
n
VVcosrr, + Ja
\j=i
V ] VxeK ) cosXidx+ o(\y\3) 2<X<4
/
= 2yij/2i / cos2 i j cos Ixi dx
+ 2 V % 2 / y / cos 2 x i cos 2 a; j dx + o(y3)
= Y
ViV2i + 5Z^y*j
+
°(lyl3)
/ u 3 cos Xi dx = I (S~^ yj cos Xj)3 cosx t dx
Jn
Jn p[
= yf
cos4 Xi dx + y^ 3j/jj/? / cos2 i j cos2 z, da; + o(y 3 )
284
Bifurcation Theory and Applications
Thus, the LyapunovSchmidt reduction equations (9.41) can be expressed as
A(A)»«  \ (§«a + ^ 4 ) yf + (3a3 + ^ 4 ) + o(y 3 )=0,
Vl
£^
2
(9.43)
4. We are now in position to prove Assertion (2). Consider the following approximate equations of (9.43) STEP
/?i(A)i/i  Viiaiyf + a2 J^Vj) = °> 1 < * < «•
(944)
where 1,3 1 2 2(2a34^Aa2)' 1. 2 2, a2= 2(3G3"2^Aa2)Ql =
It is clear that if all solutions of (9.44) are regular, then the number of solutions of (9.43) and (9.44) are the same. First, we shall show that (9.44) has 3" — 1 nonzero solutions near y — 0. For each fc(0 ^ k < n  1) the equations (9.44) has C* x 2 n  /c (C* = n ( n  l )    ( n  f c + l) N rr ) solutions as follows f Vh = 0, • • • ,Vjk = 0, 1 < ji < n, 1 < I < *, \ I/?, =••• = !/?„_, = A ( a i + ( n  f c  l ) a 2 )  1 , r i ^ j J .
(9.45)
Hence, the number of all solutions of (9.44) are 711
J2 C* x 2"* = (2 + l) n  1 = 3"  1. fc=0
Next, we show that all solutions of (9.44) are regular. The Jacobian matrix of the vector field in (9.44) are given by
Dv=
[Pih(y) 2a2y2yi .
2a2yiy2 p2h2(y).
V 2a2ynyi
2a2yny2
•••
1a2yiyn \ 2a2y2yn .
• • • Pn 
hn(y)J
(946)
285
Pattern Formation and Wave Equations
where
For any solution in (9.45), without loss of generality, we take (yo = (y°1,,y°n), J y° = 0
for 1 < i < k,
2
12
[ y° = /?J/ ( ai + (n  k  l)a2) '
forfc+ 1 < j < n.
Then, we derive from from (9.46) that
where /?=/? A P1 V
P
•An—k =
( n ~ fc)a2 V ai + ( n  f c  l ) a 2 > /
a i  a2 a ax + (n  k  l ) a 2 P l '
(^•ij)(nfc)x(n*:))
A =//?1~(3Ql + ij ~ \ 2a2(y°k+1)2
(n
"fc~1)a2)(^+l)2
fori==
^' iovi^j.
Obviously, there are only finite points A > a such that
and
detAn_fe =
—a.\
oi2
•••
a.i
a,2
—ai
•••
a.2
a2
a2
.
.
^ 0.
Q l (n _ fc)
Hence, for A — a > 0 sufficiently small, the matrices of (9.46) at the points (9.45) are nondegenerate. Therefore, by Theorem 3.9, we derive Assertion (2)The proof of the theorem is complete. •
286
Bifurcation Theory and Applications
Remark 9.2 In the same fashion as used in Theorem 4.4, one can prove that for an n—dimensional, k—homogenous vector field
\fci + +fc n =fc
/
ki + +kn=k
if the following bifurcation equations Y,
Pi{\)xi +
<kA"
• •  ^ = 0, 1 < t < n
fci+—+*:„=«!
have only finite number of bifurcated solutions, then the maximal number of the solutions is kn — 1. If the number of bifurcated solutions of (9.2) is just kn — 1, then the solutions are regular. Remark 9.3
When n — \, the following condition in Theorem 9.5 cyf}
(9.47)
aa3 > —a\
2 is replaced by 0:03 > d^. Meanwhile, we see that if we only assume that a
2 + a 3 7^ 0 instead of (9.47), then from (9.43) we can claim that the problem (9.21) with (9.23) for Q. = [0,7r]n bifurcates from (u,X) = (0,a) to exactly 3" — 1 steady state solutions, which are regular, on each side of A = a. 9.2.3
Periodic boundary condition
We consider the periodic boundary condition (9.24) where 0 = [0,2n]n. In this case, the eigenvalues PK{X) and eigenfunctions of the linear operator L\ = —A + B\ defined by (9.25) are given by •f3K(\) =
\K\2(\\K\2a),
< * = (*!•• • • ' * » ) '
ki G Z
(9.48)
i = I, ,n,
,\K\2 = kl + .. + k l and f e ) s : = cos(fc 1 a;i +  + A; n a: n ), \e%
= sin(feia;i H
h knxn).
(9.49)
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Pattern Formation and Wave Equations
It is clear that the first eigenvalue /?i(A) = A — a of L\ has multiplicity 2n, and the first eigenfunctions are as follows e] = COSXJ, e? = sinZj, 1 ^ j < n. Theorem 9.6 For the periodic boundary condition, if the constants in (9.22) satisfy (9.47), then we have the following assertions. (1) The problem (9.21) with (9.24) bifurcates from (u, A) = (0, a) on\> a to an attractor H\ homologic to S2n~1. (2) For each k (0 < k < n — 1), the attractor EA contains C^ (n — k)—dimensional tori T"~fc, which consist of steady state solutions of (9.21) and (9.24). Proof.
We denote
u = ^^ VK cosKx + ZK sinKx,
Kx = k\X\ +
h knxn,
and Vi = VK, Zi = ZK with K = (5u, • • • , Sni).
Then, by (3.62) and (3.63), we obtain the center manifold reduction equations of (9.21) and (9.24) as follows
' ^ = (A  a)Vi  yi[aiyf + e2z? + a3 £ ( y ? + z))\ + O(ly 3 M3)
' ' ' " £• = (A  a)Zi  Zi[axzf + a2yf + a3 ^)(y? + z])\ jfti
3
3
+ o(y ,N ), for 1 < i < n , w h e r e y = { y i ,    , y n ) , z = { z x ,    , z n ) , a n d CTl =
I" 3 + 2(A4a) 1 (Aa) 4 3
aa =
,
1
2
2°3 + 2(A4a)(Aa) a2 '
(9.50)
288
Bifurcation Theory and Applications
At the critical value Ao = a, the numbers aj(l ^ i ^ 3) are given by o
^
=
0 CT2 =
0 CT3 =
"
^2
4° 3 ~ 6^°2' 2
2a3~6^a2' " ^ 2 _as _ _«2;
which satisfy, by assumption (9.47), that either C7°, <7°, < 7 ° > 0 ,
or A >
if
Then Assertion (1) can be proved in the same fashion as in the proof of Theorem 9.5. Since the space of all even functions is an invariant subspace of L\ + G defined by (9.25), for the functions
v = \ J UK cos Kx, we deduce that the LyapunovSchmidt reduction equations of (9.21) and (9.24) are the same as (9.43). Therefore the problem (9.21) and (9.24) has the solutions given in (9.45) in the space of even functions. By the translation invariance, for each k(0 ^ k ^ n — 1) and a fixed index (ji, • • • ,jk), the steady state solution associated with (9.45) generates an (n — k)— dimensional torus fn~k which consists of steady state solutions of (9.21) and (9.24). For example if (ji, • • • , jk) = (1, • • • , k), the kdimensional singularity torus T* is as follows n fc
T = {u(x + B) = Y, j=k+l
y* co<xi + 9i) + o{\y\),V{ek+u • • • , 6n) G R"""}
where u{x) is a steady state solution of (9.21) with (9.24) associated with (9.45) with 2/1 = • • • = 2/fc = 0, yk+\ = • • • = ynObviously, for a fixed (ji, ,jk), the 2n~k steady state solutions of (9.21) with (9.24) associated with (9.45) are in the same singular torus Tk. Furthermore, for two different index ktuples (ji, • • • ,jk) and {i\, • • • ,ik), the two associated singularity tori are different. Hence, for each 0 ^ k ^
Pattern Formation and Wave Equations
289
n — 1 there are exactly C^ (n — &)—dimensional singularity tori bifurcated from (u, A) = (0, a). Thus, the proof of the theorem is complete. • When we consider the bifurcation of invariant spheres from general eigenvalues, the conditions the constants a2,az a n d a m (922), similar to (9.47), for different eigenvalues are different. However, if we simply assume a2=0, a3^0,
(9.51)
then we can obtain the following bifurcation results of invariant spheres from an arbitrary eigenvalue. To this end, we need to characterize the eigenvalue of (9.48) and (9.49). We say that an eigenvalue /3K of (9.48) is of type pi x • • • x pr(\ ^ p\ < • • • < Pr ^ n) if there are r indices Kj(l ^ j ^ r) such that I zr
Ai
I 7v  2
2
= •• • = \KT\ ,
Kj = (kjl,
,kjPj,0,.•,()),
kji^O Vl^i^Pj. For example, /3i(A) = A — a is of type pi = 1 with K\ = (1,0, • • • ,0) and /?fe = 25(A  2 5 a ) is of type p\ x p2 = 1 x 2 with K\ = (5,0, • • • , 0) and K2 = ( 3 , 4 , 0 ,    , 0 ) .
It is not difficult to check that if 0k is of type p\ x • • • x pr, then /3k has multiplicity m = 2PlR1CP1
+ + 2PrRrC%,
(9.52)
where Rj is the permutation number for (%i, • • • , kjPj). When kji ^ kji for i ^ l,Rj = pj\, a n d for (kji,,kjpj) = ( 1 , 1 , 2 ) , t h e n u m b e r Rj = 3!/2! = 3. Theorem 9.7 Let /3k(X) be of type pi x • • • x pr at \  \k\2a, and the condition (9.51) hold true. Then the following assertions hold true. (1) The problem (9.21) with (9.24) bifurcates to an invariant set Y,\, homologic to the (m — 1) —dimensional sphere, with m given in (9.52), from (u,X)  (0, \k\2a) on A > \k\2a if a 3 > 0, and on A < fc2a if a3 < 0. (2) For each given pj(\ < j ^ r), the invariant homologic sphere Y,\ contains C% (kpj)~dimensional tori Tkpi for 1 < k < N, N = RjC%. Proof. The proof of Assertion (1) is the same as that of Theorem 9.3, we only need to prove Assertion (2).
290
Bifurcation Theory and Applications
For simplicity, we consider the special case that /?& is of type pi =2 for n = 3, with RiCg = 2 x Cf = 6 indices Kn = (1,2,0), #12 = (2,1,0),
# 2 1 = (1,0,2), K22 = (2,0,1),
#31 = (0,1,2), #32 = (0,2,1).
Each index, is associated with 2 Pl = 4 eigenfunctions. For instance, # n = (1,2,0) is associated with e
n = cosxi cos2a:2,
e{\ = sin x\ cos 2^2,
e
ii = cosa;isin2x2,
e\x = sin 2:1 sin 2^2.
Let the eigenfunctions associated with # y be ertj (1 < j ^ 2, 1 < i ^ 3, 1 < r ^ 4). For each pair (i,j), there is an even eigenfunction in {e^\l ^ r < 4}, denoted by e^. Since the function /(w) in (9.22) is a polynomial, the space
Ek = I £ < / P K + • • • + e)hY \eIn = e\njn,yp G M1 1
(9.53)
is invariant for the operator L\ + G defined by (9.25). Obviously /3fc(A) is a simple eigenvalue of L\ restricted on E^ at A = fc2a, therefore L\ + G has a bifurcated singular point from A = \k\2a on A > A;2a if a^ > 0 and on A < fc2o; if as < 0. Let v,k(x, A) be the bifurcated steady state solution of (9.21) from A = fc2a in Ek Then, by the invariance of translation, the set T = K ( i + M )  « = (»i)fl2)«3)GR3} is a 2k—dimensional singularity torus (2k = p\k). For every fc(l ^ k ^ N,N = RiC?1 = 6), there are Cjy spaces as (9.53), hence the invariant set EA contains at least C^ singularity tori TPlfc, and Assertion (2) for n — 3 and type p\ = 2 is achieved. In the same fashion, we also derive the assertion (2) for the general case. Thus, the proof is complete. • 9.2.4
Saddlenode bifurcation
Let fi C R" be a general bounded domain. We assume that a2 # 0, a2p+i > 0 in (9.22).
(9.54)
Pattern Formation and Wave Equations
291
Let the first eigenvalue p\ of (9.27) have multiplicity m > 1, and {ei, , e m } be the first eigenfunctions. We also assume that x = (xi, • • • , xm) — 0 is an isolated zero point of the following equations m
Y^ a^xtXj = 0
VI < k < m,
(9.55)
where a
ij
=
/
e e e
ijk
dx.
Then, for the CahnHillard equation with the Neumann boundary condition, we have the following theorem. Theorem 9.8 Let the conditions (9.54) hold, and x = 0 be an isolated zero point of (9.55). Then the following assertions hold true. (1) The problem (9.21) with (9.23) has a steady state bifurcation from (u, A) = (0,api), and there is at least one bifurcated branch on each side of X = api. (2) If p\ is simple, then (9.21) with (9.23) bifurcates from A = api on A > api to an attractor consisting of one steady state, which attracts a sectorial domain Dr{6) with angle 9 = n and some radius r > 0. (3) The equation (9.21) with (9.23) has a saddlenode bifurcation point (uoi Ao) € Hi xR1 with Ao < ap\. Assertions (1) and (2) of Theorem 9.8 are a direct corollary of Theorems 4.1 and 6.5, and Assertion (3) can be proved in the same method as used in Theorem 7.16. 9.3 9.3.1
Complex GinzburgLandau Equation Setup
In this section, we study the bifurcation of attractors and invariant sets of the complex GinzburgLandau equation, which reads f " ^  (" + */3)AM + (o + ip)\u\2u  Xu = 0, [ u(x, 0) = <j> + iip,
(9.56)
292
Bifurcation Theory and Applications
where the unknown function u : ft x [0, oo) —> C is a complexvalued function, ft c R"(l ^ n < 3) is a bounded domain. The parameters a,(3,a,p and A are real number and a > O,cr > 0. Equation (9.56) is supplemented with either the Dirichlet boundary condition u\an = 0,
(9.57)
or the periodic boundary condition ft = (0,27r)n and u is ft  periodic.
(9.58)
The complex GinzburgLandau equation arises in models of fluid dynamics as the amplitude equation governing the instability waves. It is found, for example, in the study of Poiseuille flow, the nonlinear growth of convection rolls in the RayleighBenard problem and TaylorCouette flow. In this case the bifurcation parameter A plays the role of a Reynolds number. This equation also arises in the study of chemical systems governed by reactiondiffusion equations. For the boundary conditions (9.57), we set tf = L 2 (ft,C), iTi = .ff 2 (fi,c)nff 0 1 (n,c), and for the boundary conditions (9.58), we set H = {u £ L2(Q, C)\u is fi  periodic}, Hi = {u £ H2(Q,C)\u is ft  periodic}, where
L2(ft,C) = {m+ iu2\ ui,u2
eL2(Q)},
Hk(£l,C) = {«i +»«2 «i,U2 e Hk(Q)}. The operators associated with the GinzburgLandau equation (9.56) are defined as Lx + G = A + Bx + G : Hi » H,
(9.59)
293
Pattern Formation and Wave Equations
where  Au = (a + i/3)Au, B\u = Xu, Gu = — (cr + ip)\u\2u.
9.3.2
Dirichlet boundary condition
Let Afc be the k—th eigenvalue of —A with the Dirichlet boundary condition. Then we have the following bifurcation theorem for the complex GinburgLandau equation with the Dirichlet boundary condition. Theorem 9.9 For the problem (9.56) with (9.57), the following Assertions hold true. (1) When A ^ a\i,u = 0 is a global asymptotically stable equilibrium point of (9.56) and (9.57). (2) When X crosses aXi, i.e. for any aX\ < A < aAi + e for some e > 0 the problem (9.56) with (9.57) bifurcates from (u,X) = (0,aAi) a cycle attractor Y,\ = S1, which attracts H\T, where T is the stable manifold ofu = Q having codimension two in H. (3) If f32 + p2 ^ 0, then the attractor T,\ = S1 is a periodic orbit, and if 0 = p = 0, then SA = S1 consists of steady state solutions of (9.56) and (9.57). (4) If Xk has multiplicity m ^ 1, then the problem (9.56) with (9.57) bifurcates from (u,X) = (0,ajAfc) on X > aXk an invariant 2m — 1 dimensional homologic sphere Y,\. Moreover, if j32 + p2 ^ 0 then there is no singular point of (9.56) and (9.57) in Y,\. Proof.
We proceed in several steps as follows.
STEP 1. Let u = u± +iu2 The GinzburgLandau problem (9.56) with (9.57) can be equivalently written as follows
jr— = aAui — f3Au2 + Xui — a\u\2ui + p\u\2U2, ' ^=/3Au1+aAu2 (u1,u2)(x,0)
+ Xu2a\u\2u1p\u\2u1,
( 9 " 6 °)
= (
We shall apply Theorems 6.1, 5.2 and 5.10 to prove this theorem.
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Bifurcation Theory and Applications
The mappings L\ = A + B\ and G : Hi > H defined by (9.59) are also rewritten as follows
Au=(aAui~f3Au2]
\/3Aui + aAu2J '
r
_
* " —
fa\u\2u1+p\u\2u2\
I
I
19
119
I •
\—a\u\'iu2 — p\uYu\)
It is known that H1/2 = H&(Q, C), therefore G : # 1 / 2 > i? is C°°. It is easy to see that the eigenvalues of A are given by Vfc = l , 2 , "  , a\k+ip\k lim At = +oo, with corresponding eigenfunctions Zk = ek ± «efc,
efc the eigenfunctions of — A,
and {ek,iej\ 1 ^ k, j < oo} is an orthogonal basis of H. These properties implies that L\ : H\ —> H is a sectorial operator. The eigenvalues of L\ are as follows {\a\k)±i(5\k,
fc
= l,2,.
(9.61)
Hence, the conditions (6.2)(6.5) in Theorem 6.1 are fulfilled. At A = a\\ it follows from (9.60) that
I^! 2 at
=
_ f [alVw^aAilu^+aH4] dx Jn
< — a / u4cfo,
which implies u = 0 is globally asymptotically stable at A = a\\ for (9.60). Thus, by Theorem 6.1 we infer that (9.60) bifurcates from (u, A) = (0, a\\) an attractor Y,\ which attracts H \T. STEP 2. We shall prove that (9.60) bifurcates from (0, a\k) an invariant sphere SA = S2m~l for any k > 1.
Pattern Formation and Wave Equations
295
Let {ei, ,em} be the eigenfunctions corresponding to Afc. By the canonical reduction (3.54) and (3.55), for m
m
vi = ^2 x{eu
v2 = ] P yiei,
we can obtain the reduction equations of (9.60) as follows —i = (A  a\k)xi + f3yi+ \a / \v\2vxei dx at L Jn +p f \v\2v2eidx\
+o(\x\3,\y\3),
' d • r  ^ = (A  aXk)yi f3xi+\o I \v\2v2eidx dt L Jn p /  u  V e i d J +o(a; 3 ,y 3 ). Jn J K
(9 62)
"
Denote the vector field in (9.62) by V(x, y). Then we have + \y\2)a f ^4 dx + o(\x\\ \y\A). (9.63) Jn By Theorems 5.2, we deduce that (9.62) bifurcate from ((x,j/),A) = (0, a\k) on A > aXk an homologic sphere attractor Y,\ = S2m~1, therefore (9.60) bifurcates from (u, X) = (0,aAfe) on A > aXk an invariant homologic sphere Y,\. (V(x,y),(x,y)) = (\a\k)(\x\2
3. We now prove that £* has no singular points of (9.60) provided P +/3 ^O. When (3^0, the eigenvalues (9.61) of L\ at A = aXk are nonzero, therefore L\ : Hi —* H is a linear homeomorphism at A = aX^, which means that (9.60) has no steady state bifurcation. As /3 = 0 and p =fi 0, from (9.60) we have 2
STEP 2
which implies that EA has no singular point of (9.60). STEP 4. Finally, as /3 = p = 0 the equations (9.60) are as in (8.24). By Theorem 8.6 we see that near A = aXi the attractor T,\ = 511 consists of singular points of (9.60). This theorem is proved. •
296
9.3.3
Bifurcation Theory and Applications
Periodic boundary condition
For the periodic boundary condition, the eigenvalues of L\ are
f3k = (\a\k\2)
+ i\k\2f3,
where K
= \k\, • • • , kn),
fc2 =fc?+ ... + *#, and the eigenfunctions are cos kx ± i cos kx,
cos kx ± i sin kx,
sinfez± i cos kx,
smkx±isinkx,
where fcx = fci^i H h k n x n . The first eigenvalue of Lx at A = 0 is /3i = 0, which has multiplicity m = 2. In the same fashion as used in the proof of Theorem 9.9, we can obtain the following bifurcation theorems for the periodic boundary condition. Theorem 9.10 For the periodic boundary condition (9.58), we have the following assertions. (1) If A ^ 0, u = 0 is global asymptotically stable. (2) If A > 0 the problem (9.56) with (9.58) bifurcates from (u, A) = (0,0) a cycle attractor T,\ = S1, which attracts H \ F, where T is the stable manifold ofu — 0 with codimension two in H. (3) If p ^ 0 then Y*\ is a periodic orbit, and if p = 0 then S,\ consists of singular points of (9.56) with (9.58). Theorem 9.11 For the GinzburgLandau equation (9.56) with the periodic boundary condition (9.58), we have the following assertions. (1) If\>a, the problem (9.56) with (9.58) bifurcates from (u, A) = (0,a) to an invariant An — 1 dimensional homologic sphere £;\. (2) If (P + p2 7^ 0, then 'Ex contains no steady state solution of (9.56) with (9.58), and if (3 = p = 0 then H\ contains at least C^ (k + 1) —dimensional singularity tori T fc+1 for every k (1 ^ k ^ n); (3) If the eigenvalue fa of L\ has multiplicity AN (N > 1), then the problem (9.56) with (9.58) bifurcates from (u, A) = (0,aA;2) on A > A:2a: to an invariant AN — 1dimensional homologic sphere Y,\, which contains no singular points provided 01 + p2 ^ 0.
Pattern Formation and Wave Equations
297
(4) If P = p = 0, and the eigenvalue (3k is of type pi x • • • x p r (l < pi < • • • < Pr ^ n), then for each given pj(l ^ j ^ r) the bifurcated invariant sphere SA contains at least C^(kpj + 1) dimensional tori T fepj+1 for each k (1 < k < N, N = RjClj as in Theorem 9.7).
9.4
GinzburgLandau Equations of Superconductivity
The main objective of this section is to study the nature of the phase transition from normal to superconducting states, which occurs when the temperature of a sample decreases. The rigorous analysis is conducted using the bifurcation theory presented in previous chapters. Superconductivity was first discovered in 1911 by H. Kamerlingh Onnes, who found that Mercury had zero electric resistance when the temperature decreases below some critical value Tc. Since then, one has found that large number of metals and alloys possess the superconducting property. In the superconducting state once a current is set up in a metal ring, it is expected that no change in this current occurs in times more that 1010 years (see [Tinkham, 1996]). In 1933, the other important superconducting property, called the diamagnetism or the Meissner effect, was discovered by W. Meissner and R. Ochsenfield. They found that not only a magnetic field is excluded from entering a superconductor, but also that a field in an originally normal sample is expelled as it is cooled below Tc. One central problem in the theory of superconductivity is the nature of the phase transition between a normal state, characterized by an order parameter that vanishes identically, and a superconducting state, characterized by the order parameter that is not identically zero. In this section, we address this problem by conducting rigorous bifurcation and stability analysis for the time dependent GinzburgLandau (TDGL) model of superconductivity.
9.4.1
The model
Let Q, C R™ (n = 2 or 3) be a bounded open set. We consider the attractor bifurcation of the TDGL equations of superconductivity defined on Q. The following three unknown functions are involved in the mathematical formulation: a complex valued function ip : Q —> C for the order parameter, a vector valued function A : fl —> R3 for the magnetic potential and a scalar
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Bifurcation Theory and Applications
function (j> : Q —> M.1 for the electric potential. The TDGL model reads
(9.64)
+L(hiV +Afip = 0, Zms
c
J = a{At + Vci>)^\iP\2A msc c
(9.65)
— J = curl2A  curltfa,
(9.66)
^^VvvW),
where h is the Planck constant, es and ms the charge and mass of a Cooper pair, a the conductivity of the normal phase, D the diffusion coefficient, c the speed of light, J the supercurrent, Ha the applied magnetic field, and 4>* the complex conjugate of tp. The parameters a = a(T) and b — b(T) are coefficients satisfying the following conditions (see, among others, [de Gennes, 1966]):
J >0
forT>Tc,
{ <0
for T < Tc,
a = a(T) <
b = b{T) > 0. Here Tc is the critical temperature where incipient superconductivity property can be observed. In the BCS theory, for instance, they are given (see see [de Gennes, 1966]) by
{
T _T a(T)=N(0)—^,
N(O)
(9 67)

>(T) = 0 . 0 9 8 ^ . Equations (9.64) and (9.65) are the TDGL equations generalized by P. L. Gor'kov and G. M. Eliashberg [Tinkham, 1996; Gor'kov, 1968], and (9.66) is the classical Maxwell equation. The order parameter ip describes the local density ns of superconducting electrons: \ip\2 = ns. In addition, tp is proportional to the energy gap parameter A near Tc, which appears in the BCS theory.
299
Pattern Formation and Wave Equations
Nondimensional forms From both the mathematical and physical points of view, we introduce here two nondimensional forms of the TDGL equations: one of which is used often in the literature, and the other is more suitable for the study of bifurcation and stability analysis presented in this section. For convenience, we start with the dimensions of various physical quantities. Let m be the mass, L the typical length scale, t the time, and E the energy. Then we have E:L2m/t2,
h: Et,
D:L2/t,
e2 : EL,
a: 1/t,
c: L/t,
a:E,
b : EL3,
i> : l/Lz'2,
A : (£/L) 1 / 2 ,
H : (£/L 3 ) 1 / 2 
Then we introduce some physical parameters: V>o2 = \a\/b, Hc = ( 4 7 r  a  » 1 / 2 , A = A(T) =
(msc2b/4ire23\a\)1/2,
e = ^(T) = V(2msa)1/2) K = \H, j] = AnaD/c2, T
= \2/D.
Physically, V>o2 stands for the equilibrium density, Hc for the thermodynamic critical field, A = A(T) for the penetration depth, £(T) for the coherence length, and r for the relaxation time. The ratio of the two characteristic lengths K = A/£ is called the GinzburgLandau parameter of the substance. When 0 < K < 4=, the material is of the first type, and when K > 4=, the material is of the second type. We now introduce the nondimensional variables (those with prime): V = •0oV )/ ;
x = Az',
t = rt',
V2HCX A = —— A ,
DV2HC
K
K
V2H
C Ha = ——H a. K
Then we have the following traditional nondimensional TDGL equa
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Bifurcation Theory and Applications
tions (we henceforth drop the primes) i>t + IKW + «2(02  1)^ + (iV + Afij) = 0, •q(At + V
for the case where a < 0, or equivalently T
Vb .
AO =
e2
hi
feshc2\1/2
\WB)
a=
'
2aVbmsD
M=
—&'
_ 471crZe2. czn
_ 2mSD h
hD
^'
_ 4ne2s mscH
and
= /x', A = A0A',
t = TDt', 0 = <^',
x
V = r3/y, Fa = r ^ f f i .
Then we have the second type of nondimensional TDGL equations (we drop the primes too): ' V>t + i<jnl> = (ifiV + A)2ip + aip~ (3\ip\2ip, 2 2 < ((At+fiV(i>) = cm\ A + curlHa7\ip\ A
^968)
_2^! (v ,*vV'VVV'*). We shall see in later discussions that the parameter a plays a key role in the phase transition (or bifurcation), which is given in terms of dimensional quantities by ._
2VbmsDN0
TC~T
Boundary conditions A physically sound boundary condition for the order parameter is given by Cx(iW + —A)tpn = C2ihip,
on dfl,
(9.69)
Pattern Formation and Wave Equations
301
which means that no current passes through the boundary, where n is the unit outward normal vector at dQ, and C\, C^ > 0 are constants depending on the material to which the contact is made. Physically, they satisfy ([Tinkham, 1996; de Gennes, 1966])
{
Ci = 0,
C\ j£ 0
for an insulator on <9£2,
d = 0,
C2 j= 0
for a magnetic material,
(9.70)
0 < C2/C1 < 00 for a normal metal. We note that the equations (9.649.66) with (9.69) is invariant under the following gauge transformation
(V, A , ® > ^ e i e , A  W,  6» t ), where # is an arbitrary function. If we take 9 such that he — A6 = div A G
he dO
— TT = A • n
es on
in fi, on aS7,
then we obtain an additional equation and a boundary condition; see also [de Gennes, 1966; Tang and Wang, 1995]: div A = 0
in 0,
(9.71)
An = A • n = 0
on afi.
(9.72)
Another boundary condition often imposed for A is a curL4 xn = Haxn
on 3fi.
(9.73)
TDGL equations of superconductivity With the gauge taken such that (9.71) and (9.72) hold true, the nondimensional TDGL equations are ' ipt + icpip = (i/uV + Afip + aip /3\ipfip, C{At + nV) = curl 2 ^ + curlffo  j\i>\2A
. divA = 0.
(9.74)
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Bifurcation Theory and Applications
The initial conditions are given by iKO) = ifo,
.4(0) = Ao.
(9.75)
The boundary conditions are one of the following: NEUMANN BOUNDARY CONDITION.
For the case where ft is enclosed by
an insulator: dib  I  = 0,
An = 0,
curLA x n = Ha x n
on9fi.
(9.76)
DlRICHLET BOUNDARY CONDITION. For the case where ft is enclosed by a magnetic material: ip = 0,
An = 0,
curL4 xn = Haxn
ROBIN BOUNDARY CONDITION.
onffl.
(9.77)
For the case where ft is enclosed by a
normal metal: Z.+Cip = 0,
An = 0,
C/Tt
curlA xn = Haxn
on dft.
(9.78)
Remark 9.4 If the material is a loop, or a plate Cl = Q x (0, h) with the height h being small in comparison to the diameter of Cl, then it is reasonable to consider the boundary condition with periodicity either in redirection or in (x, j/)directions.
9.4.2
Attractor
bifurcation
Mathematical setting It is known that for a given applied field Ha with div Ha = 0, there exists a field Aa such that
{
curl Aa = Ha
in fi,
div Aa = 0,
in
Aa • n = 0
on d£l.
ft,
(9.79)
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Pattern Formation and Wave Equations
Let A = A + Aa. Then (9.74) are rewritten as ' V>t + %H> = (*MV + Aa)2ip + c*V  2A, • Aip  2i/j,A • VV>  \A\2ip  /SIVlV, <
C(At+^cf>) = cml2AjAa\^\2yA\^\2
2
(9.80)
 fVvv>w*),
dWA = 0, with the following initial and boundary conditions V(0) = Vo, An = 0,
(9.81)
A(0) = Ao,
curU x n = 0,
on dft,
(9.82)
together with one of the following three boundary conditions for ip: NEUMANN BOUNDARY CONDITION:
^ = 0 on
on d£l,
(9.83)
DlRICHLET BOUNDARY CONDITION:
V> = 0
on DO,,
(9.84)
ROBIN BOUNDARY CONDITION:
^ + Cip = 0 on 0^. (9.85) on Hereafter we use Hk(Cl,C) for the Sobolev spaces of complex valued functions defined on f2, and Hk(£l,M.s) for the Sobolev spaces of vector valued functions defined on fi. Let H%(Q, C) = {V> S F 2 (Q, C)
 V satisfy one of (9.83)  (9.85)},
D2(Q, R 3 ) = {A e iJ2(£7, R 3 ) 2
3
2
3
£ (fi,R ) = {.4eL (r2,R )
 div^ = 0, A satisfy (9.82)}, 
dwA = 0,An\9n = 0}.
We set tf = L 2 (n,C) x£ 2 (J7,]R 3 ), Hi = H2B{Q,C) x D 2 (O,R 3 ).
304
Bifurcation Theory and Applications
Let P:L2(fi,R3)>£2(ft,R3) be the Leray projection. Then it is known that the function <j> in (9.80) is determined uniquely up to constants by
CMW = (/  P) [yi(VW  V* WO  7M V + A*)] ,
(9.86)
where I is the identity on L2(Q,R3). Namely, for every u = (ip,A) e Hi, there is a unique solution of (9.86) up to constants. Therefore, we define a nonlinear operator $ : H± —» L2{Q) by $(u) =(/>= the solution of (9.86) with /
Jn
(9.87)
Eigenvalue problems In order to describe the dynamic bifurcation of the GinzburgLandau equations, it is necessary to consider the eigenvalue problems of the linearized equations. Let ct\ be the first eigenvalue of the following equation (i/iV + Aa)2<tp = anl)
Vx e Q,
(9.88)
with one of the boundary conditions (9.83)  (9.85). It is clear that (9.88) can be equivalently expressed as f  /i2AV>l + \Aafip!  2nAa • VV>2 = Q^l, [  M AV2 + Aa2V2 + 2ftAa • V^i = anfa,
(9.89)
where i/> = ipi + iip2It is not difficult to check that (9.89) with one of the boundary conditions (9.83)(9.85) is symmetric. Therefore, there are an infinite real eigenvalue sequence of (9.88) ( ai
{
<
a2
<
•• • ,
r
I l i m a.k = oo, k k—>cx)
(990)
and an eigenvector sequence {eneH2B(Q,C)
 n = l,2,...},
which is an orthogonal basis of L2(Q, C).
(9.91)
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Pattern Formation and Wave Equations
The eigenvalues of (9.88) always have even multiplicity, i.e. if ip is an eigenvector of (9.88), then eieip (^ e R1) ar e also eigenvectors correspondin g to the same eigenvalue. Let the first eigenvalue a\ have multiplicity 1m (m > 1) with eigenvectors e 2 / t  i = i p k i +i4>k2,
e2k = 4>k2 + i t p k i ,
l < k < m .
(9.92)
We know tha t a\ enjoys the following propertie s c*i = ai(Aa)
depends continuously on Aa, forA^O,
^ (9.92)
(g
a\(0) = 0
for the boundar y condition (9.83),
ai(0 ) > 0
for either (9.84) or (9.85).
Now, we consider another eigenvalue problem, which is also crucia l for the attracto r bifurcation of (9.80). The problem read s ( cm\2A \V(j) = pA, < divA = 0, [An\dn
= 0,
(9.94) curlA x n  a n = 0.
Here, we remar k that the boundar y condition in (9.94), i.e. (9.82), is the free boundar y condition, which can be expressed as Allan = 0,
dA ~\an = 0,
(9.95)
where r is the tangent vector on dSl. To see this, for a given point zo £ dQ, we take (T\, r 2 , n) as an orthogona l coordinat e system, where T\ , T2 ar e unit tangent vectors and n the outward unit vector at x 0 € dCl. Then, by An\gn = 0 ,we find tha t ... dAT2 8ATl (dAT2 dATl\ curU(io) = —jr^n + ^T2 + —2 _ _ _ ! !
n
Hence we have curlA(x0) x n = ~TI + ~T2 on on x=x0 which implies tha t (9.82) is equivalent to (9.95). It is known that there ar e a rea l eigenvalue sequence
f 0 < Pl < p2 < • • • , { v \
hm pk = oo,
k—too
(9.95) ( v
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Bifurcation Theory and Applications
and an eigenvector sequence {ak e D2(n,R3)
 fc = 1 , 2 ,  } ,
(9.97)
which constitutes an orthogonal basis of £ 2 (f2,R 3 ). Main theorems In superconductivity, the parameter a can not exceed a maximal value a(T) < a(0). Hence, we have to impose a basic hypothesis:
„,<„<„) = 1 ^ 5 * ,
(9.98)
where c*i is the first eigenvalue of (9.88), and No the density of states at the Fermi level. In this subsection, we consider the case where the first eigenvalue a± of (9.88) has multiplicity two. We start with the introduction of a crucial physical parameter, which determines completely the dynamic properties of the bifurcation behavior of the GinzburgLandau equations. Let e £ H2(Cl, C) be a first eigenvector of (9.88). Then there is a unique solution for
{
curl2 A + V
(999)
•Ao • n\dn = 0, curL40 x "an = 0. We define a physical parameter R as follows 0 2 L curL4o2dz
R=
7+
JJeJ
•
< 9  10 °>
It is clear that the parameter R is independent of the choice of the first eigenvectors of (9.88). Since the first eigenvector e of (9.88) and ho = curlAo given by (9.99) depend on the applied magnetic potential Aa and the geometric properties of fi, the parameter R is essentially a function of Aa, fi and physical parameters (3,7,/i. The main results in this section are the following theorems. Here, we always assume that the first eigenvalue a\ of (9.88) with one of the boundary conditions (9.83)  (9.85) is complex simple, and the condition (9.98) holds true.
Pattern Formation and Wave Equations
307
Theorem 9.12 If the number R defined by (9.100) satisfies R < 0, then for the problem (9.80)  (9.82) with one of (9.83)  (9.85), the following assertions hold true. (1) If a < ct\, the steady state (ip,A) — 0 is locally asymptotically stable for the problem. (2) The equations bifurcate from ((ifi,A),a) = (0,ai) an attractor E a for a > oc\, which is homeomorphic to S1, and consists of steady state solutions of the problem. (3) There is a neighborhood U C H of (tp, A) = 0 such that the attractor T,a attracts U \T in H, where V is the stable manifold of (tp, A) = 0 with codimension two in H. (4) Each (ip,A) £ S a can be expressed as 1/2
(
/
curl 2 ^ =  7 ^ ^ \
• [\e\2Aa + /i/m(eVe*)] .
(aai\
1
l/2\
(9101)
_ 7R/ne4tfa
I * " /ne»d* ' where e is the first eigenvector of (9.88). Theorem 9.13 If R > 0, then for the problem (9.80)(9.82) with one of (9.83)  (9.85), we have the following assertions (1) The steady state (i/>,.4) = 0 is locally asymptotically stable at a < a\, and unstable at a>_a\. (2) The equations bifurcate from ((ip,A),a) = (0, a\) to an invariant set S Q on a < ai, and have no bifurcation on a > a\. (3) S a = S1 is a circle consisting of singular points of the equations, and has a twodimensional unstable manifold. Remark 9.5
The parameter R defined by (9.101) can be equivalently
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Bifurcation Theory and Applications
expressed as follows 00
1 fC I2 2 2V(\e\ Aa+2^e2Ve1).ak\ * = ^ + — 7
7 / \e\4dx
,
(9102)
Jn
where e = e\ + ie 2 , /ofc are the eigenvalue of (9.94) given by (9.96), and {a^} the normalized eigenvectors given by (9.97) Remark 9.6 Theorems 4.1 and 4.2 show that the two cases with R < 0 and R > 0 have completely different superconduction transition characteristics, see Section 5 for further discussion. Remark 9.7 It is readily to check that if a = 0, ($,A) = 0 is globally asymptotically stable for (9.80)  (9.82) with one of (9.83)  (9.85) . If R > 0, Theorem 4.2 implies that there exists a ao(O < a 0 < a\), such that if a < a\, the equation has no nonzero singular points, and if a = oto, the equation generate at least a cycle So of singular points, and if a > ao, the equations bifurcate from S o to two cycles £„ and Y?a consisting of singular points, such that Z^ is as described in Theorem 4.2, and E2, is an attractor with dist(£ 2 ,0) > 0 at a = «i. Proof of Theorems 9.12 and 9.13 We proceed in the following several steps. STEP 1. We set the mappings La = —K + Ba and G : H1 —> H by KU
{
r'curl 2 ^ ) ' t itl>$(u) + 2Aa • Aip + 2ifiA • VV + \A\2ip + p\ip\2ip \
where u = (ip,A), <&(u) is defined by (9.87), and P the Leray projection. Thus, the problem (9.80)  (9.82) with one of the boundary conditions (9.83)
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Pattern Formation and Wave Equations
 (9.85) can be rewritten in the following operator form
t%=Lau + G(u), u=(1,,A)eHl,
(9io3)
\ «(0)  w0. We see that La : H\ —> H is a sectorial operator, and the eigenvalues of La satisfy that
{
<0
ifa
= 0
ifa = ai,
>0
ifa>ai,
(9.104)
and for j > 3,
fft(a 1 )=a 1 a fc orr 1 p J ,
{gm)
\f3j(a1)<0, for some A; > 1, I > 1. It is clear that the operator $ : # i > L2(fi, C) defined by (9.87) is C°°, and by the estimates proved in [Tang and Wang, 1995] for $, we have
r
2
[r
/ \$(u)ip\ dx< \
3
l2/3rr
\$(u)\ dx\
\
1
1 / 3
6
\i>\ dx\
< c ( H ^ i / 2 + HH1/2V'lli4)2V'lli6, where Hi/2 is the closure of Hi for the Hlnorm. Hence, it is not difficult to check that there is a number 1/2 < a < 1 such that G : Ha —> H is C°°. STEP 2. It is known that the dynamic bifurcation of (9.103) is determined by its reduced equation to the center manifold. Let
ipo E Ei = {ze  z <E C,
e the first eigenvector of (9.88)}.
Then the reduced equation of (9.103) reads ^
= /Ji(a)V>o  PiG(V>0 + ^(Vo), A(1>o)),
(9106)
where Pi : H —> E\ is the canonical projection, and $(V*o) = (ip(ipo),A(ipo)) € Hi the center manifold function.
310
Bifurcation Theory and Applications
The k multilinear operators (k — 2,3) in G are given by /
G3{u) =
2Aa • Aip + 2ifj,A • Vip
{iiP$2(u) + \A\2iP +
7CUIVI2
~[
\
m2iP\
)'
where 3>2(w) is the bilinear operator in $(w). By the first approximation of the center manifold reduction, the center manifold function <£ = (•ip(ipQ),A('>po)) satisfies that
curlM + /iV
0(1 ^ i ^ m). We assume that 9i{x,Z,Q = o{\£.\,\t\), p
\gi(x, f, C) < C{\£\ + C + 1], 1 < i^ m,
(9.138)
(9.139)
Pattern Formation and Wave Equations
325
where c > 0, p is as in (9.134). By applying Theorem 6.2, we can derive the following theorem. Theorem 9.17 Under the conditions (9.138) and (9.139), if u — 0 is locally asymptotically stable for (9.137) at A = Ai, then the problem (9.137) bifurcates from ((u,ut),X) = (0, Ai) on A > Ai an attractorT,\ withml ^ dimT,\ ^ m, and T,\ has the homotopy type of(m — 1)— dimensional sphere Remark 9.9 tial operator
If the function g(x, u) = (gi, • • • , gm) in (9.137) is a poten, . dG(x, z) gi(x, z) = — — — , 1 ^ i < m,
for some scalar valued function G(x,z)(z £ R m ), and G(x,z)^^p\z\k
+ o(\z\k),
k>l,
then u = 0 is locally asymptotically stable for (9.137) at A = Ai. 9.6
Notes
9.1 The KSE arises in several physical contexts as an amplitude equation for spatiotemporal growth of instabilities such as flame fronts [Sivashinsky, 1980], reactiondiffusion problems [Kuramoto and Tsuzuki, 1976], and thin film flow down an inclined plane [Chang, 1986]. Extensive mathematical and numerical studies have been conducted for the KSE in the last twenty years or so, including, among many others, [Tadmor, 1986] on wellposedness, [Nicolaenko et al., 1985; Collet et al., 1993; Goodman, 1994] on existence of global attractors, [Foias and Kukavica, 1995] on determining nodes, [Foias et al., 1988; Temam and Wang, 1994] on inertial manifolds, [Kevrekidis et al., 1990; Jolly et al., 1990; Michelson, 1992; Zgliczynski, 2002] on bifurcations. The work presented in this section is based on the authors' recent work. 9.2 The CahnHillard equation models pattern formation in phase transitions; see [Cahn and Hillard, 1957]. There are many studies from the mathematical points of view; see, among many others, [NovickCohen and Segel, 1984; Alikakos and Fusco, 1998; Alikakos et al., 1994; Bates and Fife, 1990]. 9.3 This section is based on [Ma et al., 2004].
326
Bifurcation Theory and Applications
9.4 This section is based on [Ma and Wang, 2005a]. 9.5 This section is based on the authors' recent work, and the results are introduced here for the first time.
Chapter 10
Fluid Dynamics
In this chapter, we study dynamic bifurcations for two classical problems hydrodynamic stability and bifurcation: one is the RayleighBenard covection problem, and the other is the Taylor problem. 10.1 10.1.1
Geometric Theory for 2D Incompressible Flows Introduction and
preliminaries
The study of structural stability has been the main driving force behind much of the development of dynamical systems theory following the program initiated by S. Smale and others; see among others [Palis and de Melo, 1982; Peixoto, 1962; Pugh, 1967; Robinson, 1970; Robinson, 1974; Shub, 1978; Smale, 1967]. We are interested in the structural stability of an incompressible vector field with perturbations of incompressible vector fields. We call this notion of structural stability the incompressibly structural stability. We proceed in two cases: a) the free boundary condition case, and b) the Dirichlet boundary condition case. 10.1.2
Structural stability theorems
Let M be a two dimensional differentiable Riemannian manifold with boundary dM and with the Riemannian metric g. In this section, unless otherwise stated, we always assume that r > 1 be an integer. Let C^(TM) be the space of all rth differentiable vector fields v on M such that V\OM € Cr(TdM), namely the restriction of any rth differentiable vector field v e Cr(TM) on the boundary dM is a rth. differentiable vector field of the tangent bundle of dM. 327
328
Bifurcation Theory and Applications
Consider a vector field v £ C^(TM). A point p £ M is called a singular point of v if v(p) = 0; a singular point p of v is called nondegenerate if the Jacobian matrix Dv(p) is invertible; v is called regular if all singular points of v are nondegenerate. For convenience, we set DT{TM) = {v£ Crn{TM)\ div v = 0}, Br(TM) = {v£ Dr(TM)\ ^\dM
= 0},
Br0{TM) = {v£ Dr(TM)\ v\dM = 0}. Let $(x, t) be the orbit passing through x £ M at t = 0 of the flow generated by v. The wlimit set w(x) and the alimit set a(x) of the trajectory <&(x,t) axe defined by w(x) = {y S M  there exist tn —> oo such that $(x, £n) —> y}, a(x) = {y £ M \ there exist tn —> —oo such that $(x,tn) —> y}. An orbit with its end points is called a saddle connection if its a and wlimit sets are saddle points. Definition 10.1 Two vector fields u, v £ Dr{TM) are called topologically equivalent if there exists a homeomorphism of