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(0, x, u) = x and being defined for all t E R. If one interprets (2.1) as a control system, the first important question is that of reachability, i.e. denote by 0+(x) 0 there are n £ N, x = x o , . . . , x n = y in M , to .. . < n - i > T and uo ■ • . u „ _ i G U with d(<^(tj,Xi,Ui),x I + i) < e for all i = 0 . . . n — 1, where d(-, •) is a Riemannian metric on M, (ii) for all x £ E there is u 6 U with y>(i, x, u) 6 E for all t £ R, (iii) £? is maximal with the properties (i) and (ii). 0. Now the property of inner pairs means that the trajectory
(p)v|. p€M 0 Ei endowed with the norm ||(y)|, |Zty| = sup max|£V(y)v|. Let W be the metric space of smooth m-dimensional manifolds W which are graphs of maps (p G $. The metric in W is defined by dist(W\, W2) = \\ (y)), y G i?i- In fact, since W = f~xW | + (b + 5) £ |Z) y>|2. Setting p0(<5) = % + e), pi(£) = (a + e)(6 + 6) + £<5, p2(<S) = (6 + 6)e, consider the function a> : R+ —► R+ given by u(r) = p0 + p\r + P2^2- Since po(0) = P2(0) = 0 and pi(0) = 06 < 1 then for 6 small enough there is ft > 0 such that a;(r) < ft for r < ft. Therefore, if |Dy>| < ft then |£>A(rt| j_i) and \D(pj\ < ft then, as easily seen, Lip((p*) < ft. Let us prove that L(Ei, E2), L(Ei, E2) being the space of linear maps from E\ to E2, equipped with the norm |u;| = sup \u(y)\ = sup max |u;(y)i;|. y€£i „Z(/V)) + DzZ(P( *(P( \A\d then l ^ " 1 ^ > d. Since / = T outside # then for y G £ 1 , \y\ > \A\d, the map h(y) = P(y, *(P( *) + ( DaP( i(u>) © $ f M © «>, ™ G Eu). Recall that W ^ and Wc}s exist also when /(0) ^ 0; W/ and Wf are locally unique and all manifolds depend continuously on / in the C 1 topology. The following Center Manifold Theorem is due to Pliss [21] and Kelley [10]. -1 is its inverse mapping. Let us note Pi{x,x') < a^(/(x,w(x)),/(x',u(x))) < aptiftx,«(»)),
= {y E M ; there is u E U and t > 0 with y>(<, x, y) = y}
the positive orbit (reachable set) from x G M , and analogously by 0~{x)
= {yG M ; there is u 6 i/ and < < 0 with y>(tf,x,u) = y} = {y 6 M ; there is u E U and t > 0 with <^>(t, y, u) = x}
the negative orbit, then we are looking for conditions such that, given x, y G M , we have y E C? + (x) (or x E (9~(y)). Problems concerning reachability under additional assumptions, like stability, optimality, etc., can be treated once some basic results in this direction are obtained. In general, the answer to this question is very difficult. For linear systems of the form x = Ax + Bu in Rd with U = Rm a satisfactory criterion can be given (Kalman's controllability rank condition): (2.2) is completely controllable (for all x , y E R d ) iff rank {B,AB,..., Ad~1B) = d. T h e proof simply uses the explicit solution of (2.2) (variation of constants for mula) and the Cay ley-Hamilton theorem. If the rank of the reachability matrix (B, AB,..., Ad~lB) < d, one can steer x into y, if b o t h points are in the smallest A-invariant subspace of Kd which contains i?, i.e. in the linear space generated by the columns of the reachability matrix. Complete controllability of linear systems (with U = R m ) is equivalent to a seem ingly weaker condition, namely int 0+(x) ^ <j> for all x E Rrf (accessibility), where 'int' denotes the interior of a set. The equivalence is proved via the observation t h a t accessibility holds iff the Lie-algebra
C = CA{Ax + Bu; ue Rm} = CA{Ax, 6 l t ..., bm}
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FRITZ COLONIUS AND WOLFGANG KLIEMANN
has rank d for all x £ Rd. (Here the 6 1 , . . . , 6 m are the columns of B, and for a set H of vector fields CA{H] denotes the Lie algebra generated by the elements of H.) For x = 0 this is exactly Kalman's criterion. In this sense accessibility and controllability are the same for linear systems (2.2). In the case of nonlinear systems (2.1) there is a gap between these two concepts: accessibility does not imply controllability. The accessibility problem is very well understood these days with the help of geometric control theory: A somewhat sharper version (local accessibility, i.e. for all neighborhoods N C M of x and all t > 0 it holds that int 0
+ S u j X i ( x ) ; (ui) = u £ U] has rank d for all x £ M.
(These are the theorems of Frobenius [44] and Chow [20].) If C has the maximal integral manifolds property then the state space of (2.1) can be reduced to such a maximal integral manifold, since these are invariant for (2.1), and there the above Lie-algebraic condition holds, see e.g. Sussmann [88]. If the Lie-algebra C is not integrable, the orbits of (2.1) in M are still immersed submanifolds; however, one has to give up the connection between accessibility and the rank criterion. For complete controllability there is — under the assumption of accessibility — a series of sufficient conditions, all of which use specific properties of the system's vector fields, see e.g. the textbooks of Isidori [51] or Nijmeijer and van der Schaft [65]. However, in general this property is not even satisfied for relatively simple classes of systems, such as systems on the unit sphere S d _ 1 C K d , which are obtained as projections of linear systems in R d onto S r f - 1 , cp. Section 6. Therefore it is important to clarify the control structure of accessible systems, i.e. to identify those regions in M , where any two points are reachable from each other. At least on compact manifolds, or in compact, invariant sets of (2.1), such regions always exist. For the rest of this paper we always assume (local) accessibility, i.e. dim CA{X0
+ XuiXi]
(ui) = u£ U}(x) = d for all x £ M
(H)
holds, and also U C R m is compact and convex. 2 . 1 . D e f i n i t i o n . A set D C M is a control set of (2.1) if (i) D C ciO+{x) for all x £ JD, (ii) for all x £ D there is u £ U with ip(t, x,u) £ D for all t £ R, (iii) D is maximal (with respect to set inclusion) with the properties (i) and (ii). (Here 'c^' denotes the closure of a set.) In the context of control theory, in particular for the construction of suitable feedback laws, only those control sets D play a role, for which int D ^
CONTROL THEORY AND DYNAMICAL SYSTEMS
127
these control sets (ii) is automatically satisfied, and we have furthermore: Control sets are pairwise disjoint, connected, ci{int D) = c£ D, and int D C G+(x) for all x £ D, i.e. each point in int D is precisely reachable from any point in D. Besides controllability the concept of limit sets is crucial for the analysis of the long term behavior as t —* ± 0 0 , such as stability, global perturbations, control of complex behavior. For this aspect, a study of the control sets is, in general, not sufficient: Reachability is a finite time concept, i.e. y is reachable from x if there i s u G W and t > 0 with
Chain control sets are pairwise disjoint, connected, and closed. The following example illustrates the difference between control sets and chain control sets. 2 . 3 . E x a m p l e . Consider the control system on M = S 1 , given by
x = — sin 2 x + a cos 2 x — u(t) cos 2 x, x £ R mod 2?r,
a > 0,
U = [A, a] C R.
This system has four control sets with nonvoid interior (see Figure 1.)
D,=
0 , a r c t a n ( a — A)1?2
D3 = Dl + 7T,
,
D2 = In — arctan(a — A ) 1 / 2 , ^ ) , DA = D2 + 7T.
However, there is only one chain control set E = S 1 , cp. [24], where a general procedure is described to determine the control sets and chain control sets on one-
128
FRITZ COLONIUS AND WOLFGANG KLIEMANN
dimensional manifolds.
Figure 1. Control sets and dynamics for Example 2.3. This example shows in particular that chain control sets do not necessarily reflect the reachability structure of a system. Therefore, the relation between the control and the limit structure plays an important role in the sequel, in particular we look for conditions, under which the limit sets are contained in the interior of control sets. This will enable us to analyze various aspects of the long term behavior also for systems that are not completely controllable. So far, we have argued with the orbits and chain orbits of the system (2.1), where the contribution of each single trajectory {<£>(<,£, u); t G R} is lost. A precise analysis of the global fine structure of control systems is not possible with these concepts alone, just as for stochastic systems, where one has to use the associated stochastic flow to be able to answer certain questions. In complete analogy we will define the control flow o n W x M, which describes the precise behavior of (2.1) and which yields via the theory of dynamical systems, results for control systems. In this way another mathematical area, which has enjoyed substantial progress over the last years, can be utilized for control theory, in addition to the theory of ordinary differential equations, differential geometry, Lie groups and Lie algebras, linearization techniques and linear algebra. A continuous dynamical system (or flow) on a metric space S is given by a continuous m a p rj): R x S —> S with (i) t/'o = id and (ii) tpt+s = ij>t o r/)a for all t,s 6 R. (Here we have used ij>t = >(*,-).) For a given, nonconstant function u G U the m a p (t,x) *-* y?(t,a;,u) does not define a flow, as u is time varying. If one considers, however, the entire family of differential equations (2.1), indexed by u € U, then one can define a corresponding control flow on S = U x M, in the
CONTROL THEORY AND DYNAMICAL SYSTEMS
129
following manner: $:RxWxM-»WxM,
$(t,u,x)
= (u(t +
-),ip(t,x,u)).
The shift 6tu(-) = u(t + •) on U, which is itself a dynamical system, guarantees the desired group property (ii), and turns $ = (0, (?) into a skew product flow. For compact and convex U C R m the set U is a compact, complete, metric space, if equipped with the weak* topology o f W c L ° ° ( R , R m ) = ( L ^ R ™ ) ) * . This also yields continuity of $ . T h e use of the weak*-topology on U is appropriate for control theoretic considerations, because convergence of un —► u in U implies uniform convergence of ip(- , i , u n ) —*■ <^( - ,£,u) on compact time intervals. 2.4- Remark. For a fixed feedback u = F(x) 6 U with F Lipschitz continuous, the equation x = XQ(X) -f T,Fi(x)X{(x) defines, of course, a dynamical system on M. But for this one has to choose a certain feedback, which restricts the possible control functions a priori. Furthermore, for nonlinear systems it is not appropriate to consider only continuous feedback laws, see e.g. Sontag [84]. Sussmann [89] and [86] describe a construction to obtain piecewise smooth feedbacks from open-loop controllability of points for controllable systems. 2.5. Remark. T h e control flow $ consists in its first component (U,B) of the time shift, for which only the concepts of topological dynamics are appropriate. In the second component, which by itself is not a flow, we find the smooth dynamics of the vector fields — and hence linearization techniques etc. can be used there, cp. Sections 6. and 7. Similar constructions for the definition of flows corresponding to non-autonomous differential equations were described in the 60's by Sell [79] and Miller [63], in particular for the almost periodic case. In the theory of stochastic differential equations with random perturbations £t instead of u(t) in (2.1), it is common place to consider the system over the trajectory space of (t. The classical construction of Kolmogorov leads, for stationary £/, t o a shift invariant probability measure P on this space, which is considered given in the stochastic theory. In contrast, for the ergodic theory of control flows one first has to construct shiftinvariant measures, e.g. via the Krylov-Bogolgubov device, see e.g. [CK4]. Let us recall the following concepts from the topological theory of dynamical systems (see e.g. Maiie [63]). In the following (.?, VO denotes a flow on a metric space. 2.6. D e f i n i t i o n . The limit set u>(s) of 5 £ S is given by w(s) = {y € 5 ; there is tk —* oo with tp(tk,s)
—» y}.
(S,ift) is topologically transitive, if there exists s G S with u>(s) = S. (S, ip) is topologically mixing, if for all open sets V\,V2 C S there exist times To G R, Ti > 0 such t h a t for all n e N we have ^{-nTx + T 0 , Vi) D V2 ^
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FRITZ COLONIUS AND WOLFGANG KLIEMANN
A closed V'-invariant subset W C S is maximal topologically transitive, if (W, il>\w) is topologically transitive, and if for each closed W D W with ( W , if)\ w) topologically transitive, we have W = W. Similarly maximal topologically mixing sets are denned. We obtain a relation between these concepts for the control flow (U x M, $ ) and the control sets of (2.1) in the following manner: Define the lift of a control set D C M with int D ^
tp(t,x,u)
6 intD for all * G R } .
(2.3)
2.7. T h e o r e m . Let U C R m be compact and convex, and assume (H). (i) If D C M is a control set with int D ~£ <j>, then V, de£ned maximal, topologically transitive (and mixing) set with int
D = int TTMT>
and
by (2.3), is a
ci D = ci KM *D.
(2-4)
Here -KM denotes the projection from U x M onto the second component. (ii) If T> C U X M is a maximal, topologically transitive (or mixing) set of {U X M, $ ) with int KM *D ^ $, then there is a control set D C M satisfying (2.4). 2.8. Remark. In fact, under the assumptions of Theorem 2.7.(i) one obtains more: The periodic points are dense in {T>, $\v) and $ has sensitive dependence on initial conditions in T>. Furthermore, one obtains immediately from this theorem: The system (2.1) is completely controllable on M iff the flow {U x M, $ ) is topologically transitive (or mixing). For a proof see [25], Theorem 3.9. Next we describe the limit set structure of a control system and define for a flow ( 5 , T/») (compare, in particular, Conley [35]): 2.9. D e f i n i t i o n . Let e, T > 0. An (e, T)-chain from x £ S to y G S is given by n G N, x = x0 ■ ■ .xn = y in S, t0,... , t n _ i > T such t h a t d(t/>(i,-, Xi),X{+i) < e for i = 0 . . . n — 1. Here d(-, •) is the metric on S. The chain limit set of x G 5 is fi(x) = {y G 5 ; for all e, T > 0 there is an (e, T)-chain from x
toy),
and the chain recurrent set is defined as CR = {x G S; x G 0 ( z ) } . The flow ( S , ^ ) is c k m recurrent, if 5 = Ci?, and chain transitive, if y G ft(z) for all x,y G 5*. For the control flow (£/ x M, $ ) we lift again the chain control sets E C M to W x M via £ = c£{(u,x) 6 W x M ; p f c s . u ) G E for all t € R } . The analogue of Theorem 2.7 is the following result.
(2.5)
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131
2 . 1 0 . T h e o r e m . Let U C R m be compact and convex, and assume (H). For a set E C W X M, the set E = KM £ is a chain control set in M iff E is a maximal, chain transitive set of (U x M , $ ) . In particular it holds: (i) For all ( u , x ) G chain control set 7 T M ^ ( U , x) C E. (ii) For all minimal, such that TTM W
U x M with {(p(t,x,u), t > 0} bounded, there exists a E C M with w(u, x) 6 E, the corresponding lift, and hence (Here the limit set is with respect to the flow {U x M , $ ) . ) ^-invariant sets W C U x M there is a control set D C M C C£ D.
For the proof of Theorem 2.10 see [25], Theorem 4.1 and Lemma 5.3. Theorems 2.7 and 2.10 clarify some basic relations between the control and limit structure of (2.1) on the one hand, and the control flow on the other hand. In the next section we analyze the connections between these two structures, aiming at the control of (complicated) limit behavior. 3 . L i m i t B e h a v i o r of C o n t r o l l e d T r a j e c t o r i e s In this section we discuss the behavior of controlled trajectories as t -> ±oo. According to Theorem 2.10, the limit sets are contained in chain control sets and their lifts. For control theoretical purposes it is important to know, under which conditions trajectories enter areas of complete controllability. A characterization of this property is the aim of this section, various applications will be discussed in the remaining parts. We start by defining (partial) orders on the sets of control sets and of chain control sets in the following way: Let D\,D2 C M be control sets of (2.1), define Dx -< D2 if there is xl g Dx with 0^(xl)
D D2 ^
(3.1)
"-<" is a partial order on the set {D C M ; D is a control set of (2.1)}, and under the hypothesis (H) we have: 3.1. Lemma. (i) Open control sets are minimal, and closed control sets are maximal elements w.r.t. -<. (ii) Invariant control sets C C M, i.e. ciO+(x) = c£C for all x G C, are a/ways closed, and hence maximal w.r.t. -<. (iii) IfM is compact, or if K C M is a compact, invariant set of (2.1), then there exist (at least) one open and one closed control set in M (or in K C M, respectively), and these are exactly the minimal and the maximal elements w.r.t. -<.
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FRITZ COLONIUS AND WOLFGANG KLIEMANN
In a similar way we define for chain control sets E, E' C M E -<E'
if there are chain control sets E = Ex . . . Ek = E' and (tii,*i).
•. {uk,xk)
i n W x M with JTMW*(wi,x,) C E{
and 7rMW(ui, x.) C £,+i for « = 1...k
- 1.
(3.2)
(Here w* denotes the limit sets for t -* - c o . ) '-<' is a partial order on the set {E C M; £ is chain control set of (2.1)}, and it holds: For each control set D C M there exists exactly on chain control set E{D) with D C E{D). If D1 X D2, then £(£>!) =$ £ ( D 2 ) , and as Example 2.3 shows, equality may hold here. The connection with Morse decompositions is given by 3.2 Proposition. The number k of chain control sets is finite iff the control Sow (U x M, $ ) has a finest Morse decomposition, i.e. iff the number k' of the connected components of the chain recurrent set is finite. In this case we also have k = k'. The proof follows from Theorem 2.10. This theorem and Proposition 3.2. show that chain control sets and their order describe the limit behavior of the trajectories. For control sets we know first of all that each minimal set of the flow $ , projected onto M, is contained in the closure of a control set (Theorem 2.10(H)). From this fact we see immediately that for all (u,x) G U x M there is a control set D with TTMU;(U, x) n clD ^
\y)
\Ul(t)
u2(t))\y)
WlthU
[ 0 '2jx^ 1,2 ''
and its projection onto the unit sphere S 1 C R 2 . We parametrize S 1 by the angle a G R mod 2TT with a = 0 for x = 1, y = 0, and obtain for the projected system four control sets with nonvoid interior
*„(*.), „=[!'2], » . * + , „ . a + , and a continuum of one-point control sets
Da = {a}, « e S 1 \ | j A j ='A.
CONTROL THEORY AND DYNAMICAL SYSTEMS
The points in A are fixed points on S 1 corresponding to u\(t) see [26], Example 5.3 and [24].
133
= 0 and u2(t)
= 1,
Figure 2. Control sets and dynamics for Example 3.3. For a G A and u(t) = (°) we have 7r§iu;(u,a) = DQ, but the projected system cannot be steered back to a after a small perturbation to ft > a, and it cannot be stabilized at the point a . Therefore, we try to find conditions, under which 7TM<*>(U,X) C int D for some control set D holds. 3 . 4 . D e f i n i t i o n . T h e pair (u, x) G U x M is called an inner pair, if there are T > 0 and S > 0 with
: = {
(i) u(u,x) consists of inner pairs, (ii) 7TAfu;(u,x) C int D for some control set D C M of (2.1). The proof of this theorem can be found in [28], Theorem 4.4. T h e remaining problem is to find simple criteria for (w,x) such that u{u, x) consists of inner pairs. In Section 5. we will demonstrate strategies for two examples, the model of a
134
FRITZ COLONIUS AND WOLFGANG KLIEMANN
chemical reaction and the Lorenz equation. Here we point at two rather general criteria. 3.6. Remark. The maps ip(t, - , u ) : M -> M are diffeomorphisms on M , forming for t > 0 a semigroup S. If S is a subsemigroup of a Lie group acting on a homogeneous space M , then (H) implies that S has nonvoid interior in (U, x) C int D for some control set D, then the system can be steered from x to any point y G int D\ in particular, trajectories with a simpler limit behavior can be realized in int D, e.g. periodic orbits with sufficiently long period. Recall that through any point y G int D there passes a variety of periodic trajectories of the control system, and vice versa all periodic trajectories of (2.1) are contained in some control set. In Section 5. we will discuss two examples, but first of all we present some consequences for the perturbation theory of ordinary differential equations. 4. T w o P e r t u r b a t i o n T h e o r e m s for O r d i n a r y Differential E q u a t i o n s In this section we consider time varying perturbations of ordinary differential equations, and we study their limit behavior using the concepts from the previ ous sections. In the literature the perturbation problem is usually formulated like this: Given a small, possibly periodic perturbation of a differential equation, which qualitative properties (e.g. the existence of periodic solutions) remain valid under the perturbation (persistence, see e.g. Murdock [64]), and which ones are changed (structural stability, bifurcation, see e.g. Ruelle [75]). Here we ask the question in the following form: W h a t is the behavior under all possible, time varying perturbations with values in a given set? It turns out t h a t under certain richness assumptions on the perturbation, the limit sets of the perturbed equations and their relation to the limit sets of the original, unperturbed equation can b e described in detail.
CONTROL THEORY AND DYNAMICAL SYSTEMS
135
As a first step we consider perturbations with values in a bounded set. We for mulate the results for compact manifolds, but all hold as well for compact, invariant sets. Consider the differential equation x = XQ(x)
(4.1)
on a compact manifold M , together with time varying perturbed right hand sides of the form m
X0(x) + £ u i ( * ) X i ( s ) ,
(4.2)
t=i
where, as in Section 2., u G U with U C R m compact and convex. We analyze the limit behavior of the trajectories y>(t,x,u) for t —♦ ± o o . According to Lemma 3.1.(iii), there are on M (at least) one open and one closed control set of (4.2), and these are exactly the minimal (and maximal, respectively) elements with respect to the order -<, defined in (3.1). Furthermore, on a compact manifold there are at most finitely many open or closed control sets with nonvoid interior. 4 . 1 . T h e o r e m . Under Hypothesis (H) for (4.2) we have: The set {(u, x) G U x M ; there is T > 0 with: for all t < -T
we have
G int C*y some open control set, and for all t >T G int C, some closed (i.e. invariant)
control
we have
set}
is open and dense in U X M. In particular we have for the limit sets ITMW*(U,X) and nMw(u,x) in an open and dense subset ofU x M: 7r^o;*(u,x) C c£ C* and TTM<^(U,X)
C
C.
This result, the proof of which can be found in [28], has various stochastic and measure theoretic analogues, see e.g. Ruelle [74] or Kliemann [54]. It says that generically all limit sets are concentrated on the finitely many open or closed con trol sets; it therefore describes a relation between limit sets and the minimal and maximal topologically transitive components of the associated control flow. Out side of these control sets there are therefore only limit sets of a "thin" subset of perturbations u G U and initial values x G M (and they are contained in the lifts of chain control sets). 4-2. Remark. If D C M is a control set (with int D ^
FRITZ COLONIUS AND WOLFGANG KLIEMANN
136
point y G
(J
D' can be in the projection of a limit set with initial value x £ X.
The converse problem, i.e. is there for a control set D and a perturbation u E W a point x € D with TTMU>(U,X) C c£ D , is a central question in the ergodic theory of control and stochastic systems, see [24]. Simple examples show t h a t this need not be the case. This property is satisfied, however, for the action of linear semigroups on spheres or projective spaces, which will be considered in more detail in Section 6. We now t u r n to the problem of small perturbations of the differential equation (4.1). We embed (4.1) into the following family of differential equations m
x = Xo(*) + 5>i(0*i(*)
(4.3")
i=i
with u = (wi)i=i... m G Up := {u: R -> R m ; \u(t)\ < p}, where p > 0, and | • | is any norm on R m . We are interested in the connection between the Morse sets of (4.1) and the control structure of (4.3 P ). Let us assume that (4.1) posses a (finite) finest Morse decomposition M. — { M i , . . . , M n } , i.e. the Mi, i = 1 . . . n, are the connected components of the chain recurrent set associated with the flow of (4.1) on M. In particular, the attractors of (4.1) are the maximal elements of M., see e.g. Conley [35] or Ruelle [75]. We obtain the following result: 4 . 4 . T h e o r e m . Assume
that the chain recurrent
set of (4.1) consists
of
£nitely
n
many
connected
components
Mi . . . M „ .
For all p > 0 and all x G (J Mi let
(0, x) G Up x M be an inner pair of (4.3^). Then we have: (i) For all i = 1 . . . n there is an increasing family Dp of control sets of (4.3 P ) with Mi C int Dp, such that Mi = f) Dp. p>0
(ii) Vice versa, if for a sequence pk —► 0 and for control sets Dpk of (4.3 Pfc ) the set of limit points L := {y G M\ there is xk G DPk with Xk —► y} is not empty, then L = Mi for some i = 1 . . . n. For a proof see [28]. This result says that the chain recurrent components of the differential equation (4.1) are 'blown u p ' to the topologically transitive components of the control flow associated with (4.3 P ), if the embedding of (4.1) into the family (4.3 P ) is sufficiently rich, i.e. if (H) holds and if the points in the Morse sets are inner pairs of the flow. We would like to point out a connection of this result with two ideas in the theory of stochastic dynamical systems: Ruelle [74] embeds a differential equation into a stochastic system with small noise, which satisfies a nondegeneracy condition, and he shows that for t —* oo this system lives basically on the attractors. If we combine Theorems 4.4 and 4.1, then we have a topological analogue of Ruelle's result.
CONTROL THEORY AND DYNAMICAL SYSTEMS
137
On the other hand, the Ventcel-Freidlin theory is concerned with small, random perturbations of dynamical systems and their transient and limit behavior, see [94]. The difference between our approach and the stochastic theories is that we consider perturbations of bounded size Up C R m , while in the latter the strength of a nondegenerate noise goes to zero. We will make a few remarks concerning the meaning of Theorem 4.4, and draw a control theoretic consequence. 4-5. Remark. One could try to draw from Theorem 4.4 the conclusion t h a t for p > 0 sufficiently small the number of control sets of (4.3 P ) coincides with the number of chain recurrent components of (4.1). In general, this is false, as the following example shows. T h e reason is that even for small p the control system can change the dynamics of the differential equation substantially. 4 . 6 . E x a m p l e . Consider the following system on the unit circle M = S 1 x =
XQ(X)
— 3ui + 6i/2 = : X(x, u\, u 2 ) ,
x £ R mod 27r,
with U = [—1,1] X [—1,1]. We construct the vector field XQ in the following way: Define for n G N xn
1
:= TT -\—
n
Jn '= k ( l n + Zn+i ) , - (xn + I n _i ) J . Then it holds for no sufficiently large and for all n > no : x neInCJn C(0,2TT), and Jn fl Jn+i = (/>• Choose a C°° vector field XQ on [0,27r) such t h a t 2 <X0(x) n
1 < - n
for x e In
2 X0(x)
<
n
for
X e Jn \ In,
and such t h a t for some yn £ Jn \ In 1 Vn < *n we have Yl
Xo(yn)
T h e n Xo(n)
10
\
=
.
n = 0 and we require furthermore that there is ZQ E (0, w) with *o(2o)-3-6>0
FRITZ COLONIUS AND WOLFGANG KLIEMANN
138
so that the system cannot be steered from x = 7 r t o y = 27r = 0. With this construction we obtain the following behavior: For x G (?r, 2TT) and u\ > 0 one has
X(x,uu0)=X0(x)-3ul <0. Take pN = i , i.e. U'» = [ - £ , #\ x [ - £ , £ ] . Then we have for all n 6 N with y < n < N, and for u\ = — jj, u2 > 0, x G I n X K - i , u 2 J = X0(x) - 3ux + 6u 2 > - - | + | r + 6u 2 > - ^ + 6u 2 > 0. Hence J n is contained in the interior of a control set D„ . Furthermore, n is in the interior of some control set DQ:
x(*,±o)=Xo(*)-^
= -l<0
and
x(*,o,i) = *>(*) + ! = ■£>(>. Let iV > 2n 0 be given and take j , n 6 N such that y < j < n < N, then one cannot steer the system from D^ to D„: The point yn is in between D^ and .D^ and we have for all ( u i , u 2 ) G l / ' w : tn
X{yn,uuu2)
o
c
1
= A"0(y„) - 3ui + Qu2 < - — + — + — < - — < 0,
and Df D .D^ =
P>o
*
J
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139
then Mi -< Mj. In particular we obtain: The maximal (and the minimal) Morse sets of (4.1) correspond exactly t o the maximal (and the minimal, respectively) control sets of (4.3 P ) for p small enough. Hence attractors of (4.1) axe in one-to-one correspondence with the invariant control sets. Additional control sets, like those in Example 4.6, can be neither open nor closed. As a last remark in this section we draw a control theoretic consequence from Theorem 4.4, concerning complete controllability with arbitrarily small controls. 4.8. C o r o l l a r y . Under the assumptions of Theorem 4.4 we have: The system (4.3 P ) is completely controllable for all p > 0 iff the £ow of the differential equation (4.1) is chain recurrent on M. This corollary generalizes results of Lobry [59], Cheng et al. [39] and Jurdjevic and Quinn [53], 5. T w o E x a m p l e s In this section we discuss examples that show how t o use the theory developed so far for control of the dynamical behavior of systems. The first example is the model of well-stirred tank reactor, which is t o b e controlled around a hyperbolic fixed point, and the second example concerns the controlled Lorenz equation, for which we want to change the (numerically observed) strange attractor t o a simpler limit set, namely a periodic trajectory. In both cases the system is not completely controllable, b u t the limit sets of the uncontrolled system are contained in t h e interior of control set, in which we can accomplish the desired changes. For the Lorenz equation we will use a generalization of Theorem 3.5. The model of a well-stirred tank ractor is given by the equations (see e.g. Golubitski and SchaefFer [45] or Poore [70]) (— xi — a(xi — xc) + Bot{\ - X2)eXl \
(X\\
UJ = (
: i2
+ a ( i_ l 2 ) e *,
..( xc - x\\
)■**){
v
. ,
u\v i ^
0 J =:*•(*)+"(')*.(*)•
(5.1) Here X\ is the (dimensionless) temperature, #2 is the product concentration, and a, a, 5 , xc are positive constants. As control we use the heat transfer coefficient, and take u(t) eU = [-0.15,0.15] C R. T h e state space for (5.1) is M = (0,oo) x (0,1). For our analysis we have taken a = 0.15, a = 0.05, B = 7.0, and coolant temperature xc = 1.0, see Poore [70] for the system behavior with different parameter values. The uncontrolled equation for u = 0 has 3 fixed points in M, namely X
°
X
'
X
=
=
\aez°/(l
[ae^/il
+ aez°)) + ae*1))
* = \ae*3/(l + aez2)J
stable hyperbolic, i.e. the linearization about x1 has one negative and one positive eigen value stable.
FRITZ COLONIUS AND WOLFGANG KLIEMANN
140
Here z < z < z are the zeros of the transcendental equation
—x —a(x-xc)
+ Ba ( 1 - ——
) ez = 0.
Figure 3. shows the phase portrait of (5.1) for the parameter values chosen above. 1
Figure 3. Phase portrait of (5.1) for a = 0.15, a = 0.05, B = 7.0, xc = 1.0, u = 0. The interesting feature of this system is that the stable fixed point x 2 with the highest product concentration cannot be realized because of technical reasons, see Bellman et al. [8], while the fixed point x 1 is unstable, and hence it cannot be used for a technical realization of the system without modification. However, if one can embed x 1 into the interior of a control set D, then we can steer the system to x 1 from all x G D, and from the entire domain of attraction of D, and stabilize the system there. In order to use Theorem 3.5 and Remark 3.7, we compute the corresponding Lie derivatives of X0 and Xu with these vector fields defined as in (5.1):
d
a^Xo X i = ( l - * i ) dx
x
adlXQX, = [1 + Ba(l - x 2 )e*i(-2 + x)] — + ( a - x 2 ) e ^ ( - l + x x ) dxi dx2
■ ■
CONTROL THEORY AND DYNAMICAL SYSTEMS
a
^x0x\
= [(-l-15a:i + 0.15 + Ba(l - x2)eXl)Ba(l
141
- x 2 ) ( - l + xx)
- (1 + Ba(l - x 2 ) e I l ( - 2 + X!))(-1.15 + Ba(l -
x2)eXl)
+ BaeXl(2
- x 2 ) ( - l + Xi)]
- x1)(-x2
+ [axi(l - x 2 )e
Xl
+ a ( l - x2)eZl)
+ Ba2e2xi(l Xl
(-1.15X1 + 0.15 + Ba(l - x2)e )
- a ( l -- x 2 )e
dx\
Xl
• (1 + Ba(l - x 2 ) e Z l ( - 2 + Xi)) + a e X l ( l - X i ) ( - x 2 + a ( l - x 2 )e X l ) + a ( l - x 2 ) e X l ( - l + xx)(l + aeXl)]
—. OXo
One sees easily that for x = ( x i , x 2 ) € (0, oo) x (0,1) with xi ^ 1 the vectorfields X\(x) and adx X\(x) span the tangent space T Z R 2 . For x\ = 1 we have
( « & . * , ) ( * ) = (1 - Bor(l - x 2 )e) 4r + 0 ^X! dx 2 '
and the second component of (ad?x X\)(x) vanishes iif 1 — Ba(l —x2)e = 0. But for the choice of the parameters above, this equation has no zero in the interval (0,1). Hence the three vector fields adx X\, i = 0,1,2, span the entire tangent space for all ( x i , x 2 ) G M. Using the results from Sections 3. and 4. we see that there exist control sets D , D2 around the fixed points x°, x 1 , x 2 with xl G int Dl for i = 1,2,3. Here and D2 are closed (invariant) control sets, and D1 is variant, i.e. for all x G there is u G U such that
D°, D° D1 4.4
Figure 4. shows the numerically computed control sets D°y Dl, D2, and Figure 5. contains the domain of attraction A(D1) of the variant control set D 1 , i.e.
A(D) = {xeM;
G+(x) n D / ^ } .
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FRITZ COLONIUS AND WOLFGANG KLIEMANN
Figure 4. The control sets Z)°, Z ) \ D2 of the system (5.1).
Figure 5. T h e domain of attraction A(Dl)
of the variant control set D 1 of (5.1).
As a result of these considerations we obtain that the reactor (5.1) can be steered to the interesting point z 1 from ^ ( D 1 ) , and it can be kept at t h a t point, as long as disturbances do not perturb the system out of ^ ( D 1 ) . Furthermore, the control set D1 and the domain of attraction A(Dl) can be characterized precisely via the stable and unstable manifolds of the hyperbolic fixed points x T ( u ) , u E U. This allows the
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143
construction of a (discontinuous) feedback law u = F(x), such that x1 — x1 (u = 0) becomes a globally asymptotically stable equilibrium on A{Dl) for the feedback system. This feedback h a s certain maximal robustness properties with respect to disturbances. T h e explicit construction can be seen in [31]. Our second example concerns the Lorenz equation
*i\
/
p{x2-xx)
\
/°\
X2 I = I -xix3 + rxi - x2 ) +u(t) £3/ \ x\x2 — hxz )
I xi I =: X0(x) \ 0 /
+ u(t)Xi(x)
(5.2)
with U = [—p,p], p = 10, 6 = f, r = 28, where p > 0 is a constant. This system serves as a finite-dimensional model of the Rayleigh-Benard convection, and we have chosen as control u the Rayleigh coefficient, i.e. the temperature difference applied at the boundary, see e.g. Lorenz [60] or Berge et al. [11]. Equation (5.2) exhibits for the parameter values above a (numerically observed) strange at tractor A C R 3 , and we want to answer the question, whether it is possible to reduce this complex limit behavior to a simpler one, e.g. to periodic trajectories, using small controls u(t) € U ('control of chaos'). We will need the following facts, which can be found e.g. in Hairer et al. [48] or in Sparrow [87]. For constant controls U(t) = u E U the set R=
{(x!,X2,x 3 ) e R 3 ; x]+xl
+ (xz-p-(r
+ u))2 < c 2 }
is positively invariant for c sufficiently large. Hence the positive trajectories {ip(t, x , u ) ; t > 0} are bounded for x £ R. If r + u > 1, the equation has three (hyperbolic) fixed points, namely the origin O G R 3 and x* = (y/b{r + u — 1), y/b(r + u — 1), r + u — 11 x** = (-y/b(r
and
+ u - 1), - y / b ( r + u - 1 ) , r + u - l\ .
In order to show the existence of a control set D with nonvoid interior, which contains the strange attractor, it would be tempting to use Theorem 3.5 and the criterion from Remark 3.7, just as in the first example. However, this approach is not successful because of the invariance of the X3-axis. Hence we will use the following generalization, which does not need the hypothesis (H). 5.1. T h e o r e m . Assume that there exists an open set L C M, such that we have for all x E L and all V > T > 0: ip(T, x, 0) G int C%T,(x) and x E intO
-xx)-z—
dx\
+ (xx -pxx
-x2)-
dx2
FRITZ COLONIUS AND WOLFGANG KLIEMANN
144
ad2XoX\ = p(x2 — xi )(1 — p — 2xi + x3 — r — x2) dx\ + [ ( - * i * 3 + rxi -x2)p
= fi(x)^—
OX\
-x\)(l-p-
xi)]
pi
Q
adxXx
+ {xi -pxi
+ / 2 ( ^ ) ^ — + P{xix2 OX2
dxn
pj
- bx3)(x2
-
2xx\
OX3
where the precise form of the functions f\ and f2 is not important. We have for
(i) x\ =fi 0, and (ii) xi ^ x2 that X\(x) and (adx0Xi)(x)
are linearly independent. Furthermore, if
(iii) xxx2 — 6x3 ^ 0, and (iv) x2 — 2xi ^ 0, then the linear span of {Xi(x),(adx0Xi)(x), (ad3x Xi)(x)} is all of the tangent 3 space T X R for those x that satisfy (i)-(iv). Hence the system is locally controllable at these points, and therefore the assumptions in Theorem 5.1 are satisfied on this set. For all points that do not satisfy (i)-(iv) and t h a t are not on the £3-axis, one verifies that the vectorfield Xo is nowhere tangential to the manifold described by (i)-(iv). Hence for all points outside the a^-axis one can steer the system with the control u = 0 immediately into a region of local controllability. This verifies the assumptions of Theorem 5.1 in the current set up. Therefore, we obtain the following result: Let x G R 3 be a point outside of the X3-axis. T h e n there exists a control that steers the system from x into the interior of a control set Z), which contains the strange attractor in its interior (actually the strange attractor intersected with any compact set t h a t does not contain the X3-axis). In ini D one can follow any periodic trajectory, which is realized in (5.2) by appropriate steering. Let us recall that through any point in ini D there exists a variety of periodic trajectories. The simplification of the limit behavior follows this recipe: Embed the attractor of dimension between 1 and 3 into a control set of dimension 3, and realize there a limit set of dimension 1. 6. A n a l y s i s of Linearized C o n t r o l S y s t e m s Linearization techniques play an important role in the theory of dynamical sys tems for the analysis of local behavior, such as stability statements, Lyapunov ex ponents, invariant manifolds, entropy theory, etc. It seems possible to use all these methods also for the analysis of control systems, but here one faces the problem that a control system always contains trajectories that are not Lyapunov regular. Furthermore, there is no 'natural' shift-invariant measure on the space U of admis sible control functions. But such a measure is needed in the theorem of Oseledec
CONTROL THEORY AND DYNAMICAL SYSTEMS
145
[67] for the characterization of the spectrum. In this section we will present briefly some ideas concerning the control theoretic analysis of linearized systems, and then state some results on the spectrum in the next part. Let us start again with a nonlinear control system m
x = X0(x) + J2 ui(i)Xi(x) =■ f (*, «(*))
(2-1)
i=l
as explained in Section 2. In control theory the linearization of (2.1) is usually done in the following way: Pick a fixed point x° = x°(u°), u° G K.m of (2.1) and linearize the system about this point with respect to x and u. One obtains a "linear control system" of the form v = Av + Bu
(6.1)
where A = Dxf(x°,u°), B = Duf(x°yu°), compare the basic paper by Brockett [13] and the survey by Sontag [84] for stability results that can be obtained via this approach. Instead of following this procedure, we will linearize the system (2.1) only with respect to x to obtain a system on the tangent bundle TM. This allows for global results on M, and will lead, as in the theory of dynamical systems, to a construction of invariant manifolds via the linearization, see e.g. [15] and [32]; compare also Ruelle [73] and Dahlke [40] in the context of stochastic flows. On the tangent bundle TM we obtain the equation m
(Tx)'
= TX0(Tx)
+ J2 Ui{t)TXi{Tx)
(6.2)
i=l
with initial values TXQ = (xo, i>o) £ TXQM, where Tx denotes points in TM, and for a smooth vector field X on M we write its linearization as TX = (X, DX). In local d
coordinates on a chart in M this means: If Xj = ]T) akj(x)-Q^-
for j = 0 . . . m ,
*=i
denote by Aj(x)
= ( V
a
g' ; x
)
'
its Jacobi matrix.
W i t h this notation we have
/ i,k
TXj(x) = (aj(x)yAj(x)v), where a , is the column (ackj)k=i...dlocally a pair of coupled differential equations
T h e n (6.2) is
m
x = a0(x)
+
y^Uj(t)ai(x) *=i "Im
v = A0(x)v
+
(6-3) y^Ui(t)Ai(x)v.
i=l
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146
If x° € M is a fixed point of the control system (2.1), linearization about x° yields a so- called bilinear control system m
v = A0(x°)v
+ ^2ui(t)Ai(x°)v
inR'.
(6.4)
i=i
Its analysis gives information about local stability and stabilization for (2.1) in a neighborhood of x°. For this reason, bilinear systems play an important role for control theoretic problems, when one uses this linearization approach. Before we are able to make statements about the structure of (6.2), we need some further concepts: (6.2) induces a control system on the projective bundle P M , which is given by m
(Pa:)' = PJTo(Px) + J^ Ui(t)PXi(Px)
(6.5)
t=i
with initial values fx0 = (X0,SQ) G P X O M , the projective space in TXoM. Here FX is the projection of a vector field TX from TM onto P M , i.e. for j = 0 . . . m the FXj are given locally by FXj(x,s)
=
(aj(x),h(Aj{x),s)) - sTAj{x)s
h {Aj{x), s) = (Aj{x)
■ Id) s,
^'^
where T denotes transposition and Id is the d x d identity matrix. In the following we denote the solutions of (6.2) by T
xFM
->Ux
T$((u,Tx) =
PM,
(0tu,T
P$*(u, Fx) = (6tu, Py?(t, Px, u ) ) .
^6'^
First we describe the chain control sets of the projectivized system. We have to introduce some notation. For a chain control set E C M and its lift S C U x M (cp. (2.5)) let TE = {(u,(x,v))
eUx
TM;(u,x)
£ €}
F£ = {u,{x,Fv))
€Ux
P M ; ( u , x ) G S).
Note that TS is T$-invariant and the dimension of the fibers is equal to dim M = d. In fact, TE -► S is a vectorbundle. For a chain control set &E C P M denote the lift t o W x P M by P £ and define the lift to U x TM by TE
= {(u,Tx)
£Ux
where Z is the zero section in
TM-,Tx TM.
$ Z implies (u,F(Tx))
e
P£},
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147
6 . 1 . T h e o r e m . Let E C M be a compact chain control set of system (2.1). (i) There are 1 < / < d chain control sets pJ5, of the projective system (6.5) on the projective bundle FM with Fn(pEi) C E. The order on these sets (cp.(3.2)) is Unear and we enumerate in such a way that pE\ -< ■ • • -
= {Tx e TM; Tx £ Z implies F(Tx) e p-D,}.
(6.8)
For a control set pD and its extension TD we define the lift toWx FM (and U x TM) by pD = c£{u, Fx) e U x FM] Py?(t, Fx, u) e int P D for all t e R}, (6.9) and analogously for j - P . Now we are faced with a question, which is particularly important for spectral theory and related problems: Do the T^i define a decomposition of (U x TM, T3>) into invariant subbundles? The following example shows that this cannot be ex pected in general. 6.2. Example. Consider again the system from Example 3.3, i.e.
ft)-(JS)-+-w(! 0— «(!!)• with U = [0, | ] x [1,2]. For the projected system on the projective space P 1 in R2 we obtain 2 main control sets D\ and Z>2, see Example 3.3., which are connected by a continuum of control sets with void interior. For constant controls u\(t) = a, u2(t) = /3we get the eigenvalues of the system matrix I
„ J as
All2 = \(l + 0) ± yjcfi-fl + \{l + W. Hence for a ^ 0, /? ^ 1 the eigenspaces are one-dimensional, but for a = 0, 0 = 1 the (generalized) eigenspace is R2. Subbundle decompositions necessarily have constant dimension, hence 7 ^ 1 , T^2 cannot yield such a decomposition.
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FRITZ COLONIUS AND WOLFGANG KLIEMANN
Below we will show that this is an 'exceptional' situation. We embed - as in Section 4 — the control system with fixed control range U into a family of systems with varying control range pU, p > 0. Then we will show t h a t under an inner pair assumption the control sets with nonvoid interior and the chain control sets coincide up to closure for all p > 0, with the possible exception of at most d ^-values. For U C R m compact and convex with 0 € U denote Up =pU = {pw, u 6 U} for p > 0. To complete the picture we also consider the case of unbounded perturbations. Let U°° = | J Up = Loo(R, R m ) and denote the corresponding control system by (2.1)°°. Also for other quantities a superscript p will indicate their dependence on the control range Up for 0 < p < oo. Note that for p = 0 we simply obtain the differential equation x = Xo(x). Let E° be a chain recurrent component for (2.1)°. For p > 0 there are unique chain control sets of (2.1) p with E° C Ep. Under an inner pair assumption (cp. Theorem 4.4) there are also unique control sets Dp with E° C int Dp. Similarly, we consider the chain recurrent components pE® in the projective bundle FM (with Fir(ipE°) C E°) of the uncontrolled projective system (Pa;)- = FX0(¥x),
t G R.
Here i € / , some index set according to Theorem 6.1. for p = 0. Let C(PM) denote the set of compact subsets of FM with the Hausdorff metric, and consider for i £ I the following maps associating to p a chain control set of (6.5) p P£i:[0,oo]-»C(PM),
p»FE?
where FE? C pEf for p > 0. Similarly we define for i E I maps associating to p control sets of (6.5) p by PDi
: [0, oo] -> C(FM),
p h-> c£FDP
where pD{(0) =pE^ and pDf is a control set with pE? C int $>DP for p > 0. The following theorem shows in particular, that these maps are well defined. Note that some of the pE{ (and p£),, resp.) may coincide. 6.3. T h e o r e m . (6.5)p are locally controlled system set of (2.1)p with
Fix 0 < pi < oo and assume that for all p e (0,/>i] the systems accessible. Let E° be a chain recurrent component for the un (2.1 f and for p € (0,pi] let Ep denote the unique chain control E° C Ep. Suppose that there is a compact set L C M such that
CONTROL THEORY AND DYNAMICAL SYSTEMS
EPl C L and the set L is positively invariant under all controls u G Upl. that the following p-p'-inner pair condition holds: For all p\p
149
Assume
G [0,p\) with p' > p and all chain control sets pE?
every (it, Par) G p£j* is an inner pair for (6.5)'\P'
(i) Then for all p G (0, pi] there are unique chain control sets Ep with E° C Ep and £(p) chain control sets PE? with FTT(PEP) f) E° ^ 0; their number £{p) satisfies 1 < i(p) < d and one has Fn(pE?) = Ep. (ii) For all p G (0,pi] there are unique control sets Dp of (2.1)p with E° C int Dp and control sets PDP of (6.5)p such that p E j C int FDP; for all but at most d p-values int Dp = int Ep and int FDf = int Furthermore,
for all p G [0, p\] the projections
P PE .
onto M satisfy F7r(Dp) = Dp.
The first part of this theorem is a consequence of Theorem 6.1. The proof of the second part which, additionally, uses control theoretic methods is given in [32] (a proof for the bilinear case is given in [30]). In the next section we will use these control sets and chain control sets in the projective bundle in order to describe the Lyapunov exponents of the linearized system. 6.4- Remark. Local accessibility of the projective linear system can be guaranteed if Hypothesis (H) holds for the system on P M , i.e. dim£.4{PA'o + S i / i P A " , ; ^ ) = u6 U}(x,v)
= 2d-l
for all ( x , v ) G P M ,
compare San Martin and Arnold [82] and San Martin [81] for a discussion of this hypothesis. Furthermore, [82] shows uniqueness of the invariant control set in P M over an invariant control set in M . In [14], Barros and San Martin give more precise information about the number of control sets in bilinear systems. 6.5. Remark. In the bilinear case, the relation between control sets and chain control sets of the projected system (6.5) can be described precisely, without the inner pair condition, see [26]. Here we only mention that each chain control set pEj contains (at least) one control set pZ),- with nonvoid interior. Furthermore, control sets and chain control sets are described by eigenspaces that belong to the fundamental matrices of solutions which are periodic on P d _ 1 and associated with piecewise constant periodic controls.
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7. S p e c t r a l T h e o r y of N o n l i n e a r C o n t r o l S y s t e m s For the analysis of dynamical and of stochastic systems via linearization spectral theory (Lyapunov exponents) play an important role. As far as we know, there is no approach to a systematicspectral theory for nonlinear control systems, although the well-known concepts of the Oseledec spectrum [67], the dichotomy spectrum (see Sacker and Sell [76]) and the 'topological' spectrum (based on the condition that the zero section is isolated, see Selgrade [78], Salamon and Zehnder [91]) are applicable to this situation. We want to start a systematic investigation in this section and analyze the Lyapunov spectrum of nonlinear control systems, as well as its connections with other spectral concepts. There are several results for bilinear systems (see [21], [23], and the survey [34]), which, in particular, have applications to the stabilization of linear, uncertain systems via output feedback (see [22], [27]). Let us mention here, that spectral theory for control systems is first of all an openloop theory, and one has to construct appropriate feedbacks from a detailed analysis afterwards. Let us consider again the control system (2.1) and its linearization (6.2), where we can write, according to (6.3), the solutions of (6,2) as Ttp(t,Tx,u) = (
xTM
^>R
A(u,x, v) = limsup - log ||Z)
v ^ 0
t
where R is the extended real axis. (7.1) defines the forward Lyapunov exponent, i.e. for t —> oo, and in a similar way the backward exponent A - : U x TM —^ R. is defined for t —► — oo. In general it holds even for linear systems: There is, for (x, u) fixed, no decompo sition Rd — ©Vi into linear subspaces with \(u,x,v) = A ~ ( u , x , u ) for all v 6 UVi; the limsup need not be a lim; and (exponenetial) stability of a system does not necessarily imply stability under small perturbations of the vectorfield, since con trol systems have time dependent right hand sides, see e.g. Cesari [19] or Hahn [47] for examples in the context of ordinary differential equations. Lyapunov [61] introduced the concept of regularity, which guarantees the three properties above. The main result in this direction is the theorem of Oseledec [67], for which one has to assume stationarity of the underlying flow $ (i.e. the existence of a ^-invariant probability measure P ) , together with an integrability condition on the cocycle D(p. Then there is a ^-invariant set of full P-measure, such t h a t all points in this set are Lyapunov regular, see e.g. Ruelle [73] for a presentation of this theory, and also Mane [62] for the time discrete case. (Note that from a topological point of view the set of Lyapunov regular points, although of full measure, may be quite 'thin', namely residual, see e.g. Marie [62].)
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For control systems the situation is somewhat more complex, since one always has to consider non-regular points as well: Let the vectorfields XQ .. .Xm in (2.1) be linear, i.e. the system is bilinear x = AQX + Euj(<)Aj:r, with non-zero matrices Ao ... Am. Let the space U C R m of admissible control values be a product of intervals. T h e n there is an admissible control function u £ W, such that the matrix function Ao + Eu;(tf)Aj£ is not Lyapunov regular, (examples are easily constructed using the ideas in Cesari [19] or Hahn [47]). We are interested in the full Lyapunov spectrum E L j , = {A(u, x, v); (w, x, v) e U x TM) (7.2) and therefore we cannot use Oseledec's Theorem. In t h e following we will analyze S using the projected linearized system (6.5) on the projective bundle P M , and we will discuss relations with other spectral concepts. For applications to the stability theory of control systems we refer to the papers mentioned above. In the following we construct for the Lyapunov spectrum an 'inner' approxima tion, the Floquet spectrum, and an 'outer' approximation, the Morse spectrum. Then the Lyapunov spectrum is sandwiched in between. Additional assumptions will imply t h a t the (closure of) the Floquet spectrum and the Morse spectrum and hence the Lyapunov spectrum all coincide. In this situation, semicontinuity prop erties of the Floquet and the Morse spectrum yield continuity properties of the Lyapunov spectrum. We start by defining the Floquet spectrum, which is based on periodic coefficient functions for the linear part. Clearly, this can only be guaranteed, if the corre sponding control function together with the trajectory in the base space M is peri odic. Then the Floquet exponents of the corresponding linear periodic differential equations will be special Lyapunov exponents. Essentially, the Floquet spectrum will consist of these Floquet exponents with some convenient additional properties, which we specify in the following formal definition of the Floquet spectrum of the system (2.1). 7 . 1 . D e f i n i t i o n . Let pD be a control set with nonvoid interior of the system (6.5) on the projective bundle FM induced by the linearization (6.2) of system (2.1). T h e Floquet spectrum of the system (6.5) over pZ) is defined as £F/(I
•£) = {
A(w, x); (it, Px) 6 U x int pD, u is piecewise constant, 1 periodic with period r such that P y ( r , Fx) = Fx J
T h e Floquet spectrum over a control set D in the base space M of the system (2.1) is
(D) = \J{
S f ^ ( p i ) ) ; pD is a control set with nonvoid interior 1 of the system (6.5) with FTT(PD) CD j '
Obviously, one has for every control set D C M the inclusion ZFl(D)
C
ZLy.
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Thus the Floquet spectrum furnishes an inner approximation to the Lyapunov spec trum. Next we introduce an 'outer' approximation of the Lyapunov spectrum given by the Morse spectrum. This concept is based on topological considerations. Chain control sets in the projective bundle FM or the chain recurrent components of the corresponding control flow P $ on W x FM are an appropriate generalization of sums of generalized eigenspaces. (On the other hand, the Floquet spectrum is based on control sets corresponding to topologically transitive components of the control flow and to eigensapces of periodic matrix functions, see [26]). Observing that chain recurrent components are denned via chains, it may appear natural to base a corresponding concept of exponential growth rates on chains. This leads us to the following definition. For e , T > 0 an (e,T)-chain C of P $ is given by n G N , T 0 , . . . , T n > T, and (u0,FxQ),..., (un,Fxn) inU xFM with d(F${Ti,(ui,Fxi)), (ui+1,Fxi+1)) < e for i = 0 , . . . , n — 1. Define the finite time exponential growth rate of such a chain C, (or 'chain exponent') by
( where x{ e
n-l
\
Y,Ti) i=0
- 1
n-l
^(logWTi.x.-.uOI-logM) /
i=0
F~1(Fxi).
7.2. Definition. Let $>C C U x FM be a compact invariant set for the induced flow P $ on U x FM and assume that P$|p£ is chain transitive. Then the Morse spectrum over p£ is £MO(P£) = -
A G R; there are ek -► 0, Tk -+ oo and (ek,Tk) - chains C* \ in fjC with A(£fc) —► A as k —* oo
For a compact invariant set C C U x M define the Morse spectrum over C as
where the union is taken over all chain recurrent components $>£ of P$|(P7r) - 1 £. For C = U x M we call T,M0{U X M) the Morse spectrum of the system (2.1), denoted by SMOOf particular interest is the Morse spectrum over a control set D C M with nonvoid interior and compact closure. Then P £ = (P7r) _1 D cUx FctDM cUx FM is compact, where V is the lift of D. Define via this compact invariant set i n W x M XMO(D)
:=
HMo{V)
CONTROL THEORY AND DYNAMICAL SYSTEMS
153
Similarly, we define the Morse spectnun EMO(-E) over a chain control set E C M via the lift to U X M. Theorem 3.7 in [29] implies that the Lyapunov spectrum is contained in the Morse spectrum, i.e. Ef7
C ELJ, C
?>MO-
Hence the Lyapunov spectrum lies in between the Floquet and the Morse spec trum. We are mainly interested in the Lyapunov spectrum, hence we present condi tions which imply that the closure of the Floquet spectrum and the Morse spectrum, and hence the Lyapunov spectrum all coincide. 7.3. Theorem. Let the assumptions of Theorem 6.3 be satisfied. Then the fol lowing assertions hold: (i) For all p 6 [0, p\\ and for each i = 1 , . . . , I, the Morse spectrum has the form
s Mo (p^f) = [«*(p£f), *( -raf)] with K*(VE?) < K*( P £p and K(PE?) < K(FE?) if^Ef ^ VE? and i < j . (ii) For each i = 1 , . . . ,£ the sets of continuity points of the two maps p *-*■ c£T,Fi(pD?) and p t-+ Y,M0(pEf) agree. At each continuity point we have c£Sir/(pZ)f) = YiMo(vEi )• In paraticular, if p is a continuity point for all t = 1 , . . . , 1 , then e
i
clY,Ft{D<>) = | J c*EF*(pD?) = VLy{D>) = (J EMo(p£f) = EM,(£?'). i=i
i=i
Note that there are at most countably many points of discontinuity. This theorem gives - under the p-p'-'mner pair condition - a complete character ization of the Lypunov spectrum and the corresponding 'eigenspaces' for bilinear control flows on vectorbundles. The proof follows from the separate analysis of the spectra together with the inner pair condition and the results in [28], see [32]. For the special case of bilinear systems, see [30]. We also note that the strict in equalities in assertion (i) follow from an important (but apparently little known) theorem which appears in Bronstein and Chernii [9], see also [15]. It sheds new light on Selgrade's Theorem [78] and states that a decomposition of a linear flow into subbundles corresponding to attractor-repeller pairs in the projective bundle is equivalent to a decomposition into exponentially separated subbundles. 7.4- Remark. Suppose, in addition to be assumptions of Theorem 6.3, that M is compact. Then the set {(w,x,u) £14 x TM\ A(w,x, v) £ M{£f7(pC), pC is an invariant control set}} contains an open and dense subset of U x TM. Thus, generically, the Lyapunov exponents will belong to an invariant control set in PM. The spectra Hiy^E) corresponding to different chain control sets in FM can overlap, as the following simple example shows:
FRITZ COLONIUS AND WOLFGANG KLIEMANN
154
7.5. Example. Consider the following 2-dimensional linearized (bilinear) system
with J7 = [0,2] x (^,1] x [1,1]. The control sets of the projected system on the projective space P in R2 are given by pD1=UF{(Vv1 PD2
= n
P
J € R 2 ; v2=avu
ae
\-y/2,--j=\)
{ f M € R 2 ; u 2 = a u 1 , a e H § » ^ },
where lip is the projection of R2 onto P. The chain control sets are PEI = ci pDi,
pE2 =p D2,
see [CK4].
For constant u €U denote the right hand side of the system equation by
Furthermore, let Xi(u), X2(u) be the real parts of the eigenvalues of A(u) with Ai(u) < X2(u). Then we have (Ai(u); u € U) = [—1,|] and {A2(y); u € u} = [ | , 3 ] . Since these intervals are contained in the spectral sets over p£?i, and pE2 respectively, we obtain [£, | ] C ^Ly(pEi) D ELy(wE2). Little can be said about the relation of the different spectra at the p-discontinuity points of Theorem 7.3. The next example shows that while the spectral intervals can be different at these points, the closure of the entire Floquet spectrum and the Morse spectrum may still agree. 7.6. Example. Consider the bilinear system
+uw
- C s)" (-°i :)•• with u € U = [A, a] C R. The projection of this system onto the unit circle S 1 C R2 yields Example 2.3. If we identify the projective space with the angle
S ( F A ) = (0, v ^ A ) , S(pD 2 ) E ( ? £ ) = [ - V ^ m , y/a=A\
=(-v^TA,0), = I(pE) = E.
CONTROL THEORY AND DYNAMICAL SYSTEMS
155
Hence the control spectrum decomposes the Morse spectrum I(pE) into two in tervals, corresponding to the controllability structure of the projected system. In particular, we have: • {A(u,t>); ueU}
= c£?:(pDi) for v € P \ (PD2 U {TT - arctan(a - A ) 1 / 2 } ) ,
•
= E(IPJE) for v €
{A(u, v);
ueU}
?D2,
• {A(u,t>); u € U) = c £ E ( P D i ) U { - v ^ a ^ A } for v = 7r — arctan(a — A ) 1 / 2 . Hence only for v € n»i?2 U {7r — arctan(a — A ) 1 / 2 } is there an u G 14, such t h a t the Lyapunov exponent of the solution is negative, i.e. such that the system is open-loop stabilizable at the origin 0 from v. Finally, we briefly discuss the relation to the dichotomy spectrum (or dynami cal spectrum) introduced by Sacker and Sell [76], cp. also Selgrade [78], and the Oseledec spectrum [67]. We note that a number of interesting connections between the dichotomy spectrum, the Oseledec spectrum and the Lyapunov spectrum have been derived in [52]. T h e dichotomy spectrum is based on exponential dichotomies of the flow and, over a chain recurrent base space, this is equivalent to the topological spectrum. The Lyapunov spectrum is always contained in the dichotomy spectrum.
T h e intervals of the dichotomy spectrum are obtained by taking the
unions of overlapping intervals of the Morse spectrum, cp. [29]. Example 7.5 above shows t h a t the intervals of the Lyapunov spectrum corresponding to different con trol sets (or chain control sets) may overlap. Hence the bundle decomposition of the Morse spectrum can be strictly finer than the one induced by the dichotomy (or the topological) spectrum. Furthermore, the bundle decomposition associated with the Morse spectrum is equivalent to a decomposition into exponentially separated subbundles. And exponential separation is equivalent to the existence of a scalar cocycle such t h a t the linear flow multiplied by this cocycle admits an exponential dichotomy, cp. Palmer [68] and, in particular, Bronshtein [15]. For the case of a base space which is not chain transitive, we note t h a t the topological spectrum is still associated with attractor-repeller pairs in P M , (cf. [91], Theorem 2.7.), and the dichotomy spectrum corresponds to a subbundle de composition via the projectors of exponential dichotomies, but the decompositions corresponding to the Morse spectrum are not defined globally. The Oseledec spectrum is defined for invariant measures fi on the base space. Hence, for a given /i, the associated (measurable) bundle decomposition can be finer t h a n the one induced by the Morse spectrum. One of the main results of [52] shows t h a t for all ergodic \i the Oseledec spaces are contained in the subbundles induced by the dichotomy spectrum. Combining this with Bronshtein's result on
FRITZ COLONIUS AND WOLFGANG KLIEMANN
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Sys
SHADOWING IN DISCRETE DYNAMICAL SYSTEMS Brian A. Coomes, Huseyin Kogak*, and Kenneth J. Palmer * Department of Mathematics and Computer Science University of Miami Coral Gables, FL 33124 U.S.A. Abstract: In this paper three main aspects of shadowing in discrete dy namical systems are addressed. First, a new proof of the classical infinitetime shadowing theorem for hyperbolic diffeomorphisms is given. Second, a finite-time shadowing theorem for maps which do not satisfy the hyperbolicity requirement is presented. The use of this theorem in numerical computations is demonstrated. Finally, a new theory of periodic shadow ing for maps is developed. This theory allows one to verify the existence of periodic orbits, including long and unstable ones, and estimate their Lyapunov exponents from numerical computations.
1. Introduction Shadowing is a property of hyperbolic sets of diffeomorphisms which states that pseudo or approximate orbits in the set are near (or shadowed by) true orbits. It is closely related to classical results from the theory of ordinary differential equations (conf. Coppel [16]) stating that if a nonautonomous system has a bounded solution, the variational equation of which admits an exponential dichotomy, then a perturbed system has a bounded solution nearby. In its modern form, to paraphrase Bowen [7, p.88], the idea of pseudo-orbit has probably occurred to many people. The shadowing lemma is first exphcitly proved in [6], though earlier similar statements are in [5] and for Anosov diffeomorphisms in [3]. Sinai [33] stated explicitly the shadowing lemma for Anosov diffeomorphisms. Since then the shadowing lemma has been reproved many times, see, for exam ple, [11, 17, 24, 25, 27, 28, 29, 32]. Also it has been extended to nonautonomous, * Supported in part by NSF grant DMS 9201951. 163
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B. A. Coomes & Huseyin Kogak k
Kenneth J. Palmer
noninvertible, and infinite-dimensional maps as in [8, 35, 36], r a n d o m diffeomorphisms [10], ordinary differential equations [13, 19], skew-product flows [26], and elliptic differential equations [2]. In Section 2 we give a new proof of the classical shadowing lemma for diffeomorphisms modeled on the one given for ordinary differential equations in [13]. As usual, the basic assumption is that the diffeomorphism has a compact hyperbolic set. It turns out that the hyperbolicity assumption is rarely satisfied by the attractors arising in practice. For example, for certain parameter values the Henon map has an apparent strange attractor but it does not appear to be hyperbolic although it seems that most orbits are hyperbolic for a finite time, t h a t is, there does seem to be a splitting of the tangent space along the orbit into stable and unstable subspaces. In Section 3 we prove a finite time shadowing theorem for maps without the hyperbolicity assumption. It is clear that computed orbits of maps are, in fact, pseudo orbits. Hence this theorem can be used to show the existence of true orbits near computed orbits of a m a p . Hammel, Yorke and Grebogi [21, 22] were the first to show the existence of true orbits near computed orbits. Later Chow and Palmer [9] and Sauer and Yorke [30] both proved finite-time shadowing theorems. The theorem proved here is similar to the one in [9]. However, the handling of roundoff error in the computations has been considerably streamlined. One gains many clues to the structure of chaotic attractors by studying the periodic orbits and their Lyapunov exponents; see, for example, [4, 20, 23]. There are few effective theorems which can be employed to establish the existence of periodic orbits of specific maps, especially if they are of long period and unstable, as is the case with chaotic maps. Usually one has to rely on numerical evidence. However, how can one be sure that an apparent computed periodic orbit is, in fact, near a true one? In Section 4, inspired by our earlier work [12] (see also [31, 34]) on periodic shadowing for differential equations, we show t h a t similar techniques to those used in our finite-time shadowing theorem yield a theorem which allows one to verify that a computed periodic orbit is near a true one. This theorem can be applied to prove the existence of unstable orbits of possibly very long periods. We also give a method to rigorously estimate the Lyapunov exponents. In conclusion we should like emphasize that Section 2 on infinite-time shad owing in hyperbolic sets and Section 3 on finite-time shadowing in nonhyperboUc sets are expositions of previous works. The section on periodic shadowing, however, contains new and surprising results on the existence and stability types of periodic orbits, including a rigorous verification of the existence of an unstable periodic orbit of period 100003 of the quintessential chaotic m a p of Henon.
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2. T h e Infinite-time Shadowing Theorem In this section we first give a precise notion of shadowing of a pseudo orbit by an associated nearby true orbit. Then we present a general shadowing theorem and its proof for hyperbolic sets of difFeomorphisms. Throughout this paper, unless otherwise indicated, we use the Euclidean norm for vectors and the relevant operator norm for matrices and linear operators. We now proceed with the definition of a pseudo orbit of a C 1 diffeomorphism / : IRn —> IR n . Definition. A sequence {yk}t^°-oo of points in JRn is said to be a 6 pseudo orbit
off if
||y*+i - /(y*)ll < s forfcezs Next we introduce the notion of shadowing of a pseudo orbit by a true orbit. Definition. A true orbit {x^t^-oo °f f> ^na^ to e-shadow the 6 pseudo orbit {yjfc}j£fcl00 if ||xjfc-yjfc||<e
1S
> x *+i = /(xJfc) f°r *H &>
for
2S s a j
d
keTL.
In preparation for the statement of the Shadowing Theorem, we next recall the definition of a hyperbolic invariant set. Definition. A compact subset S of R n is said to be a hyperbolic set for f if S is invariant, that is, f(S) = S, and for each x € S there is a continuous splitting
Rn = ££ 0 E£ such that dim E£. and dim E£ are constant. that is,
D/(x)(J55) = EJ (x) ,
Moreover, the splitting is invariant,
Df(x)(EZ) = EJf(x)
and there are positive constants K, X with A < 1 such that for all k > 0 and x G 5 \\Dfk(x)t\\
for
(£E°X
(1)
\\Df-k(x)Z\\
for
£€£*.
(2)
and
Now we state the Infinite-time Shadowing Theorem for hyperbolic sets of diffeomorphisms.
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Brian A. Coomes & Huseyin Kocak & Kenneth J. Palmer
Shadowing Theorem. Let S be a compact hyperbolic set for the C1 diffeomorphism f : IRn —► lR n . Then there is a positive constant M depending only on f and S such that if8>0 is sufficiently small every 6 pseudo orbit of f in S is e-shadowed by a unique true orbit with e = 2M6. Our setting for the proof of this theorem is as follows. Given a 6 pseudo orbit {y*}jfc^-oo °f /> w e wish t ° show that {yk}t=°-oo 1S e-shadowed by a true orbit {x*}jfc^-oo' th a * ^s> w e w a n ^ Xjt+i = /(xfc)
and
||x* - y*|| < e
for k € TL.
To this end, we introduce the Banach space ^°°(2Z,IR") with norm I K x * } f = - J | = sup ||x*|| te2Z and the function Q : r°(2Z,IR n ) -+ £°°(7Z,JRn) given for x = { x * } ^ ^ by [a(x)]fc=xJk+i-/(xifc),
for k£2L.
(3)
The Shadowing Theorem will be proved if we can find a unique solution x = {x*}it=^-oo °f ^ne equation £(x) = 0 in the closed ball of radius e about y = {yk)t^To solve this equation, we use the following lemma which is an abstract varia tion of Newton's method: Lemma. Let X be a Banach space and let Q : X —* X be a C1 function. y is an element of X for which \\G(y)\\ < S and the derivative L = DQ(y) is invertible with \\L-i\\<M for some positive constant M. Then if
IIDQ(x) - DG(y)\\ <
1
2M
holds for ||x - y || < 2Mb, there is a unique solution of the equation £(x) = 0
Suppose
Shadowing in Discrete Dynamical Systems satisfying ||x - y|| <
167
2MS.
Proof of Lemma. Define the operator F : X —> X by F ( x ) = y - L - ^ W - W ( y ) ( x - y)]. Clearly Q(x) = 0 if and only if F ( x ) = x. Moreover, if ||x - y|| < e = 2M£, ||F(x) - y|| < HZ,"11| ||a(x) - Q(y) - DQ(y)(x - y) + £(y)|| < M ( | M - 1 e + <5) = e. Also, if ||x - y|| < e and ||z - y|| < e, then ||F(x) - F(z)\\ = WL-'lGix) - Q(z) - DQ(y)(x - z)]|| < M | M _ 1 ||x-z|| = i||x-z||. So F is a contraction on the ball of radius e, center y, and thus the lemma follows from the contraction mapping principle. Proof of the Shadowing Theorem: Let X = £°°(7L,JRn). We apply the lemma to the C1 map Q : X —► X defined in Eq. (3) and y = {yjtj^.oo, a 6 pseudo orbit of / in S with the size of 6 to be determined later. It is easy to see that DQ(y) — L, where the linear operator L : X —> X is denned as follows: if £ = {^jfc}^-.^ 6 X, then (LC)k = fc+i - Df(yk)£k for ktTL. To apply the lemma we must show that when 6 is sufficiently small, the operator L is invertible and we must also find a bound for | | L - 1 | | . Invertibility of L and estimation of | | L - 1 | | : First we introduce some notation. Denote by ^ ( x ) the projection with range £ £ and kernel £?*. It follows from the invariance of the subspaces that V{x) satisfies the identity Df(x)V(x)
= V(f(x))Df(x).
(4)
Also, since the splitting is continuous, ^ ( x ) is continuous on the compact set S and hence bounded. So there are positive constants M3 and Mu such that for all x € S \\V(x)\\ < M\
||/-7>(x)||<Mu.
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Brian A. Coomes & Huseyin Kogak & Kenneth J. Palmer
Then it follows from Eqs. (1) and (2) that for all k > 0 and x € S \\Dfh(x)V(x)\\
< KMa\\
\\Df-k(x)(I
- V(x))\\ < KM»\k.
(5)
To motivate the argument that follows, let us consider first the case that {yk}kL-oo is a true orbit of / , that is, 8 = 0 and y*+i = /(yjfc) for all k. Now if L is invertible and f = L"1g, then {C*}*L-oo 1S a bounded solution of the difference equation fr+i = Df(yk ) 6 + gfc for keTL. Set uk =
Pk(k,vk=(I-Pk)Zk,
where Pk =
V{yk).
Note that from Eq. (4) we have the identity Df(yk)Pk
= Pk+1Df(yk).
(6)
From this it follows that uk and v* are bounded solutions of the difference equations u fc+ i = Df(yk)uk
+ Pjt+ig*
for k <E 7L
(7)
and vk+i = Df(yk)vk
+ (I-Pk+1)gk
ioik€7L,
(8)
respectively. By repeated application of Eq. (7), we find that for any nonnegative integer q ufc = D/(y f c _i W ( y * _ 2 ) • • • I > / ( y * - , - i )u Jk _,_ 1 +
£
(9)
^/(yjfc-iW(yfc-2)---JD/(ym)Pmgm-i.
m=k—q
Now if k > m, ||D/(y f c _ 1 )jD/(y f c _ 2 )... J D/(y m )P m || = ||D/*- m (y m )7>(y m )|| < So we may let q tend to infinity in Eq. (9) to obtain k
u fc = Y, m = —oo
Df(Yk-i)Df{yk-2)...Df{ym)Pm%m_l.
KMa\k~m.
Shadowing in Discrete Dynamical Systems
169
We proceed similarly with Eq. (8) except that we use the difference equation back wards to obtain oo
v
^/(y*)~12>/(y*+ir1-^/(ym-i)-1(/-i,m)gm-i.
*=- £ m=fc+l
is a true orbit, we conclude that if L _ 1 exists it must
Hence when {y^kL-oo be given by the formula k
(L-1g)k=
£
^/(y*-iW(yw)-D/(ym)P«gm-i
m=—oo oo
^/(yjtr1^/(yfc+ir1---^/(ym-i)-1(/-Pm)gm-i.
- £ m=fc+l
Now suppose {yfcjjfc^-oo is just a 6 pseudo orbit of / in the hyperbolic set 5 . Then for suitably chosen positive integer p consider the linear operator T defined by k
(Tg)it =
£
Ak-1Ak-2---AmPmgm-i <«»
-
E
A
1
A
J
p
* ^r-- m-i( -- "»)8»-i.
m=Jfc+l
where Ak=Df(yk). We shall show for 6 sufficiently small that \\LT-I\\<\,
||TL-/||
(11)
The first of these inequalities implies that LT is invertible with inverse Q satisfying IIQH < 2. Then TQ is a right inverse for L. Now the second inequality in Eq. (11) implies that L is one to one. Hence L is invertible with inverse L - 1 = TQ and thus ||L-1||<2||T||.
(12)
To establish the first inequality in Eq. (11), we calculate (LTg)k
- git = (Tg)ifc+1 - Ak(T&)k = -AkAk-i - AkllAkU
- gfc
• • • Ajb_p+iPjfc_p+igjfc_p • • * *k+p(I ~ Pk+p+l)gk+p.
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Brian A. Coomes & Euseyin Kogak & Kenneth J. Palmer
Hence we have \\LT-I\\
< sup{||A*A f c _ 1 ---A f c _ p + iP f c - p +i|| *€E A
+ U1XI
M
(13)
•••4SF(J-A+F+i)l|}-
To proceed further, we need bounds for expressions of the forms ||Ajb_iAjfc-2"-A m P m || and | | V ^ f c - i i - - ' ^ m - i ( i ' " pm)l In order to bound ll-Afc-i-Afc-2 • • • -^-mPm || we estimate the difference dk = \\Ak-xAk-2
■ • • AmPm
- J5/fc-m(ym)^(ym)||
(U)
for m < k < m + p recursively by deriving an inequality for dk+i in terms of dk. For m
■ ■ ■ AmPm
< \\Ak\\dk + \\Df(yk)
Df(fk-m(ym))Dfk-m(ym)V(ym)\\
-
- Df(fk-m(ym))\\
\\Dfk-m(ym)V(ym)\\
(15)
fk-m{ym)\\)KM3,
< M1dk + u(\\yk -
where u> is the modulus of continuity u(u) = sup{||D/(x) - Df(y)\\
: x € IR n , y € 5, ||x - y|| < a)
and Mx = max{sup||Z>/(x)||, sup H^D/"1 (x)||}. xes xes Now we assert that when k > m l|y* - / * " m ( y m ) | | < ( l + Mi + • • • + M i " " 1 " 1 ) *
(16)
where both sides are interpreted as zero when k = m and Mx = max{ sup{||/(y) - / ( x ) | | / | | y - x|| : x <E 5, y € 5, x ± y } , supdir^y) - r
1
W | | / | | y - x|| : x G 5, y e 5, x ^ y} }.
This inequality follows by induction on k using the estimate ||y* + i - / * + 1 - m ( y m ) | | < ||y fc +i_- /(y*)|| + ||/(y fc ) - / ( / * - • » ( y r o ) ) | |
<S + Mx\\yk-f*-™{yin)\\. Combining the inequalities in Eqs. (15) and (16), we find that for m < k < m + p - 1 dk+i < Midjt + n ,
Shadowing in Discrete Dynamical Systems
171
where rx = KMsu((Ml
- l)-
1
^"
1
- 1)*).
Since c?m = 0, it follows by repeated use of the last inequality that \im
<{M1-\)-\M[-l)rl.
'
Finally, it follows from Eqs. (5), (14) and (17) that for m < k < m + p \\Ak-iAk-2
• • • A m P m | | < (M1 - \)-\M{
We now turn to the estimation of \\A^1 A^^ k < m, we define dk = WA^A^l,
- l)n + W A f c - m
(18)
• • • A^^I — Pm)\\- For m — p<
• • • A-^I - Pm) - Dfk-m(ym)(I
- V(ym))\\-
(19)
We estimate dk recursively by deriving an inequality for dk-i in terms of dk- For m — p+ 1 < k < m, using the chain rule, dk-! = WA-^A-1 - • • A ^ i l - Pm) -[i?/(/fe-™-l(ym))]-lp/*-™(ym)(J-P(ym)||
< IIA^IIllAj^Si *' • A m-i( J " P - ) " ^/*- m (ym)^ - V(ym)\\ + iiAji, - [i>/(/*-",-I(y»))]-1|l P/ fc - m (y,n)(/ - P(ym))|| < Mxd* + ^ / ( y i t - ! ) - 1 - [P/(/*- m - 1 (y m ))]- 1 |I^M« < Mxd* + p/- 1 (/(y*-i))||||i>/(/ fc - m - 1 (ym)) - Af(y*-i)ll \\Drl(fk-m{ym))\\KM* < Midk + ^ w ( | | y t - i -
fk-m-\ym)\\)KM\
Now we assert that when k < m
||y* - /*- m (ym)|| < (Mi + M\ + • • • + M?~k)6, where both sides are interpreted as zero when k = m. This inequality follows by backward induction on k using the estimate
||y*-i - /*- 1 " m (ym)|| < ||y*-i - /"'(yOll + lir'Cy*)^ /" 1 (/ k " m (y m , < \\T\f (y_k-i.)) - /_1(y*)ll + Mi ||y* - / fc - m (y m ))|| <M 1 ^ + M 1 ||y Jt -/*- m (y T
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Brian A. Coomes k Huseyin Kbcak k Kenneth J. Palmer
So when m — p+1 < k < m, dk-i
<M1dk+r2,
where r2 =
- l ) _ 1 ( M j - 1)*).
KM^MULJ(M1(M1
Since dm = 0, it follows by repeated use of the last inequality that M?-k~l)r2
dk < (1 + Mi + • • • + <{Mx-l)-\M[
(20)
-l)r2
for m — p < k < m. Finally it follows from Eqs. (5), (19) and (20) that - l)r 2 + KM*\m~k
1 4 ^ 1 • • ■ 4 - i ( ^ " Pm)\\ < (M1 - IY\M{
(21)
for m — p < k < m. Then it follows from the inequalities in Eqs. (13), (18) and (21) that \\LT - I\\ < (M1 - l ) _ 1 ( M f - l ) ( n + r 2 ) + K(M9 + Mn)\*.
(22)
We now proceed to estimate \\TL — I\\. We calculate: it
(TLOk=
Y,
A^Ak-2---
Amxm
\Q,rn
-^m — 1 sro —1J
m=k— p+1 k+p m=k+l k =
Y
[Ak-lAk-2
•■AmPmirn
~ Ak-\Ak-2
■ ••
Am-lPm-\£m-l]
m=k—p+1 k+p
[A:1A:l1---A^_1(I-Pm)U-A^A:l1---A-1_2(I-Pm-l)U-i]
~ E m=k+l k
+ . 2J
Ak-lAk-2'--Am[Am-iPm-l
— PmAm-l}£m-l
m=k—p+l k+p +
/
Afc
Afc+1 ■ • • A m _ 1 | A m _ i . F m _ i — i^mAm_iJ^m_j
m=Jk+l = Pfcf jfc - Ak-l
Ak-2
■ ■ ■
Ak-pPk-pik-p
Shadowing in Discrete Dynamical Systems
173
- V ^ i i • ■ • 4T+P-i(7 - p*+p)t*+p+c - p o a k +
J^
Ak-lAk^-'-AmlAm-xPrn-!
-
PmAm_i]fm_i
m=fc—p+1 T
fc+P / v -^jfc A f c + 1 • • • A m _ 1 [ A m _ i P m _ i — P m A m _ i J ^ m _ i . m=ik+l
Hence, using the inequalities in Eqs. (18) and (21), \\TL - I\\ < KMsXp + (Mi - l ) _ 1 ( M f - l ) n + KMu\r k
k+p
m
Yl M*- + J2 ^r~k}
+ sup| fc€2Z
m=fc-p+l
m=Jfc+l
SUp ||Afc_iPjfc_i keTL
PjfcAjfc_i||.
Thus we need to estimate \\Ak-iPk-i Eq. (4), we proceed as follows: WA^Pk-,
+ (Mi - l ) _ 1 ( M f - l ) r 2
- PkAk-i\\
=
— PjfcAfc_i||.
Using the invariance in
\\Df(yk-i)V(yk^)-V(yk)Df(yk^)\\
= ||7>(/(y*_i))Z?/(y fc _i) -
Viy^Dfiyt.y
<\\nf(yk-i))-nyk)\\\W(yk-i)\\ < Mi^s(«5), where u ; » = sup{||P(x) - V(y)\\ : x , y € S, ||x - y|| < a}. Then we conclude that \\TL - I\\ < K(MS + MU)\P + (Mi - l ) _ 1 ( M f - l ) ( n + r 2 ) + Mi(Mi + l)(Mi - l ) _ 1 ( M f - l)u;a((5).
(23)
We choose p as the smallest positive integer such that K(M3 + Mu)\p
< \
and then we assume 6 > 0 is so small that (Mi - 1)~\MP
- l ) ( n + r2) < \
(24)
and Mi(Mi + 1)(M! - l ) _ 1 ( M f - \)u\6)
< |.
(25)
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Brian A. Coomes & Huseyin Kocak k Kenneth J. Palmer
Then it follows from the inequalities in Eqs. (22) and (23) that both inequalities in Eq. (11) are satisfied. So L is invertible and the inequality in Eq. (12) holds. Now, referring to the definition of T in Eq. (10) and using the inequalities in Eqs. (18), (21) and (24) we see that K
\\T\\ < sup {
[KMa\k-m
V
+ (Mi - l ) _ 1 ( M f - l)ri]
m=k— p-\-l
k+p
+ Y, [KMu\m-k+
{M1-l)-l{M*-l)r2\}
m=k+l
< KMS{\
- A)" 1 + p(M 2 - l)~l (M{ - l ) n
+ KMU A(l - A ) " 1 + p ( M i - I)-1 (Ml - l)r 2 < K(M9 + M u A)(l - A)" 1 + \p. Hence L~l || < M,
(26)
where M = 2iC(M s + Mu\)(l
- A)" 1 + \p.
Completion of the proof of the Theorem: Let Q : X —+ X be the C 1 map defined in +oo Eq. (3). Then if {y*}Jf!oo k=—oo is a ^-pseudo orbit
\\S(y)\\ < 6. Also if 8 satisfies the inequalities in Eqs. (24) and (25), DQ(y) = L is invertible and HZ -1 !! satisfies the inequality in Eq. (26). For x = { x * } ^ ^ and f = {6}j£L-oo in X, we calculate the derivative [Dg(x)t)k
= £k+1 - Df(xk)£k
for
keTL.
Then if | | x - y | | < 2M8, \\DQ(x) - DQ(y)\\ < sup \\Df(xk) fees
- Df(yk)\\
So provided 8 satisfies the inequality 2Mu(2M8)
<1
< u(2M8).
Shadowing in Discrete Dynamical Systems
175
in addition to the inequalities in Eqs. (24) and (25), the lemma can be applied. Thus the equation Q(x) = 0 has a unique solution x with ||x — y|| < 2M8. So if we take e = 2M8 this means that x = { x f c } ^ ! ^ is the unique true orbit which e-shadows the 8pseudo orbit y = {y*}£fioo3. Finite-time Shadowing 3.1. The Finite-time Shadowing Theorem Despite the ubiquity of hyperbolic maps, it is practically unfeasible to establish the hyperbolicity of a particular map. Moreover, numerical computations seem to indicate that many of the well-studied examples such as the Henon map are at best hyperbolic only on a large subset of the invariant sets. Thus, the most one should expect in the investigation of a specific map is to be able to shadow finite segments of pseudo orbits. In this section we formulate the notion of shadowing for finite time and prove a Finite-time Shadowing Theorem for a C2 map / : IRn —> IRn without the requirement of hyperbolicity. Definition. A sequence of points {yjfc}^L0
1S s a j c
' *° De
a
^ Vitua%0 orbit of f if
||y*+i - / ( y * ) l l < 6 for fc = 0 , . . . , J V - i . Definition. The 8 pseudo orbit {yk)k=o , s sajc ^ *° De c-shadowed by a true orbit {xfc}£L0, that is, Xfc+i = /(xfc) for k — 0,... ,N - 1, if ||xfc-y*||<e
for fc = 0,...,iV.
Finite-time Shadowing Theorem. Let f : IR" —► lRn be a C 2 map and {yjfc}JL0 be a 8 pseudo orbit. For a given right inverse L~l of the linear operator L : ( I R " ) ^ 1 ^ ( H n ) N , defined for u = {uk}%=0 G 0Rn)N+1 by {Lu)k = uk+l-Df{yk)uk
forfc = 0 , . . . , i V - l ,
(27)
set e = 2\\L-i\\8, where the norm of L~l is the operator norm with respect to the supremum norm on ( H n ) N + 1 and (JRn)N. Next, let M = sup{||Z> 2 /(x)|| : x 6 ]R n , ||x -yk\\<e
for some k =
0,...,N}.
Then if 2M\\L-1\\28<1, the 8 pseudo orbit {yk}k=0
off
is e-shadowed by a true orbit {x f c }^ = 0 of f.
To prove this theorem, we use the following lemma.
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Brian A. Coomes & Kuseyin Kocak & Kenneth J. Palmer
L e m m a . Let X and Y be Unite dimensional normed spaces and let Q : X —» Y be a C2 map. Suppose for some y € X, the derivative DQ{y) has a right inverse K. Set e = 2\\JC\\6 and M = sup{||£>2S(x)||:||x-y||<e}. Then if 2M||/C||2<5 < 1, the equation
0(x) = 0 has a solution x with ||x — y|| < e. Proof of Lemma. Define the continuous operator F : X —► X by
F(x) =
y-IC[G(x)-Dg(y)(x-y)}.
Clearly if F(x) = x, then Q(x) = 0. Moreover, if ||x — y|| < e, ||F(x) - y|| < \\JC\\ ||0(x) - Q(y) - 2>0(y)(x - y) + G(y)\\ <||/C||[±M||x-y|| 2 +<S]
<W|M £ f + ||iCp <M\\>C\\28e + \\K;\\6 <\e + \e = e. Now, the lemma follows from Brouwer's fixed point theorem. Proof of Theorem. We apply the lemma t o I = ( I t " ) * * 1 and Y = {WC)N with the norms Hxll = ll{x*}fcLoll= max: ||xfc|| 0
for x € X and
tall-lltelCs1! = 0<^_1llg*ll for g € Y. We define the C 2 function Q : X -> Y by [a(x)]fc = Xit+i - / ( x f c )
for
k=
0,...,N-l.
Shadowing in Discrete Dynamical Systems
177
If y = {yjfc}jbL0 is a ^ pseudo orbit of / , we see that
Il0(y)ll < 6Clearly DQ(y) = L. Also note that if x = {xjfc}£L0, u = {vik}jf=0 and v = {vfc}JL0 are in X, [D2g(x)uv]k
= -D2f(yLk)ukvk
for k =
0,...,N-l
and so if ||x — y|| < e,
||£>2£(x)|| < M. Hence, by the lemma, if 2M||L- 1 || 2 <5< 1, the equation
0(x) = 0 has a solution x with ||x — y|| < e. Thus the theorem is proved. The main problem in applying the Finite-time Shadowing Theorem to a specific map is the choice of the right inverse L _ 1 and the estimation of its norm. In the remaining part of this section we present a method of solving these problems for one and two dimensional maps. The details of the higher dimensional case are similar to those of the two dimensional case and thus are omitted here (see [14]). 3.2. Implementation of the Theorem in Dimension One Let {yjfc}jfcLo b e a pseudo orbit of / : IR —► IR. In preparation for the estimation of ||L _ 1 ||, let ak be the machine computed value of Df(yk). To account for the inevitable machine error let <$i be a positive number such that \ak - Df(yk)\
< Si forfc = 0,...,JV.
(28)
We approximate L by the operator T defined for u = {ujt}^_ 0 £ IR^"*"1 by (Tu)fc = ujfc+i - akuk
for k = 0,...,iV - 1.
In view of the inequality in Eq. (28), it follows that
\\L-T\\<61. Suppose that we find a right inverse T - 1 for T with ffi||T-1||
(29)
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Brian A. Coomes & Huseyin Kogak k Kenneth 3. Palmer
Then (/ - T~\T
- L ) ) - ^ - 1 is a right inverse L'1 for L satisfying the estimate \\L-1\\<(i-s1\\T-1\\r1\\T-1\\.
(30)
To find a right inverse T _ 1 for T we have to solve the difference equation ujfc+i = ajkUjt + git for k = 0 , . . . , N - 1 to get u = {ujt}fcL0 = T _ 1 g , where g = { g * } ^ 1 € RN. We expect to be dealing with chaotic maps so that \ak\ > 1 for most k. Since we want \\T~l || and hence ||u|| to be as small as possible, we construct the right inverse obtained by solving the equation backwards starting with u/^ = 0 and using the recursion MN = 0,
Ufc = alkJfe1(ujt+i -gfc)
for k = J V - 1 , . . . , 0 .
Clearly, if ||g|| < 1, then |u*| < uk where the iijt's are given by the backward recursion UN = 0,
Uk= lajtT^wjt+i + l)
for fc = i \ T - l , . . . , 0 .
So, IIT"1!! = maxfllT-^H : ||g|| < 1} < max uk.
(31)
0
Computing Uk from this recursion would involve storing all the cut's and this may not be possible when N is large. So we truncate the computation as follows. We choose p so that ti = 0
Jaklakii
•••«*+,! < 1 -
(32)
Note that Uk* = =
K * l K t i l - • • \"klP\uk+P+i+
Y,
K * l K i l l • • • l«ml
for k = 0 , . . . , JV-p-1
m=k
(33) and
N-l
"*= Z)K1IK|il---|aml m=fc
iork = N-p,...,N.
(34)
Shadowing in Discrete Dynamical Systems
179
To facilitate the estimation of ujt we introduce k+p a
w*=El fc 1 IK+il-"l a m 1 l
for
* = 0,...fJV-p-l,
(35)
m=k
and let iik — v>k for k — N — p , . . . , N. We see from Eq. (32) that max Uk < u max wt + max Uk 0
~
0
0
and so, from Eq. (31), IIT -1 !! < (1 - fi)-1
max uk.
(36)
Now we describe the computation of the quantities fi and maxo<jt<jv itjt and hence an upper bound for ||T _ 1 ||. Since Uk depends on both k and N-, we denote it by Uk,N-, and since fi depends on iV, we denote it by fi(N). Then we define cr(N) = max Uk N0<*<JV
(37)
'
Now from Eq. (36) we see that our upper bound for | | T - 1 | | (which also depends on N, of course) is given by
iir-isa-MwrM*)-
(38)
We calculate the quantities fi(N) and cr(N) recursively with respect to N. From the definition (32) of fi(N) we see that fi(N) = 0 if N < p and that for N > p l*(N + 1) = max { fi(Nl
\a^_pa^_p+1
■ ■ ■ a^11 } .
(39)
To calculate cr(N), we see from Eqs. (34) and (35) that (uk,N, uk,N+i=
Since
U*,JV+I
I «fc,N + K l o J f c + i - - " a w | .
ifN-p
llaJJ 1 !,
iik = N.
(40)
= Uk,N for 0 < k < N — p — 1, we define
H0
1) = y
max 0<Jfc
uk,N
and
cr(iV,2)=
max N-p
ufc,;v
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Brian A. Coomes & Huseyin Kogak k Kenneth J. Palmer
so that since UN,N — 0 cr(N) = max{ <x(AT, 1), <x{N, 2) }.
(41)
To calculate a(iV, 1) recursively, we observe that a(0,1) = 0 and that for N > 0 a(N + 1,1)= max {
.
(42)
We also note that
max
Uk,N+i-
(43)
N—p+l
On the basis of the relations in Eqs. (39), (42), and (43) we describe an al gorithm to calculate fi(N), cr(iV, 1), cr(iV, 2), and hence obtain an upper bound for T - 1 | | from Eqs. (41) and (38). For this purpose, we introduce the arrays 2l(iV,z) =
-i -i a N-p+i-l N-p+i
a
'''
-i N-l
for i = 0,. . . , p
a
and b(N,i) = UN-P+i-i,N
for t = 0 , . . . , p ,
where any entry with N — p + i — 1 < 0 is interpreted as zero. Starting with the initialization at JV = 0 using 21(0,0 = 0
/K(0) = 0,
b(0,i) = 0,
we compute the necessary quantities iteratively using the algorithm defined by the formulas
[\aN
11
p(JV + 1) = max{2l(JV + 1,0),
if * = p, p(N)},
unv^i -\ f b W t + 1 ) + 21(^ + 1,0, b(JV + l,t) = ^ I |aAT | >
ifi = 0 , . . . , P - l . if « = p ,
Shadowing in Discrete Dynamical Systems
181
3.8. Example: The Logistic Map We apply the Finite-time Shadowing Theorem to the well-known logistic map f(x) = ax(l - x), where 0 < a < 4. To find <$, let f(x) be the computed value of f(x). If -2~d is the maximum relative error the computer makes when adding, subtracting, multiplying or dividing two numbers, then it follows from [38, 39] that f(x) = ax(l - x)(l + e), where (l_2-<*) 3 < l + e < ( l + 2 _ d ) 3 . Hence | 7 W - f(x)\ < |ax(l - x)\ \e\ < ^((1 + 2~df
- 1)
if 0 < x < 1. So provided our computed orbits lie in [0,1], we may take * = | ( ( 1 + 2 - " ) 3 - 1). To find <$i, note that Df(x)
= a{\ - 2x)(l + e),
where (1 - 2~df
< |1 + e| < (1 +
2~d)2.
So if 0 < x < 1, \Dj{x) - Df(x)\
< \a(l - 2x)e\ < a((l + 2~df
- 1).
Hence for computed orbits in [0,1], we may take <$! = a((l + 2~d)2 - 1). Using a machine with d = 63 and with a = 4 we computed an orbit {yjk}fcL0 with yo = 0.125 and N = 100000. Also we calculated the approximations ak to Df(yk) and using the algorithm given above found that with p = 50 | | r _ 1 | | < 8.682005275757906681 x 10 3 ,
Brian A. Coomes & Huseyin Kocak & Kenneth J. Palmer
182
where we rounded upwards in sums and products of positive quantities in order to ensure that a rigorous upper bound was obtained (any platform conforming to the IEEE standards [1] provides such rounding mode control). Since 61 ||T _ 1 1| is clearly less than 1, from Eq. (30) HIT11| < 8.682005275758004751 x 10 3 . Clearly here M = 2a = 8 and so 2M||L- 1 || 2 £ < 6.537931273473288422 x 10" 1 0 < 1. Therefore our theorem allows us to conclude the existence of a true orbit {xjfc}j^?°00 such that |x* - y * | <£ = 2||£ -1 ||<$ < 9.413048981507439755 x 1 0 - 1 5 .
3.4- Implementation of the Theorem in Dimension Two Let {yjk}^_0 be a 6 pseudo orbit of a C2 map / : H 2 —> 1R2. In preparation for the estimation of | | L - 1 | | , let Yk be the computed value of Df(yk) and, to account for the inevitable machine errors, let £1 be a positive number such that ||n-Af(y*)||<*i
forfc = 0 , . . . , i V - l .
We define the approximating operator L : (JRn)N+1 (Zu)jb = ujfc+i - Ykuk
(44)
—* (IR")^ by
for k = 0,...,N-
1.
Note that ||Z-L||<*i. So if we have found a right inverse L~x for L with
then L'1 = (I — L~l{L — L ) ) _ 1 Z - 1 is a right inverse for L with IIL-^i^a-^HL-1!!)-1!!!-1!!.
(45)
In order to find a right inverse for L we triangularize the Yk as follows. We begin with an orthogonal matrix So and then we define orthogonal matrices Sk
Shadowing in Discrete Dynamical Systems
183
and upper triangular matrices Ak by performing a sequence of QR factorizations recursively: YkSk = Sk+1Ak forfc = 0 , . . . , i \ T - l . Of course, in the computer, what we actually find are matrices Sk satisfying \\StSk ~ I\\ < Si
forfc = 0,...,iV
and upper triangular matrices Ak satisfying \\Ak - S*k+1YkSk\\ <6i
for k = 0 , . . . , N - 1,
where a rigorous upper bound for <$i here and in Eq. (44) can be found using the techniques in [38, 39]. Our aim is to find a right inverse for Z, that is, given g G (IR 2 )^, we have to solve the equation Lu = g for u 6 (1R 2 ) N + 1 . This equation can be written as the difference equation Ufc+i = l*u* + gft
forfe = 0 , . . . , J V - l .
(46)
As verified below, if 6\ < 1, Sk is invertible and so we may make the transformation nk = SkVk
iork = 0,...,N.
(47)
Then Eq. (46) becomes VM-i = SjTiYibSfcVfc + S j ^ g *
forfc = 0 , . . . , i V - l .
(48)
First we find a right inverse for the corresponding operator L : ( I R n ) N + 1 —» (IR n ) N defined by (Lu)jfc = u f c + 1 - S^YkSkUk for k = 0 , . . . , N - 1. To do this we approximate L by the operator T defined by (Tu)* = u fc+1 - Akuk for k = 0 , . . . , N - 1. Note that ||I-T||<
max
WS^YkSk - Ak\\.
To estimate this latter quantity, note first that if
(49)
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Brian A. Coomes & Huseyin Kogak & Kenneth J. Palmer
then S%Sk is invertible and
IKSjs*)-1!! < (i - hT1,
\KSiSk)-1 -1|| < ^(i - tfi)"1.
So 5fc is also invertible and
||^1||<(l~^1)-1V/i+^" and
us*-1 - stt = WKSIS,)-1 - i\s*k\\ < 61y/nr1(i - s1rl.
(50)
(si)
Then tSft&Sk
- A*|| < ||[5fc-i - 5I + 1 ]n5 f c || + \\S*k+1YkSk - Ak\\
< 0(1-S^il+ 6^+6!, where 0 = max ||n||. Hence, from Eq. (49), \\L-T\\<[0(l-61)-1{l
+ S1) + l]61.
Now suppose we have found a right inverse T _ 1 for the operator T with
[^(l-^rHi+^^+i^iiiT- 1 )!
(52)
Then if we define g by gfc = ^+18* for* = 0 , . . . , J V - l we see from Eqs. (46), (47), and (48) that (L'lS)k
=
Skiing)*
defines a right inverse for L and \\L-1\\<(l-61)-\l
+ 61)\\L-%
(53)
Shadowing in Discrete Dynamical Systems
185
What remains now is to find a suitable right inverse for T. The operator T is given by (Til)* = Ujfc+1 - Akuk for k = 0 , , . . .N - l, where
For g = { g c l j j j 1 equation
m , A ujfc = \vk] and Ak = \ak bk wk\ [ 0 ck e (IR)) iV , to find u = T * g , we have to solve the difference
Ufc+i = AfcUfc + gfc
for k = 0 , . . . , N - 1
which, in components, reads Ufc+i = akvk +fefc^jt+ g£1} 2)
wk+i = Ckwk + g[
for A; = 0 , . . . ,iV - 1
for fc = 0 , . . . , N - 1.
(54)
For a chaotic map, we expect |e*| < 1 and \ak\ > 1 for most k. So we construct the right inverse obtained by solving the second equation in Eq. (54) forwards starting with w0 = 0 and the first equation backwards starting with vN = 0. If ||g|| < 1, we get \m\<wt where the tDfc's are given by the recursion wo =0, wk+1 = \ck\wk + 1 forfA= 0 , . , . . J V V - 1 and
\vk\
where the V s are given by the backward recursion
Vk = K 1 K^+i + \b^k + 1) for k = N - 1,...,0. Then, as we are using the Euclidean norm in IFr, ll^-MI < max 0
Jvl+wl V
As with uk in the scalar case, we have to truncate the calculation of the vk. The quantities p and fi are defined as in Eq. (32) and we take k+p Vk=J2
K ' t K + l l - ' - ^ m l d ^ m N m + l)
for
k = 0, . . . , N - p - 1
186
Brian A. Coomes k Huseyin Kogak & Kenneth J. Palmer
and iv-i V* = YJ l a A 1 M a * + l l - - - l a m 1 K I 6 ^ l ^ + 1 ) i0Tk =
N-P,...,N.
m=k
We find that max Vk < v(1 — n)
o
~
max Vk-
o
and hence IIT" 1 II <
4
/ ( l - / i ) - 2 ( max u f c ) 2 + ( max wk)2.
(55)
Now we describe the computation of the quantities fi, maxo
;
= max Vk N o
and
0S(N) = max Wkv ; O<*<JV *
(56) v
;
Now from Eq. (55) we see that our upper bound for ||T _ 1 || (which also depends on N, of course) is IIT"11| < y/(l-fi(N))-**''(Ny+V(Ny.
(57)
We calculate the quantities n(N), cra(N), and cru(N) recursively with respect to N. Prom the definition (32) of p(N) we see that /u(iV) = 0 if TV < p and that for N >p H(N + 1) = max { fi{N), \ajf1_pd^1_p+1 ■ ■ • a^11 } . Also from the definition (56) of a3(N) and the fact that w0 = 0, we see that a J (0) = 0 and that for N > 0 a9(N + 1) = max { as(N),
wN+1 } .
To calculate <7U(JV), we proceed as with cr{N) in the scalar case (see Eq. (40)) and define au(N, 1) =
max 0
vk,N
and au(N,2)=
max
'
N-p
vk N '
so that au(N)
= max{<7 u (JV,l), cr u (iV,2)}.
(58)
Shadowing in Discrete Dynamical Systems
187
To calculate
}.
We also note that cru(iV + 1 , 2 ) =
max
Vk,N+i-
N-p+l
'
T
Finally, we describe an algorithm to calculate fi(N), au{N, 1), <xu(iV,2), cr3(N) and hence obtain an upper bound for | | T - 1 | | from Eqs. (57) and (58). For this purpose, we introduce the arrays 2t(JV, t) =
a
N-p+i-laN-p+i
' ' * aN-l
for i = 0 , . . . ,p
and b(N,i) = VN-p+i-itN
for i = 0,. . . , p ,
where any entry with N — p + i — 1 < 0 i s interpreted as zero. Starting with the initialization at N = 0 using 2l(0,i) = 0 /z(0) = 0,
b(0,i) = 0,
a u ( 0 , l ) = 0,
u>0 = 0,
<J9(0) = 0,
we compute the necessary quantities iteratively using the algorithm defined by the formulas
\aN
|,
if i = p,
/i(JV + 1) = max{Ql(iV + 1,0), /*(JV)}, fb(JV,i + l ) + «(iV + l,t)(|6jv|wN + l ) , {\aN \(\bN\wN+ 1), (TU(AT 4-1,1) = max{au(iST, 1), b(N + 1,0)}, (7u(iV + 1 , 2 ) = max b(N + 1, i), l
WJV+1 = | C ^ | ^ N + 1,
or*(JV -h 1) = max{crs(JV), tSjv+i}.
ifi = 0 , . . . , p - l ifz=p,
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Brian A. Coomes & Huseyin Kogak & Kenneth J. Palmer
S.5. Example: Tinkerbell Map Consider the Tinkerbell map / ( x , y) = (x2 - y2 + cxx + c2y, 2xy + c 3 x + c 4 y) where we take c\ = 0.9, c2 = -0.6013, c 3 = 2, c4 = 0.5 and compute an orbit starting at (—0.72, -0.64). This computed orbit stays in the region R = {(x, y) : -1.23312 < x < 0.461052, -1.54655 < y < 0.544313}. Let / ( x , y) be the computed value of / ( x , y). Then if (x, y) is in the region R, the techniques in [38, 39] show on our machine that, using the Euclidean norm in IR 2 , ll/(s, y) - f{x, y)\\ < 7.4972377002276131 x 10" 1 3 .
(59)
Note that 2x 4- ci
Df(x, y) = 2y + c3
- 2 y + c2 2x + C4
Using the operator norm in IR2, we find on our machine that \\Df(x, y) - Df(x,
y)\\ < 1.715908067801535 x 1 0 - 1 5 ,
(60)
when (x, y) £ R. Our computed orbit is {(x*, yk)}i^=o with (xo, yo) = (—0.72, —0.64) and N = 100000. Also we calculate the approximations Yk = -D/(xjt, yk) to Df{xk, yk) and we find 9 = max ||YJb|| < 3.1677088472709594. 0
-
We compute the matrices Sk and Ak and find that \\S*kSk - I\\ < 1.8296455365185333 x 1 0 - 1 5
for k = 0 , . . . , N
(61)
and ll^fc - Si+1YkSk\\
< 7.3000907661013553 x 1 0 - 1 5
for k = 0,...,JV - 1.
(62)
We take 6 to be the number on the right side of Eq. (59) and #i as the maximum of the three numbers on the right sides of Eqs. (60), (61) and (62).
Shadowing in Discrete Dynamical Systems
189
Fig 1. A pseudo orbit of length 100,000 for the Tinkerbell map. The Finite-time Shadowing Theorem guarantees that this pseudo orbit is shadowed by a true orbit within a distance of at most 7.1895888272521938 x 10" 9 .
Using the algorithm given above with p = 100 we find that | | r _ 1 | | < 479482.50973728485. We check that <$i < 1
and that o = [0(1 - <5i) _1 (l + <$i) + ljtfiHT"11| < 1.4588088916838136 x 10~ 8 < 1. Then it follows from Eqs. (52) and (53) that | | £ - 1 | | < (1 - £ i ) - 1 ( l + *i)(! -
)~ 1 |l r _ 1 H ^ 479482.51673202560.
We check that 61\\L--1l\\<1
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Brian A. Coomes k Huseyin Kocak & Kenneth J. Palmer
and so conclude from Eq. (45) that HIT11| < 479482.51841034199. Now ||Z? 2 /(:r,y)ll<2. Hence we may take M = 2. Then 2M||L _1 || 2 (5 < 0.0068945643144514785 < 1. Therefore our theorem allows us to conclude the existence of a true orbit which e- shadows the computed one with e = 2||L - 1 ||£ < 7.1895888272521938 x 10" 9 . 4. Periodic Shadowing Periodic orbits play a significant role in the dynamics of maps; for example, their number and Lyapunov exponents can give important information about the structure of chaotic attractors [4, 18, 20]. Usually it is rather difficult to locate periodic orbits especially the ones with large periods. Even in the case of asymp totically stable periodic orbits if the period is large the computer simulations can be inconclusive. So, from the mathematical viewpoint the question remains: if something looks like a periodic orbit on the computer is it a periodic orbit? In this section we present a general theorem which establishes the existence of a true periodic orbit near an apparent one in suitable circumstances. Also we present a method for rigorously estimating the Lyapunov exponents of such periodic orbits. 4-1. Periodic Shadowing Theorem For a C2 map / : IRn —► IR n , we now formulate the notion of a good approxi mate periodic orbit. Definition. A sequence {yk}k=o
IS s a j
'^ to be a 6 -pseudo periodic orbit of f if
||y*+i-/(y*)ll<* forfc = o,...,iv-i and
||yo - / ( y * ) | | < & Definition. For a given e, a 8 pseudo periodic orbit {yk}k=o *s sa*d to be eshadowed by a true periodic orbit if there axe points {xjfc}£L0 with Xfc+i = /(xjt) for k = 0 , . . . , N — 1 and Xo = / ( x # ) such that ||xjfc-y fc || <e
for fc = 0 , . . . , TV.
Shadowing in Discrete Dynamical Systems
191
a
—►
If {y*}JtLo is n iV+1 (H ) by
^ pseudo orbit, we define the linear operator L:(TRn)N+1
(Lu)k
= uk+1 - Df(yk)uk
(Lu)w = uo -
for k = 0 , . . . , N - 1
Df(yN)uN.
Periodic Shadowing Theorem. Suppose f: IRn —> IRn is a C2 map. Let {yk}k=Q be a 8 pseudo periodic orbit for f and suppose that the operator L is invertible. Set e = 2\\L-1\\6, where the norm of L~x is the operator norm with respect to the supremum norm on(JRn)N+1. Now, let M = sup{||D 2 /(x)|| : x € l R n , ||x-yjfc|| < e for some fc = 0 , . . . , J V } . Then if 2M\\L-1\\26<1, the pseudo periodic orbit {yk}k=.0
is e-shadowed by a unique true periodic or-
bit {x.ltoProof. We need to find a solution of the difference equations xjfc + i=/(x f c )
for fc = 0 , . . . , i V - l ,
x0 = / ( X J V )
such that ||x*-y*||<e
for fc = 0,...,iV.
We define a C2 map Q : (IPT)"* 1 (JR B ) N + 1 , then [£(x)]* = xk+i
(IR")"* 1 as follows: if x = {xfc}f=0 €
- /(x f c )
for k = 0 , . . . tN - 1,
[a(x)]jv = xo - f{xN). Then the theorem will be proved if we can show that the equation
0(x) = 0 has a unique solution x = {x*}£L0 with ||x* - y*|| < e for k = 0 , . . . , N. To prove this we use the following lemma.
192
Brian A. Coomes & Huseyin Kocak & Kenneth J. Palmer
Lemma. Let X be a Banach space and Q : X -* X a C2 function. Suppose y is an element of X for which \\Q(y\\ < 6 and DQ(y) = L is invertible. Set e
= 2||L"1p
and M = sup{||£2£(x)||:||x-y||<e}. Then if 2M\\L-1\\2S<1, there is a unique solution of the equation Q(x) = 0 satisfying ||x — y|| < e. Proof. Define the operator F : X —► X by F(x) = y - L-'lGix)
- DQ(y)(x - y)].
Clearly Q(x) = 0 if and only if F(x) = x. Moreover, if ||x - y|| < e, ||F(x) - y|| < \\L-i\\ ||0(x) - Q(y) - DQ(y)(x - y) + Q(y)\\ < | | L - 1 | | [ | M | | x - y | | 2 + ^] <M\\L-i\\\e2
+ \e
= 2M||L-1||2% + i£ <e. Furthermore, if ||x — y|| < e and ||z — y|| < e, then \\F(x) - F(z)|| < IIL^IHI^x) - Q{z) - Z>G(y)(x - z)|| <\\L-l\\Me\\x-z\\ = 2M||L-1||^||x-z||. Since 2M\\L~X ||2<5 < 1, F is a contraction on the ball of radius e, center y, and thus the lemma follows from the contraction mapping principle. To prove the theorem we apply the lemma to the map Q : X —► X defined before the lemma with X = ( J R n ) N + 1 , where we use the norm ||x|| = |K**}*Ull = max
|M.
Shadowing in Discrete Dynamical Systems First note that if y = {y*}£Lo
1S o u r
193
^ pseudo orbit, then
ll
for k = 0 , . . . , N .
Hence sup{|p2a(x)||:||x-y||<£}<M, with M as in the theorem. Now we calculate
[DQ(y)u]k = Ufc+i - Df{yk)uk [ W ( y ) u ] N = uo - Df
for k = 0,...,JV - 1,
(yN)uN.
So DQ(y) = L and the theorem follows. 4.2. A Necessary and Sufficient Condition that L~x Exists In order to apply the theorem to a given pseudo periodic orbit we need to verify that L~l exists and estimate | | L - 1 | | . Note that L~l exists if and only if for each g = {gjfc}£L0 in (IR71)^"1"1 the equations ujt +1 = Df {yk)uk uo = Df(yN)uN
4-gjt
for k = 0 , . . . , J V - 1,
+gN
have a unique solution u = { u * } ] ^ in (IR n ) ; v + 1 . We see that for k = 0 , . . . , N k
uk = £>/(y*_i) • ■ • Df{y0 )u 0 + £
J5/(yit-i) • • • Df{ym ) g m _ ! .
(63)
m=l
Hence N+l
uo = Df(yN)
■ • • £>/(yo)u 0 + J2 Df(y")'"'
Af(y».)gm-i
m=l
and so N+l
[I - Df(yN)- • • Df'(y0)]u0 = X ! Af(yw) •' ■ Df(ym)gm-i ro=l
Since the right hand side is arbitrary (we can take go = ■ ■ • = g w - i = 0 and g/v as any vector), it follows that if L is invertible then I-Df(yN)---Df(yQ)
(64)
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Brian A. Coomes & Huseyin Kogak & Kenneth 3. Palmer
is invertible. Conversely, if this matrix is invertible we get N+l
uo = [I - Df(yN)
• • • Df (yoT
1
£
Df(yN)-
■ ■ Z?/(y m )gm-i
(65)
m=l
and for k = 1 , . . . ,N the vector ii* is given by Eq. (63). The conclusion is that L is invertible if and only if the matrix in Eq. (64) is invertible and, in this case, u = £ - 1 g is given by Eqs. (63) and (65). Moreover, we easily obtain the estimate |11| L _ 1 | | < " "
max
0
Ujt,
where Uk is obtained starting with N+l
u0 = \\[I- Df(yN)-
1
• • ^/(yo)]- !! £
\\Df(yN)-
■ ■ Df(ym)\\
m=l
and using the recursion u fc+ i = \\Df(yk)\\uk
+ 1 for k = 0 , . . . , TV - 1.
However, except in the case of an attracting orbit, this method of computing an upper bound for | | £ - 1 | | does not work well for long period orbits. For example, if / is a chaotic one-dimensional map, we expect |.D/(yfc)| > 1 for most k. So if N is large, the computation of Z?/(y^/) • • • Df(yo) will lead to overflow. So we have to estimate H^ - 1 1| in a different way, as in the following two sections. We also indicate how round-off error in the calculation of Df(yk) can be accounted for. 4-8. Implementation of the Theorem in Dimension One Let {yjfc}fcL0 D e a ^ pseudo periodic orbit of the C2 map / : IR —► IR. Let a* be our computed approximation to Df(yk). Consider the linear operator T: IR N + 1 —► IR N + 1 defined for u = {u f c }^ = 0 by (Tu)fc = ujt +1 - ajfcUfc (Tu)yv = U 0 -
for fc = 0 , . . . ,iV - 1,
aNuN.
Note that \\T-L\\<6U if \ak-Df(yk)\<61
for k = 0 , . . . , N .
(66)
Shadowing in Discrete Dynamical Systems
195
To find u = T _ 1 g , we have to solve the equations Ujk+i = a*Ujk + git u 0 = CLNUN +
for k = 0 , . . . , N - 1,
gw.
(67)
For chaotic maps we expect |afc | > 1 for most k. So we solve backwards to find that N u
for k = 0,...,N
* = ak1'-'
a
iv l u o - 5 Z afc * *" * a « 8 m
- 1. Then
u0 = a^1 • • • a^uo - J j a^1 • • • a^g™. m=0
Hence if
l^-aji^l,
(68)
N 1
1 -1
2 ^ O.Q1 ■ ■ ■ a ^ g m ,
u 0 = - ( 1 - a^ • • • a ^ )
m=0
the other Ufc's being obtained from Eq. (67). So if Eq. (68) holds, T is invertible. Moreover if we solve the backward recurrence relation UN = \aNl\(u0 + l) Uk = lajt^Kufc+i + 1) for fc = J V - 1 , . . . , 1 starting with 1
1
N
1
u0 = |(l-a^ ---a^ )- | Y, l ^ 1 - " ^ 1 m=0
then |11| T _ 1 | | < max " -
0
uk.
Finally, if in addition to Eq. (68) the inequality
MT-M|
l|L-|| < ( i - « i p- 1 |)- 1 H3r l |.
Brian A. Coomes & Huseyin Kocak & Kenneth J. Palmer
196
4-4- Example: The Logistic Map We now illustrate the use of our Periodic Shadowing Theorem on the onedimensional logistic map f(x) = 4x(l — x). By a method described in [15], we find a 6 pseudo periodic orbit {y*}*=o with y 0 = 0.004 and 6 = 1.088376 x 1 0 - 7 . We verify that Eq. (68) holds. Then using the method of estimation of | | T _ 1 | | just described, we establish that £ i | | T - 1 | | < 1 and hence that WL'^l < 103.5182, where 8\ was estimated using the methods in [38, 39]. Clearly here M can be taken as 8 and so 2M||L- 1 || 2 £ = 8||JC— x || 2 ^ < 0.01866089 and e = 2\\L-1\\6 < 2.253335 x 10" 5 . Now the theorem implies the existence of a true periodic orbit of period 997 with initial point within e of 0.004. 4-5. Implementation of the Theorem in Let {yfc}£L0 be a 6 pseudo orbit of the estimation of | | L - 1 | | , let Yjt be the for the inevitable machine errors, let 6i
Dimension Two a C2 map / : 1R2 —»■ IR2. In preparation for computed value of Df(yk) and to account be a positive number such that
\\Yk-Df(yk)\\<61
forfc
= 0,...,JV.
(69)
We define the approximating operator L : ( I R n ) N + 1 —» (IR")^"1"1 by (Zu) fc = u fc+1 - Ykuk (Lu)N
= u0
for fc = 0 , . . . , J V - l ,
-YNuN.
Note that ||£-£||<*i. So if we have found an inverse Z - 1 for L with
then X - 1 = ( I — Z - 1 ( Z - L ) ) - 1 ! - - 1 is an inverse for L with
Il^ll^a-^IIZ- 1 !!)- 1 !!!- 1 !!.
(70)
Shadowing in Discrete Dynamical Systems
197
In order to find an inverse for L we triangularize the Y* as follows. We begin with an orthogonal matrix 5 0 and then we define orthogonal matrices Sk and upper triangular matrices Ak by performing a sequence of QR factorizations recursively: YkSk = Sk+1Ak
forfc = 0,...,JV.
We repeat this procedure with SN+1 as the new S0- After several repetitions we obtain matrices Sk satisfying ||5J5ifc-/|| < * i
forfc = 0,...,iV
and upper triangular matrices Ak satisfying
IIA* - SJ+xttSiH <*i for f = 0,,.., N - 1, \\AN - SSYNSN\\ < *i, where a rigorous upper bound for Si here and in Eq. (69) can be found using the techniques in [38, 39]. Our aim is to find the inverse of L, that is, given g <E (IR 2 2", we have to solve the equation Lu = g for u <E ( I R 2 ) ^ + 1 . This equation can be written as the difference equation ujfc+i = Ykuk + gk
for Jfc = 0 , , . . , i V - 1 ,
U0 = YNuN + gN.
(71)
As in Section 3.4, if tfi < 1, Sk is invertible and we may make the transformation uk = Scvk
forfc = 0,...,iV.
(72)
Then Eq. (71) becomes v fc +i = S^YkSkVk v 0 = S^YNSNVN
+ SjJi» +
for k = 0 , . . . , N - 1,
S^gN.
First we find the inverse of the corresponding operator L : (IRn)N+1 defined by (Lu)k
= uk+1 - S^YkSkUk
(Lu)N
= uo -
S^-^YNS N uN.
for k = 0 , . . . , N - 1,
(73) -+ ( E T ) " * 1
198
Brian A. Coomes & Hiiseyin Kogak & Kenneth J. Palmer
To do this we approximate L by the operator T defined by (Tu)jfc = uk+1 - Akuk (Tu)iv = u 0 -
forfc = 0 , . . . , i V - l ,
A Nu N.
If *i < 1,
(74)
it follows as in Section 3.4 that I I L - T I I ^ K l - ^ ) - 1 (1 + ^ ) 0 + 1]*!, where 9 = max IIYfcll. 0
So if [ ( l - ^ O - ^ l - h ^ ^ + l^iHT" 1 !! < 1,
(75)
then L is also invertible and WL-'W < (1 - [(1 - 61)-\1 + S1)0 + l ^ U T - l r i T - 1 ! ! .
(76)
Then we see from Eqs. (71), (72), and (73) that L~l is given by (Z^g)* =
Sh{L-lg)ki
where g* = Sk+iSk for fc = 0 , . . . , W - l , gN = So^gNSo L-*\\<{\-t{r\\ -Hi ^ n a \ - l / i +, h)\\ir% r Mlr-1
(77)
Then, referring to Eq. (70), if ^HZ- 1 !! < 1,
(78)
l^ll^a-^illl-1!!)-1!^-1!!-
(79)
the map L is invertible and
Putting Eqs. (75), (76), (77), (78), and (79) together, we conclude that if the inequality in Eq. (74) holds and
a- = [(i - Si)"1 (i + Si)(i + 0) + lj^nr- 1 1| < i,
(so)
Shadowing in Discrete Dynamical Systems
199
then L is invertible and 1-1
L-1\\<{1-
(81)
What remains now is to give conditions that T is invertible and find an upper bound for | | T _ 1 | | . We write Ak =
ajt 0
bk cjfc
for k = 0 , . . . , JV.
If / is a chaotic map, we expect |a&| > 1 and \ck\ < 1 for most k. Now if T * exists Ufc = (T - 1 g)jt is a solution of the difference equation Ujt+1 = AfcUfc + gjt u 0 = ANUN
In components u* =
Vk
wk
for k = 0 , . . . , J V - 1.
+ g7V.
our equations read
Vk+i = akVk + bkwk + gfcX) VQ = CLNVN
for & = 0 , . . . , N - 1, }
(82)
+ bNwN + g}J
and Wk+i = ckwk + gfc
for fc = 0 , . . . , J V - 1, (2)
(83)
w0 = c^w/v + gN We solve Eq. (83) forwards to find that
N+l WQ = (1 - CN- • • Co) 1 ^
C
N- • • cmZmm-l»
m=l
where we make the assumption | C J V - C O | < 1.
Hence if g is a unit vector,
|lWo| < (1 - \CN- • ' Co|)
1
N+l ^2 l CN " " " C m l ' m=l
(84)
200
Brian A. Coomes & Hiiseyin Kogak & Kenneth J. Palmer
Then if we set N+l W0 = ( 1 -
|CjV • ' • C o | )
_1
^ \CN • • ' Cm\ m=l
and use the recursion Wk+i = \ck\wk + l for * = 0 , . . . , J V - 1 we obtain upper bounds Wk for |u>jfc|. Since we expect |ajk| > 1 for most k, we solve Eq. (82) backwards as in the scalar case to find N V0 = - ( 1 - flo"1 • • ' a^1 ) _ 1 J ^ Go"1 • • *
flmi^^
+ gff).
m=0
where we are assuming -1
~~1
a^|
\a0
(85)
So if g is a unit vector, N 1
1
1
1
\vo\<(l-\ao --a^ \)- J2\a^---a- \(\bm\wm
+ l).
m=0
Then we set N
vo = (1 - K
1
1
1
•••a^ |)- ^
K" 1 •••a- 1 |(|fc m |t5 m + 1)
m=0
and use the backwards recursion VN = \o>Jf'\(pO + |Mt«JV + 1), Vfc = |ajfc"M("*+i + W ® * + 1)
forfc= i V - l , . . . , l
to calculate upper bounds Vk for |vfc|. Hence if the conditions in Eqs. (84) and (85) hold, T is invertible and
\\T-^<maxNy/{vky+(wky. Then provided the inequalities in Eqs. (74) and (80) hold, L is invertible and we may use Eq. (81) to obtain an upper bound for \\L-1||.
Shadowing in Discrete Dynamical Systems
201
4.6. Example: The Henon Map Consider the Henon map
given by
/ ( * i , x 2 ) = (1 - ax\ + x2,
-bxi)
with a = 1.4 and 6 = —0.3. By a method described in [15], we find a 6 pseudo orbit {y*}fc=o°2 with i n i t i a l d a t a (0.53201104688235434, -0.22573297271273646) and 6 = 6.5115307479103834 x 10 - 1 5 We check that the conditions in Eqs. (84) and (85) hold and obtain an upper bound for | | T - 1 | | . Then we check that the inequalities in Eqs. (74) and (80) hold and use Eq. (81) to obtain an upper bound for | | L - 1 | | . We find that 2M||2T 1 || 2 £ = 4a||£- 1 || 2 £ < 0.000474776016367796, and e = 2 | | L " 1 p < 1.486010629876734 x 1 0 - 9 . Now the theorem implies the existence of a true periodic orbit of period 100003 near the computed one. 4-7. Lyapunov Exponents for Periodic Orbits of Maps The Lyapunov exponents measure averaged contraction and expansion rates of vectors along orbits under the linearized dynamics. For a C 2 map / : Htn —>• IR n , let Xo be a periodic point with period N + 1 and let Aj be the ith (i = 1 , . . . ,n) eigenvalue of DfN+1(x.o). Then the Lyapunov exponents of Xo are the numbers
log 1 A, I
Hi = -j——
.
. _
for i = l , . . . , n .
Note that Xo is hyperbolic if and only if m ^ 0 for all i. Furthermore, in this case Xo is stable if and only if fii < 0 for all i. In this section we give a method for computing the Lyapunov exponents of the periodic orbits found in our Periodic Shadowing Theorem. Write xjt = /*(xo) for k = 0 , . . . , N. We suppose as in the Periodic Shadowing Theorem that we have a 6 pseudo periodic orbit {y*r}£L-o s u c ^ ^ a t ||xjfc-y*||<£
forfc = 0,...,JV
and our aim is to calculate approximations to the Lyapunov exponents of xo us ing computable quantities. As in the previous sections, we suppose we have the
202
Brian A. Coomes & Hiiseyin Kocak k Kenneth J. Palmer
Fig 2. A pseudo periodic orbit consisting of 100,003 points of the Henon map. The Periodic Shadowing Theorem guarantees the exis tence of a true periodic orbit (hyperbolic and unstable) of period 100,003 within a distance of at most 1.486010629876734 x 10~ 9 of this pseudo periodic orbit.
sequences { S f c } ^ and {Ak}k=o a positive number i such that
wnere tne
||5l5fc-7|| <£i
A^s are upper triangular and there is
for k =
0,...,N,
\\Ak - St+1YkSk\\
< 61 forfc = 0 , . . . , J V - l ,
\\As - S!YNSN\\
< h,
and
the Yk being the computed values of -D/(yjt) satisfying \\Yk -Df(yk)\\
< Si fork =
0,...,N.
Our first aim is to find the eigenvalues of DfN+1(x0) = D / ( x ^ ) • • • JD/(X0). With a view to triangularizing Df(xk), using the fact that the matrices S^+1YkSk
Shadowing in Discrete Dynamical Systems
203
are almost triangular, we write Ck = S^Dfix^Sk CN = SQ
forfc = 0 , . . . , J V - l ,
D/(XN)SN-
Then CAr.-.C0=50-1Z)/;v+1(xo)50 So the eigenvalues of DfN+1(xo) are the same as those of CN • • • CO- TO facilitate calculation of the eigenvalues of CN • • • Co, we triangularize the matrices Ck using the fact that the known matrices Ak are upper triangular. First we estimate for k = 0 , . . . , N (interpreting SN+I as SQ) \\Ak - Ck\\ = \\S*k+1YkSk -
S^Dfix^Sk]
< ll-SJ+i " ST+ill linil ||5fe|| + | | S £ j \\Yk - Df(yk)\\ \\Sk\\ + \\Sk~li\\W(yk)-Df(xk)\\\\Skl Then, assuming 61 < 1
(86)
and using Eqs. (50) and (51), we find for k = 0 , . . . , N that
\\Ak - Ck\\ < W l + *i(l " S1)-19y/l + 61+(l - a O - V l + M i V f + t f T + (1 - Si)'1 y/1+ 6^6^1 = (l + 61)(l-61)-1[61$
+
+ 6! 61+Me],
where 0=
max IIYjfc 0
and M = sup{||D 2 /(x)|| : x € l R " and | | x - y f c | | <e
for some k = 0 , . . . ,iV}.
So \\Ak-Ck\\<62
forfc = 0,...,7V,
where 62 = (1 + *i)(l - ^ r ^ M + *i + Me].
(87)
Now we restrict consideration to the two-dimensional case. We triangularize the matrices Ck using the following lemma:
204
Brian A. Coomes & Huseyin Kocak & Kenneth J. Palmer
F i r s t L e m m a . For k = 0 , . . . , N let dk bk 0 ck
Ak = and let Ck be 2 x 2 matrices with Ak-Ck\\
<62
for k =
0,...tN.
Define the linear operator £: I R ^ 1 -> JtlN+1 for e = {e*}£L0 in JRN+1 (£e)k = akek+i - Cktk
by
for k = 0 , . . . , N - 1,
(£e)i\f = a^eo — cjve;v-
(88)
Then if l~^ exists and, with respect to the maximum norm on 4||r1||(l + l l ^ l l ^ m a
\bk\ + 62))62 < 1,
(89)
there exist matrices Vk =
1
0 1
ek
such that a* 0
Vk+iCkVk =
for k = 0 , . . . , iV
h
for k = 0 , . . . , N
Ck
(90)
with VN+I =
V0.
Moreover, |ajfc-a*| < £3,
|cfc-cjt|<^3
for k = 0,...,i\T,
where 63 = [l + 2\\r^(max
\bk\ + 62)]62.
(91)
0
Proof. We write
ck =
a_k
bk
dk
Ck
(92)
In order to fulfill Eq. (90) we need Ok
h
"1
dk
Ck
e
. *
0' 1_
1 ejt+i
0' 1
a* 0
h Ck
for k = 0 , . . . , N
Shadowing in Discrete Dynamical Systems
205
with ejv+i = e o -
That is, for k = 0,...,JV flfc 4- bkek = a*,
b k — b k,
dk + c^e* = efc+ia*:,
c* = ek+ibk + ck
and
We solve dk + c*ejt = ejb+i(afc + 6fcefc)
for k = 0 , . . . , N
ew+i = e 0 .
(93)
for the ejfc and then set bk = bky
dk = ak + frfce*, c* = ck - ek+ih
for A; = 0 , . . . , N.
(94)
Approximating Ck by Ak, we can write the system in Eq. (93) as afcejt+i - Cfcefc = (afc - afc)ejt+i + (c* - Cfc)ejt + dk -
ek+\bktk
for fc = 0 , . . . , J V - l ,
(95)
a^eo — c^ew = (aw — a;v)eo + (CAT — c^)ew + d;v — eo^ve^. To solve Eq. (95) we define the nonlinear mapping g: 1RN+1 —> IR^"1"1 for e = { e * } ^ in ] R " H b y fa(e)]jfc = (ajt - ajt)ejk+i + (c* - ck)ek + dk - ek+ibkek
for k = 0 , . . . , N - 1,
[0(e)]N = (ON — aiv)eo + (CAT — Cflr)eN + aV — e06;ve;v. Then solving Eq. (95) is equivalent to solving the equation £e = g(e) or e = T(e) =
rxg(e).
We use Brouwer's fixed point theorem to show that T has a fixed point in the set B=\eeJRN+1 ^
:\\e\\=
max
" "
0<Jb<JV
|cfc|
206
Brian A. Coomes & Huseyin Kocak k Kenneth J. Palmer
with
< JK—1||[«52o- + 82a + S2 + <x(S2 + &)*] = IK"11| [(262 + (b + h)*)* + h], where b = max l&jfc 0
Then using Eq. (89), \\T(e)\\ < \\r1\\(262
+2\\e-1\\(b + 62)62)a + a/2 < a.
Thus Brouwer's theorem can be applied to show that the matrices Vjt do exist. Finally, using Eqs. (92) and (94) we estimate for k = 0 , . . . , N fijfc - "Jfcl = |a/fc + bktk - a*I < \ak-ak\
+ (\bk-h\
+ \bk\)\ek\
<62 + {b + 62)2\\r1\\62 = £3.
Similarly, \ck — Ck\ < 63 for k = 0 , . . . , JV. This completes the proof of the first lemma. Now, we establish the existence of I-1 in the following second lemma. Second Lemma. Let the operator £ : IR N + 1 —> 1R N+1 be defined as in Eq. (88). If |ci\r — Co| < 1 and la^ 1 • ■ -a^l < 1 (96) then l~x exists.
Moreover, Hr1!!^
max gfc,
(97)
where the ejt 's are obtained from the recurrence relation N+l 1
e0 = (1 - la^c^v • ■ • a^col)'
5 Z \a~NCN • ■ ■ a * l c * a * - i Jfc=i
ejfc+i = la^cjfelefeH- la^ 1 !
for k = 0 , . . . ,JV - 1.
(98)
Shadowing in Discrete Dynamical Systems
207
Proof. Let g = {gk}o G IR^"1"1. If e = l_1g equation
exists, it is the unique solution of the
£e = g.
Writing e = {ejfc}£L0 this equation takes the form akek+1-ckek
= gk
for k = 0 , . . . ,N - 1, (99)
aweo — cyve;v — 9NSetting jk = aj^Cfc, note that Eq. (96) implies that JN-'-JO
^
1.
So we can solve Eq. (99) as N+l
e0 = (1 - JN •'• To)-1 5 3 7JV " ' 7*°fc-i0*-i fc=i
with the other e*'s being found from the recursion relations ejt+i = ^k^k + ajT 1 ^ for k = 0 , . . . ,N — 1. The estimate for ||^ - 1 || is immediate. This completes the proof of the second lemma. Now, we return to our original Ajt and Ck- It follows from the second lemma that if the condition in Eq. (96) holds, then £ - 1 exists. So if 62 in Eq. (87) satisfies the inequality in Eq. (89), we may use the first lemma to deduce the existence of the matrices V*. Then we see that V^CN
■ • • C0V0 =
V^CNVN
So the eigenvalues of DfN+1(x.o) exponents are log|ajv-a0|
Q>N • • •
• • • Vf'CoVo =
*
CLQ
0
CJV- • • CQ
are a^ • • • Q>o and CN * ■ *Co- Then the Lyapunov EfcLo lo sl a fcl
,
„
£r=olQgl^l
Note that log|ajt| — log |ajb I =log—^T
la*l and so by the first lemma log(l -6»/\ak\)
< log|dfc| - l o g | a * | < log(l + < V M ) .
(100)
What we compute for /ii and ^2 are
^1=
j\r + i
'
/i2_
iv + i
•
(101)
Now, using Eq. (100) and its analogue for c*, we can summarize our main result regarding the estimation of Lyapunov exponents in dimension two.
208
Brian A. Coomes & Huseyin Kocak & Kenneth 3. Palmer
Proposition. Suppose that a periodic point of period N + 1 of a two dimensional map satisEes the Eqs. (86), (96), and (89). Then the two Lyapunov exponents y.\ and fi,2 satisfy the inequalities
- , £L>i°g(i-<Wkl) < „ < - , ELo'°g(i + ^/KI) Mi + MI Ml+
-
wn
wn
and
« +
j^i
<« <w +
j
^
,
where p,\ and p>2 are as given in Eq. (101) and S3 is as given in Eq. (91). 4-8. Example: The Henon Map Reconsider Example 4.6 using the Henon map. The quantities 6, 61, 6, M and e were all calculated there. We verify that Eq. (86) holds and then we calculate 62 from Eq. (87). We check that the conditions in Eq. (96) hold and then an upper bound for ||^ - 1 || can be calculated from Eqs. (97) and (98). Then we verify that the inequality in Eq. (89) holds and calculate £3 from Eq. (91). We compute
- ] = lXoi°gM _ 041952329i ih = £ * = ° l o s | c t | = -1.62349609. Then it follows from the proposition that Hi = 0.4195 ± 0.0008,
1*2 = -1.623 ± 0.006
where we have also allowed for the roundoff error in the calculation of ji\ and p.2 ( w e used the BSD 4.3 logarithm functions log and loglp which are based on algorithms described in [37]). Thus the periodic orbit of period 100003 of the Henon map whose existence is guaranteed by our Periodic Shadowing Theorem is hyperbolic and unstable. 6. References 1. ANSI/IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754-1985. Institute of Electrical and Electronic Engineers, Inc., 345 East 47th Street, New York, New York 10017, U.S.A.
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2. S.B. Angenent, "The shadowing lemma for elliptic PDE," in Dynamics of In finite Dimensional Systems, ed. S-N. Chow and J.K. Hale (Springer-Verlag, 1987) 7-22. 3. D.V. Anosov, "Geodesic flows and closed Riemannian manifolds with negative curvature," Proc. Steklov Inst. Math., 90 (1967). 4. D. Auerbach, P. Cvitanovich, J.-P. Eckmann, G.H. Gunaratne, and I. Procaccia, "Exploring chaotic motion through periodic orbits," Phys. Rev. Lett, 58 (1987) 2387-2389. 5. R. Bowen, "Periodic points and measures for Axiom A diffeomorphisms," Trans. AMS, 154 (1971) 377-397. 6. R. Bowen, "u;-limit sets for Axiom A diffeomorphisms," J. Differential Equa tions, 18 (1975) 333-339. 7. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomor phisms, Lee. Notes in Math. 470, Springer-Verlag, New York (1975). 8. S.N. Chow, X.B. Lin, and K.J. Palmer, "A shadowing lemma with applications to semilinear parabolic equations," SI AM J. Math. Anal., 20 (1989) 547-557. 9. S.N. Chow and K.J. Palmer, "On the numerical computation of orbits of dy namical systems: The higher dimensional case," / . Complexity, 8 (1992) 398423. 10. S.N. Chow and E.S. Van Vleck, "A shadowing lemma for random diffeomor phisms," Random and Comp. Dynamics, 1 (1992) 197-218. 11. C.C. Conley, "Hyperbolic invariant sets and shift automorphisms," in Dynam ical Systems Theory and Applications, Ed. J. Moser, Lecture Notes in Physics 38 (1975) 539-549. 12. B.A. Coomes, H. Kogak, and K.J. Palmer, "Periodic shadowing," in Chaotic Numerics, ed. P. Kloeden and K.J. Palmer, AMS, Providence, Rhode Island (1994) 115-130. 13. B.A. Coomes, H. Kogak, and K.J. Palmer, "A shadowing theorem for ordinary differential equations," ZAMP, 46 (1995) 85-106. 14. B.A. Coomes, H. Kogak, and K.J. Palmer, "Rigorous computational shadowing of orbits of ordinary differential equations," Numer. Math., 69 (1995) 401-421. 15. B.A. Coomes, H. Kogak, and K.J. Palmer, "Computatibn of long periodic orbits in chaotic dynamical systems," preprint. 16. W.A. Coppel, Stability and Asymptotic Behavior of Differential Heath Mathematical Monographs, 1965.
Equations,
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Brian A. Coomes & Huseyin Kogak & Kenneth J. Palmer
17. I. Ekeland, "Some lemmas about dynamical systems," Math. Scand., 52 (1983) 262-268. 18. V. Franceschini, C. Gilbert, and Z. Zheng, "Characterization of the Lorenz attractor by unstable periodic orbits," Nonlinearity, 6 (1993) 251-258. 19. J.E. Franke and J.F. Selgrade, "Hyperbolicity and chain recurrence," J. Dif ferential Equations, 26 (1977) 27-36. 20. C. Grebogi, E. Ott, and J.A. Yorke, "Unstable periodic orbits and the dimen sion of chaotic attractors," Phys. Rev. A, 36 (1987) 3522-3524. 21. S.H. Hammel, J.A. Yorke, and C. Grebogi, "Do numerical orbits of chaotic dynamical processes represent true orbits?," J. Complexity, 3 (1987) 136-145. 22. S.H. Hammel, J.A. Yorke, and C. Grebogi, "Numerical orbits of chaotic dy namical processes represent true orbits," Bull. Amer. Math. Soc., 19 (1988) 465-470. 23. A. Katok, "Lyapunov exponents, entropy and periodic points for diffeomorphisms," Pub. Math. IHES, 51 (1980) 137-173. 24. O.E. Lanford III, "Introduction to the mathematical theory of dynamical sys tems," in Chaotic Behavior of Deterministic Systems, Les Houches, 1981, North-Holland, Amsterdam (1983) 3-51. 25. O.E. Lanford III, "Introduction to hyperbolic sets," in Regular and Chaotic Motions in Dynamical Systems, ed. G. Velo and A.S. Wightman, Plenum Press, New York (1985) 73-102. 26. K.R. Meyer and G.R. Sell, "Melnikov transforms, Bernoulli bundles, and almost periodic perturbations," Trans. A.M.S., 314 (1989) 63-105. 27. S.E. Newhouse, "Lectures on dynamical systems," in Dynamical Systems, CIME Lectures, Bressanone, Italy, June 1978, Birkhauser, Boston (1980) 1114. 28. K.J. Palmer, "Exponential dichotomies, the shadowing lemma and transversal homoclinic points," Dynamics Reported, 1 (1988) 265-306. 29. C. Robinson, "Stability theorems and hyperbolicity in dynamical systems," Rocky Mountain J. Math., 7 (1977) 425-437. 30. T. Sauer and J.A. Yorke, "Rigorous verification of trajectories for computer simulations of dynamical systems," Nonlinearity, 4 (1991) 961-979. 31. LB. Schwartz, "Estimating regions of existence of unstable periodic orbits using computer-based techniques," SIAM J. Numer. Anal. 20 (1983), 106-120. 32. M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York (1987).
Shadowing in Discrete Dynamical Systems 33. Ya.G. Sinai, "Gibbs measures in ergodic theory," Russian Math. Surveys, 27 (1972) 21-64. 34. Ya.G. Sinai and E.B. Vul, "Discovery of closed orbits of dynamical systems with the use of computers," J. Stat. Phys., 23 (1980) 27-47. 35. H. Steinlein and H.O. Walther, "Hyperbolic sets and shadowing for noninvertible maps," in Advanced Topics in the Theory of Dynamical Systems, Ed. G. Fusco, M. Iannelli, and L. Salvador:, Academic Press, Boston (1989) 219-234. 36. D. Stoffer, "Transversal homoclinic points and hyperbolic sets for nonautonomous maps I," ZAMP, 39 (1988) 518-549. 37. P.T.P. Tang, "Table-driven implementation of the logarithm function in IEEE floating point arithmetic," ACM Trans. Math. Software, 16 (1990) 378-400. 38. J.H. Wilkinson, Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. 39. J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
PERTURBATION OF INVARIANT MANIFOLDS OF ORDINARY DIFFERENTIAL EQUATIONS 1 GEORGE OSIPENKO Laboratory of Nonlinear Analysis, State Technical University, St.Petersburg, 195251 Russia and EUGENE ERSHOV Department of mathematics, University of Architecture and Civil Engineering, St.Petersburg,
198005 Russia
Abstract The lecture presents the results on the perturbation problem of invariant manifolds of smooth dynamical systems given by a general autonomous ordi nary differential equation in Rn. The submitted results cover the problem of perturbation of equilibrium points, periodic orbits, locally invariant manifolds, normally hyperbolic compact invariant manifolds and compact invariant ma nifolds which are not locally unique. Various phenomena occurring under a perturbation of invariant manifolds are illustrated by examples.
1. Introduction Let us assume that a dynamical system given by a differential equation, by a flow, or by a mapping has an invariant manifold. Roughly speaking, an invariant manifold is a surface contained in the phase space of a dynamical system that has the property that orbits starting on the surface remain on the surface throughout the course of their dynamical evolution, i.e., an invariant manifold is a collection of orbits which form a surface. The importance of the notion of invariant manifold for the study of dynamical systems has increased significantly in recent years. Invariant manifolds are of crucial importance in understanding of the behavior of dynamical systems for a broad variety of applied dynamical systems in different areas of science and engineering. An up-to-date list of areas in science and engineering to which invariant manifolds permeate can be found, e.g., in Wiggins [26]. In general, a dynamical system has many invariant manifolds. Certain of them are of prime importance for the study of the dynamical evolution of a system. The most commonly encountered types of such invariant manifolds are: 1. Equilibrium points; Supported by the Russian Fund of Fundamental Investigations under Grant 94-01-00294 and in part by Grant NWJOOO from the International Science Foundation.
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2. Periodic orbits; 3. Invariant tori, which are the closure of quasiperiodic orbits with two or more basic frequencies; 4. Stable, unstable and center manifolds associated with equilibrium points, pe riodic and quasiperiodic orbits. Invariant manifold theory begins by assuming or by proving that a dynamical system has some invariant manifold. The question arises at once of whether or not this invariant manifold persists under perturbation of a dynamical system. The mat ter is that in most cases a dynamical system, which is the mathematical model of a real phenomenon, is known with a certain degree of inaccuracy. Therefore, the existence of an invariant manifold is of moderate significance if this manifold can be destroyed by a perturbation of the initial dynamical system. The question of whether or not a persistent invariant manifold maintains or loses differentiability also arises. The problem of perturbation of invariant manifolds has a long history in dynamical systems theory. Historical descriptions can be found in Bogoliubov and Mitropolsky [2], Mitropolsky and Lykova [16], Hirsch, Pugh, and Shub [7]. In the present paper we consider the problem of perturbation of invariant mani folds of smooth dynamical systems given by a general autonomous ordinary differen tial equation defined on Rn: x = ^ = F0{x),
x e Rn,
(1)
where F0 : RJ1 —► RJ1 is a C 1 vector field. Hereafter, we will use the term "smooth" as a synonym for C1 smooth. Since in the sequel Eq.(l) will be considered in a bounded part of the phase space, there is no loss of generality to assume that the function FQ is bounded. Let us denote the solution of Eq.(l) passing through the point p £ R? at t — 0 by X(t, p). By the fundamental theorems of differential equations theory the hypotheses imposed on F0 guarantee existence, uniqueness, and smoothness of X(t, p) for all p £ R" and all t G R. Thus, the vector field F 0 generates the smooth flow Xt : RJ1 —► if*, Xt(p) = X(t, p) , satisfying the group property: Xt+a(p) = Xt(Xs(p)) for all p 6 #*,
t, s e R.
Definition. 1 A set M0 C RP is said to be invariant under Eq.(l) if Xt(M0) = M0 for all t G R, i.e., M 0 consists of orbits of Eq.(l); M0 is said to be locally invariant under Eq.(l) if it consists of arcs of orbits, i.e., for every p G M0 there exist, depending on p, 7 \ , T2, T\ < T2, TXT2 < 0 such that Xt(p) e Mo for t 6 p i , T2). An invariant (locally invariant) set M0 is called a smooth invariant (locally inva riant) manifold of Eq. (1) if MQ is a smooth submanifold of RJ1.
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The main problem dealt with in this paper is the following. Let us assume that Eq.(l) has a smooth invariant (locally invariant) m—dimensional, 0 < m < n , manifold Mo. Consider a perturbation of Eq.(l) of the form x = F(x),
(2)
where the smooth vector field F is C1 close to Fo. The problem is to find sufficient and necessary conditions such that each Eq.(2) has a smooth invariant (locally invariant) m-dimensional manifold Mp, C1 close to Mo, i.e., conditions such that Mo is preserved under C 1 perturbations of the initial equation (the precise definitions will be given in Section 2). We will refer to this problem as the perturbation problem of invariant manifolds. The following list of papers related to the perturbation problem of invariant ma nifolds is in no way intended to be exhaustive or complete, but involves only papers very closely connected with the questions considered in the paper. Poincare was likely the first who perceived the importance of the perturbation problem and began to study conditions ensuring the preservation of periodic orbits under a perturbation of differential equations. At the same time Liapunov actually constructed locally invariant manifolds passing through an equilibrium point of an analytic differential equation with the matrix of linear approximation having one zero, or two pure imaginary eigenvalues. Perron [20] proved the existence of saddle invariant manifolds for a hyperbolic equilibrium point and, in fact, showed their preservation. Various properties of toroidal invariant manifolds of differential equations in a stan dard form, the preservation property among them, were investigated by Bogoliubov [2]. In Pliss [21], Kelley [10] and in the papers of many other authors (for references see Kirchgraber and Palmer [11]) the various versions of the "reduction principle" were obtained. In accordance with this principle for the critical part of the spectrum of a differential equation, linearized at an equilibrium point (or a closed orbit), there corresponds a locally invariant manifold (so-called center manifold). The behavior of orbits on a center manifold determines the behavior of a dynamical system in a whole neighborhood of an equilibrium point (a closed orbit) and this manifold is preserved under perturbations. Samoylenko [23] obtained profound results on the preservation of invariant, exponentially stable tori. In the late 1960s and throughout 1970s, the theory of perturbations of invari ant manifolds began to assume a very general and well-developed form. Sacker [22] and Neimark [15] proved independently that a normally hyperbolic compact invari ant manifold is preserved under C1 perturbations. Fenichel [3] extended this result to invariant manifolds with boundary and proved that the normal hyperbolicity of an invariant manifold is maintained under C1 perturbations. Mane [14] showed that a locally unique, preserved, compact invariant manifold is necessarily normally hy perbolic. The modern state of the theory of normally hyperbolic invariant manifolds is available in Wiggins [26]. Sufficient conditions for the preservation of locally non-
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unique compact invariant manifolds were found by Osipenko [17]. These conditions are presently the most general ones that guarantee the existence of a perturbed inva riant manifold Cl close to the original one. Conditions necessary and sufficient for the preservation of locally invariant manifolds passing through an equilibrium point have been recently obtained by Osipenko and Ershov in [18]. A study of the necessity of conditions for the preservation of arbitrary compact invariant manifolds is available in Osipenko and Ershov [19]. The present paper is organized as follows. In Section 2 we introduce basic notions and give motivational examples. In Section 3 we consider the perturbation problem for the simplest case, where an invariant manifold is an equilibrium point. In Sec tion 4 we present results on a perturbation of linear pseudo-hyperbolic maps, used in Sections 5, 6 for a study of the perturbation problem of locally invariant manifolds through an equilibrium and a periodic orbit. In Section 5 the perturbation problem for locally invariant manifolds passing through an equilibrium point is studied. The results related to the perturbation of closed orbits and locally invariant manifolds passing through them are presented in Section 6. In Section 7 we study the pertur bation problem for an arbitrary locally unique compact invariant manifold. Here we present the results of Sacker, Neimark and Mane on persistence of compact normally hyperbolic invariant manifolds. In Section 8 we study the perturbation problem for compact invariant manifolds which are not necessarily locally unique. Here various phenomena occurring under a perturbation of invariant manifolds are illustrated by examples. Acknowledgments We are grateful to the Russian Fund of Fundamental Investigations and to the International Scientific Foundation for support. We are particularly grateful to Pro fessor Bernd Aulbach for stimulating discussions. This course was given by George Osipenko during Spring 1993 in Pohang Institute of Science and Technology (South Korea) and Dr. Seunghwan Kim made a number of useful comments. Special thanks are due to Egor Osipenko who did an excellent job of computer typing the manuscript.
2. Basic notions and motivational examples Assume that Eq.(l) has an m (0 < m < n)—dimensional compact invariant mani fold M0 without boundary. In order to study the behavior of M0 under a perturbation we identify Eq.(2) with the vector field F and introduce the topologies in the space of smooth vector fields and in the space of smooth compact manifolds C 1 close to M0. We will also obtain a representation of Eq.(l) suitable for the subsequent discussion.
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2.1 Topology in the space of smooth vector fields Since we study the flow generated by F only in a neighborhood of Mo and M 0 is compact, there will be no loss of generality to assume that F has the bounded derivative DF : TR* —> TRn. In the space T of smooth bounded vector fields defined on .ft" we introduce the topology degenerated by the C1 norm | | F | | = supn \F{p)\ + supn peR
peR
m*x\DF{p)v\,
M=i
where \p\ stands for the Euclidean norm of p G Rn1 DF(p) is the Jacobian matrix of F at p. Therefore, a neighborhood of F0 in T includes all smooth vector fields F C1 close to FQ. 2.2 Tubular neighborhood of a submanifold To obtain a representation of Eq.(l) in coordinates near Mo and to introduce a topology in the space of m—dimensional compact submanifolds of RJ1 C 1 close to Mo we use the construction of a tubular neighborhood of Mo- Roughly speaking, a tubular neighborhood of Mo is one in which we can introduce the local coordinates (p, v), where p G Mo and v is a vector in the linear subspace 7VP, dim Np = codimMo, transversal to Mo at p. The precise construction of a tubular neighborhood is follo wing. For a point p G Mo denote by TPMQ the tangent space to Mo at p. The linear space Np C TpRJ1 is said to be transverse to Mo at p € Mo if TpRn = TpM0 + Np. From the results of Whitney [24] it follows that there exists a smooth field {Np, p G M 0 } of linear spaces Np, dimiVp = k = n — m, transversal to Mo at every p G M0. Thus, we obtain the smooth k—dimensional vector bundle iV C TiT^Mo w ^ h Mo as a base and Np as fibers. Elements of iV are denoted by the pairs (p© u), where p G Mo and v £ Np. We will refer to N as the transversal bundle. To construct a tubular neighborhood of Mo let us consider the map h : N —> RJ1 defined by h(p® v) =p + v. Obviously, the map h makes possible to identify the zero section iVo = {{p® v) G N : v = 0} of N with M 0 . For e > 0 let N£ = {(p © v) G N : \v\ <e}, where \v\ is the norm on Np induced from R", be the neighborhood of the zero section N0. The following result is due to Whitney [25]. Proposition. 1 There exists e0 > 0 such that for 0 < e < £o the map h is a C1 diffeomorphism from Ne to a neighborhood VE of M 0 in R1.
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The transversal bundle TV is called a tubular neighborhood of M 0 . By construction, TV is a smooth vector bundle. However, by Whitney's Extension Theorem it is possible to introduce on TV a compatible C°° structure, i.e., a C°° structure which C 1 agrees with the initial one (see Hirsch [6], or Abraham and Robbins [1]). Thus, there will be no loss of generality to assume the transversal bundle TV as C°° smooth. By Proposi tion 1 one can transfer a C°° structure of the vector bundle from Ne to Ve. We will not make distinctions between Ne and Ve and consider Eqs.(l), (2) as defined on the vector bundle TV. The vector bundle TV is locally trivial. This means that for every p £ M0 there exist a neighborhood U of p in M 0 and a smooth basis {ei(y), e2(y), ....,ek{y)} in Ny, y G U, such that the map o : U X Rk -> N\u, ^(y, z) = y + £ i = i z,e,(y) is a C 1 diffeomorphism. However, in general, TV is not trivial, i.e., the basis described above does not exist globally. In the case when TV is not trivial we can include it in a trivial vector bundle using the following result of Hirsch, Pugh, and Shub [7], Proposition. 2 For any vector bundle TV there exists a vector bundle N\ such that TV © N\ = TV* is diffeomorphic to Mo x Rk+l, i.e. N* is a trivial vector bundle. To construct a vector field on N* we apply the technique of [7]. The vector field F defined on N can be extended to F* on N* by setting F*(u©v) = F(u)®0. The bundle N may be identified with N © {0}. Clearly, N is F*-invariant and F*\N = F © 0. The restriction F*\N obeys the same hypotheses as F. The flow on N is of the form (y(<,y,z©0), Z(t, j/,z©0)©0), y € Mo, z E Ny. To avoid complication of notations it is reasonable to omit ©0, that is equivalent to the triviality of TV. Therefore, without loss of generality we will assume N to be trivial. Taking into account C°° smoothness and triviality of TV, it is not difficult to see that Eq.(l) in coordinates (y, z), where y is a coordinate on M 0 and z is a coordinate on TVy, takes the form y= z=
G0(y,z), H0(y,z).
(3)
where Go, #o are C 1 functions, Ho(y,0) = 0. 2.3 Topology in the space of submanifolds C1 close to MQ Let M C i f be a submanifold C 1 near to M 0 . In the tubular neighborhood of M 0 one can uniquely represent M in the form of a graph (see Fig.l): M = {(PMP))
:
P e M0,
(4)
where
Perturbation of Invariant Manifolds of Ordinary Differential Equations
219
Figure 1: R e p r e s e n t a t i o n of a manifold M C1 close t o M0. and M2 = {(p,
P e M 0 ,
we define the distance by dist{MuM2)
= \\
(5)
where = sup \
pGMo
M=l
(6)
That is, the distance between Mi and M2 is the C 1 norm of y>\ —
George Osipenko & Eugene Ershov
220 Consider the differential equation x = x2,
x G R,
having the equilibrium x = 0. Clearly, Mo is destructible since the perturbed equation x = x2 + 6 has no equilibria whatever 6 > 0 be. Example. 2 Indestructible locally unique invariant manifold. Consider the differential equation i = i,
i6i?
(7)
and its perturbation
x = x + f(x),
(8)
where f(x) is C 1 small in a neighborhood of the origin x = 0. The equilibrium 0(x = 0) of Eq.(7) is an indestructible invariant manifold. Actually, near x = 0 the equation x + f(x) = 0 has a unique solution provided / is C1 small. Hence, Eq.(8) has a unique equilibrium point Of near O and dist(Of, 0 ) —► 0 as | | / | | —► 0. If the invariant manifold MQ is indestructible then, according to Definition 1, there exists a map H : W —> Ai, defined in a neighborhood W of the vector field FQ in T and associating to the vector field F E W an invariant manifold Mp of Eq.(2), i.e., H{F) = Mp, H{FQ) = Mo, which is continuous at F0. In Example 2 the map H associating to Eq.(8) its equilibrium point is uniquely determined. This property of the map H is due to the local uniqueness of the equ ilibrium point O of Eq.(7). Roughly speaking, an indestructible manifold is locally unique if it is a maximal invariant set in its neighborhood and remains so for C1 perturbations. The precise definition is following. Definition. 3 An indestructible invariant manifold Mo of Eq.(l) is said to be locally unique if there exist neighborhoods V of MQ in R* and W of FQ in T such that for every F 6 W the maximal invariant set of Eq. (2) contained in V
IF = 0
Xt(V),
teR
where Xt is the flow generated by F, coincides with the perturbed invariant manifold MF, i-e. , IF = MF-
Perturbation of invariant Manifolds of Ordinary Differential Equations
221
Clearly, if Mo is an indestructible locally unique invariant manifold then the map H is uniquely determined and H(F) = Ip- An indestructible locally unique invariant manifold is said to be via Mane [14] persistent. Thus, the equilibrium point of Eq.(7) gives an example of a persistent manifold. However, there are many simple equations possessing indestructible invariant manifolds which are not persistent. Example. 3 Indestructible locally non-unique invariant manifold. Consider, on R2, the differential equation il
=
*2'
(9)
x2 = - a ? i ,
which is a linear center, and its perturbation Xi = X2 + fl(xi,X2),
x2 = -xi+
,-QV
f2(x1,x2),
where / i , f2 are C1 small in a neighborhood of the origin O(0,0). As an invariant ma nifold M 0 of Eq.(9), we consider its equilibrium O. Equilibria of Eq.(lO) are solutions of the algebraic system x2 + f\{xi,x2) = 0, , . ~X\+ f2{xX,X2) = 0. By the Implicit Function Theorem for Banach spaces (see Section 3), Eq.(ll) has a unique solution near O. This means that O is an indestructible invariant manifold of Eq.(9). On the other hand, O is not locally unique since in any neighborhood of O there are periodic orbits of Eq.(9). Thus, 0 is an indestructible invariant manifold which is not persistent. From the Implicit Function Theorem it follows also that the unique solution of Eq.(ll) depends continuously on fi, f2. Thus, the uniquely determined map H is continuous not only at the initial vector field but in its whole neighborhood as well. The following example shows that this is not necessarily the case in general. Example. 4 Indestructible locally non-unique invariant manifold. Consider the differential equation i = FQ{x) = x 3 ,
xe R
(12)
having the invariant manifold M 0 which is the equilibrium 0(x = 0). M 0 is indestruc tible since, as easily seen, every equation C1 near to (12) has at least one equilibrium point near O. Clearly, the map H associating to a perturbed equation its equili brium is not uniquely determined since a perturbed equation can have more than
George Osipenko & Eugene Ershov
222
Figure 2: Perturbation of F0{x) = ar one equilibrium point. Let us show that H cannot be chosen continuous in a whole neighborhood W of F0 whatever small it be. Given 6 > 0 consider the C 1 function (see Fig.2) (x + a) 3 , x < —a F(x) = < 0, -a < x < a 3 x > a, { (*-<*) , where a > 0 is small enough to ensure | | F — Fo|| < 6. We assert that the map H is not continuous at F. In fact, the equation x = F(x) + a for cr > 0 has the unique equilibrium 0\ with x-coordinate XI(<J) < —a. and for cr < 0 the unique equilibrium 0 2 with x-coordinate X2{cr) > a. Hence, H(F + cr) = xi(a) for a > 0 and # ( F + a) = £2(0") f° r cr < 0. From this it follows that lim H(F + a) = - a , lim H(F +
cr—f-(-0
a—►-()
Since the left and the right limits do not coincide then H is not continuous at F independently of the choice of H(F). From the practical point of view it is important to investigate those indestruc tible invariant manifolds for which the map H can be chosen continuous in a whole neighborhood of the initial vector field. In this case small C 1 perturbations of Eq.(2) involve small C1 perturbations of its invariant manifold Mp = H(F). This makes natural the following Definition. 4 An invariant manifold MQ of Eq.(l) is called strongly indestructible if there exist a neighborhood W of Fo in T and a continuous map H : W —► M. such
that
Perturbation of Invariant Manifolds of Ordinary Differential Equations
223
(i) H(F) is an invariant manifold of Eg. (2), (ii) if Mo is an invariant manifold of Eq.(2) then H(F) = Mo. The last condition means that if Mo remains invariant for the perturbed vec tor field F then the map H associates to F the initial manifold Mo, in particular, H(FQ) = Mo. Evidently, a strongly indestructible invariant manifold is indestructible. As Example 4 shows, an indestructible invariant manifold is not necessarily strongly indestructible. From what follows (see Section 3) it will be clear that the equilibrium points in Examples 2 and 3 are strongly indestructible invariant manifolds. Definition. 5 An indestructible invariant manifold which is not strongly indestruc tible is said to be weakly indestructible. The equilibrium of Eq.(12) gives an example of a weakly indestructible invariant manifold.
3. Perturbation of equilibria Let us consider Eq.(l) and suppose that O is an isolated equilibrium point of it, i.e., i'o(O) = 0. The goal is to study the perturbation problem of the equilibrium O. As we study perturbations of Eq.(l) only near O, the restriction of Eq.(2) on some neighborhood, say V, of O is essential for our consideration. However, for convenience it will be useful to construct an equation defined on R" which coincides with Eq.(2) on V and has the same properties in i?" as Eq.(2) on V. Let us introduce some notations. For d > 0 denote by Vd = {x £ R1 : |x| < d] the ^-neighborhood of the origin. Recall that a function f : R" -+ K" satisfies a Lipschitz condition in R* if there is a constant / such that |/(x1)-/(x2)| R* be a smooth function such that /(0) = 0, Df(0) = 0. Then for every I > 0 there exist d > 0 and a smooth function g : R1 —► R* with the properties: (i) g = / on Vd, (ii) g = 0 outside V$d, (Hi) Lip(g) < /.
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George Osipenko & Eugene Ershov
Proof. If is not difficult to define a C°° function A : R -» [0,1] with the following properties (see, Hirsch [6]): A(r) = l if 0 < A(r) < 1 if A(r) = 0 i/ |A'(r)| < 1 if
|r|
The function A is sometimes called a bump function. For r G i?+ we set a(r) = m a x | D / ( x ) | . |a;|
Obviously, \Df(x)\ < oc(\x\), and a(r) - > 0 a s r - > 0 because £ / ( 0 ) = 0. Since
/(*) = £ §-t(f(tx))dt = [I Df{tx)dt then \f{x)\ < OL(\X\) \X\. Given / > 0, choose d > 0 so that 4a(3d) < / and define the smooth function g by 9{x) = A ( ^ )
/«•
The properties (i), (ii) hold by definition of g and properties of A. It remains to verify (iii). For x 6 iT1 we have
mx)\<^\D9(X)\<^{\D(x(^))\\f(X)\
+ |A(M)| |Z>/(*)|}
< ^3rf a(3d) + a(3d) = 4a(3d) < /, a which gives (iii). 0 Eq.(l) can be written in the form x = A0x + / 0 (a;),
(13)
where AQ = DFo(0), f0(x) = F0(x) - AQX. Applying Proposition 3 to the function fo(x) we obtain the equation x = Aox + g0(x), (14) where g0 is smooth , g0 = /o on V£, Lip(g0) is small provided d is small. Consider now the perturbed equation. Eq.(2) can be written in the form x = A0x + fo(x) + A F ( i ) ,
(15)
Perturbation of Invariant Manifolds of Ordinary Differential Equations
225
where AF = F — FQ. Thus, the perturbed equation corresponding to Eq.(14) takes the form x = A0x + g0(x) + AF(x) = A0x + LF(x). (16) Eq.(16) coincides with Eq.(2) in Vd and Lip{LF) < Lip{g0) + Lip(AF) < Lip(g0) + \\F - F0\\,
(17)
i.e., Lp has a small Lipschitz constant provided d and \\F — F 0 || are small. 3.1 Hyperbolic case Definition- 6 An equilibrium point 0 of Eq.(l) is called hyperbolic if the eigenvalues of the matrix DFo(O) have nonzero real parts. If so, the linear equation x = DF0(O)x is called hyperbolic. Proposition. 4 A hyperbolic equilibrium point is indestructible. Proof. Since the matrix A0 = DFo(0) has no zero eigenvalues then there exists the inverse matrix AQ1. Choose d > 0 such that
1-4—X| Lip{g0)
= AF(i).
(18)
It is easy to see that the operator AF : if1 -* K1 is a contraction: for a?i, x2 G if* we have
| A F ( * i ) - M * a ) l £ | V | \LF(xi)-LF(x2)\
<\Ao'\
Lip(LF)
|*i-*a|
< | A o ' | (£«>(#>) + 11^ - Fo\\) | * i - * a | < J | * i - * a | . Therefore, Eq.(18) has a unique solution xF, i.e., Eq.(16) has the unique equilibrium OF(X = XF)- From (18) we obtain \xFl - 2F2\ = l A ^ ^ f i ) -
AF2(XF2)|
226
George Osipenko & Eugene Ershov < \AFl(xFl)-AFl(xF2)\+
\AFl(xF2)
-
AF2(xF2)\
< I \xFl - xF2\ + | V | l*i(*A) " *a(**)l i which leads to dist(0Fl,
0F2) = \xFl -xF2\<
^ y l l - F i - Fall-
(19)
Putting in (19) Fi = F,F2 = F 0 and taking into account that a?|?0 = 0, we obtain
A-11
dist{0F, 0) = \xF\ < —2-1 ||F - Fo||. 1-/
(20)
From (20) it follows that given g, 0 < e < d, there exists 8 > 0 such that for every F , ||F — F 0 || < 6 equilibrium 0F of Eq.(16) is contained in the e—neighborhood of the origin. Since Eq.(2) coincides with Eq.(16) on Vd and e < <2, then 0F is an equilibrium of Eq.(2) and 0 is indestructible. 0 Remark. 1 As follows from the proof, if O is a hyperbolic equilibrium then the map H : W —► V associating to F 6 W the equilibrium 0F of the perturbed equation is uniquely determined and, in virtue of (19), is continuous not only at FQ, but in a whole neighborhood W. That is, O is strongly indestructible. Remark. 2 It is not difficult to verify that the assertion of Proposition 4 remains valid if the condition of hyperbolicity of O is replaced by det(DF 0 (O)) ^ 0. We show now that 0 is locally unique. To do this we will use one of the versions of the Grobman-Hartman Linearization Theorem. Consider the equation x = Ax + G(x),
x e iT,
(21)
where A is a constant matrix, G satisfies a Lipschitz condition in Rn, and the linear equation y = Ay ,y£Rn. (22) Let Xt, Yt be the flows generated by Eq.(21) and Eq.(22), respectively. The following theorem is due to Grobman [4] and Hartman [5]. Theorem. 1 Let Eq.(22) be hyperbolic. There is I > 0 such that if Lip(G) < I then there exists a homeomorphism h : R? —> R*1 which takes solutions of Eq. (21) onto the solutions of Eq.(22), i.e., for all y 6 RJ1, t G R h(Yt(y)) = Xt(h(y)).
(23)
If (23) holds one says that Eq.(21) is conjugate to the linear Eq.(22), i.e., the flow of Eq.(21) in coordinates x = h{y) becomes linear.
Perturbation of Invariant Manifolds of Ordinary Differential Equations
227
Proposition. 5 A hyperbolic equilibrium point is locally unique. Proof. By Theorem 1 and in virtue of (17) there are d > 0 and, independent of d, 6 > 0 such that for every F , \\F - F0\\ < 6, Eq.(16) is conjugate to Eq.(22). Assuming 6 small enough to ensure OF € Vd and taking into account that Eq.(16) coincides with Eq.(2) on Vd we obtain OF = h(0) and Eq.(2) on Vd is conjugate to the linear Eq.(22) on the neighborhood U = h~l(Vd) of 0. It is well known that for a hyperbolic linear equation
fl wn = o.
By conjugacy, we have
f | Xt(Vd) = f | h(Yt(h-\Vd))) t£R
t€R
= M f | Y*(U)) = h(0) = 0F. tGR
The last relation means that O is locally unique. 0 From Propositions 4, 5 it follows Corollary. 1 A hyperbolic equilibrium point is persistent. 3.2 Non-degenerate case Definition. 7 An equilibrium point O of Eq. (1) is said to be nondegenerate if the matrix DFo(O) is nonsingular, i.e., det(DF0(O)) ^ 0; otherwise it is called degene rate. From Remark 2 it follows that O is strongly indestructible. However, it seems useful to give a more direct proof of this result based on the following version of the Implicit Function Theorem for Banach Spaces [12]. Theorem. 2 Let X, Y be Banach spaces and a : Y x X —► X be a smooth map. Assume that cr(b,a) = 0 at a point (6, a) G Y x X and the partial differential with respect to the second variable Dx V is continuous. Proposition. 6 If the equilibrium point O of Eq.(l) is nondegenerate then 0 is stron gly indestructible.
George Osipeako & Eugene Ershov
228
Proof. Equilibria of Eq.(2) are solutions of the equation F(x) = 0.
(24)
Let us show that Eq.(24) has a unique solution in the neighborhood of O provided ||F - Foil is small. To this end we use Theorem 2, where Y = T, X = BT and the map a : T xif 1 is given by
o(F,x) = F(x). Clearly, *(F0,O) = F0(O) = 0 and Dxa{F0,O) = DF0{0) : K1 -> R1 is a linear isomorphism since det(DF0(O)) ^ 0. By Theorem 2, there exist neighborhoods W of F0 and V of O such that for F G W, Eq.(24) has a unique solution H(F) G V, i.e., Eq.(2) has a unique equilibrium OF — H(F) G V, and the map H : W —► V is continuous. In addition, if F 6 W and F(O) = 0 then H(F) = Oby the uniqueness of H(F). Thus, he map H is continuous and satisfies the properties (i), (ii) of Definition 4. Hence, O is strongly indestructible. 0 Remark. 3 Clearly, if an equilibrium is nondegenerate, but is not hyperbolic then it is necessarily locally non-unique in the sense of Definition 3. Therefore, vector fields with strongly indestructible equilibria form a set wider than that formed by vector fields with persistent equilibria. 3.3 Index Theorem Let us recall the notion of the degree of a map; in doing so we restrict ourselves to the case of smooth maps and smooth submanifolds of TC1 (for more details see, e.g., Hirsch [6] ). Let $ : M —► N be a smooth map, where M, N are closed submanifolds of R? of the same dimension, N is connected. Intuitively, the degree of $ is the number of times $ wraps M around N. The precise definition requires the notion of a regular value of $. A point x G M is called regular if D$(x) : TXM —* T^X)N is an isomor phism, i.e., det(D$(x)) / 0. We say that y G N is a regular value if $ _ 1 (y) consists of regular points or, $~l{y) = 0. The degree of $ at a regular point x G M is defined by degx $ = sign det.D$(a:), and the degree of $ over a regular value y G N by deg($,y)=
2 ^ d e g x $ = V^ sign det D$(x). xe*-1(v) zeQ-1^)
Thus, deg($,y) is the number of points in $ _ 1 (y) at which D$ preserves orientation, minus the number of points at which D$ reverses orientation. It turns out that
Perturbation of Invariant Manifolds of Ordinary Differential Equations
229
deg($,y) is independent of y G N and therefore, is an invariant of $ called the degree of $ and denoted by deg$. The degree of $ defined for a smooth $ can be extended to continuous maps and it is known to be an invariant of a homotopy class of a map. We give now the notion of the equilibrium point index. Let F be a vector field defined on IV1 and 0 is an isolated equilibrium point of it. Denote by DT the closed disk of radius r centered at 0. Assuming r small enough so that there are no equiUbria of F in Dr \ 0 , define the map $ r : 5 n _ 1 -+ S n _ 1 , where Sn~l is the unit sphere, by •,(*) =
F{rx)
\F(rx)\ ■
The degree of $ r is said to be the index of the vector field F on the sphere Sr = dDr. It turns out that deg $ r is independent of r and therefore it is said to be the index of F at 0 , or simply the index of the equilibrium 0 if this does not lead to confusion. The index of F at 0 if denoted by indp(0) or ind(0) and can be easily found. If 0 is a nondegenerate equilibrium then indpiO) = sign(det DF(0)). For the vector field defined on R2 : X! = P ( X ! , X 2 ) ,
x2 = Q(xi,x 2 ), where P, Q are C1, the index of the isolated equilibrium 0 coincides with the Poincare index of 0 and is given by the contour integral PdQ - QdP ind(0) = —
= deg $ r =
/^.n
/
det
D$rdv,
where v(Sn~1) is a (n — l)-dimensional volume of 5 n _ 1 . The following result is due to Hopf [9]. Theorem. 3 / / the index of a vector field F on a sphere S = 3D is nonzero then F has at least one equilibrium point inside D. From this theorem it follows Proposition. 7 Let 0 be an isolated equilibrium point of Eq.(l). then 0 is indestructible.
If indp0{0) ^ 0
230
George Osipenko & Eugene Eishov
Proof. Let e > 0 be such that there are no equilibria of Eq.(l) in De except O. Since the index of F on S£ = dDe depends continuously on F and is an integer then there exists 6 > 0 such that for F € T, \\F- F0\\ < <5, the index of F on Se coincides with that of F0 and, hence, is nonzero. By Theorem 3, Eq.(2) has an equilibrium point OF £ De, i.e., O is indestructible. 0 Proposition. 8 If the equilibrium point O of Eq.(l) is degenerate then for every 6 > 0 there exists a vector field Fx £ T, Fx{0) = 0, \\FX - F0\\ < 6, such that the equilibrium point 0 of the equation x = Fi(x) is destructible. Proof. Since det(DF0(O)) = 0 the matrix DF0(O) has a zero eigenvalue. By a linear change of variables x —> (#i,£2), where x\ £ R is the coordinate along the eigenvector corresponding to the zero eigenvalue, x-i € iT 1 - 1 , Eq.(13) is reduced to the form
ii =
fi(xi,x2),
x2 = Bx2 + /2(zi, x 2 ), where / i , f2 vanish together with their partial derivatives at O. Given 6 > 0, choose a C1 vector field F%, \\F\ — F0\\ < <5, and a neighborhood V of O such that Eq.(15) on V in coordinates (xi,x2) takes the form ii = 0, x2 = Bx2.
(25)
For this it is sufficient to choose V such that | | / i | | , ||/ 2 || < 6 on V. Clearly, the equilibrium point O of Eq.(25) is destructible since the equation i i = p, x2 = Bx2 has no equihbrium points in V for p ^ 0. 0 Corollary. 2 If the degenerate equilibrium point O of Eq. (1) is indestructible then it is weakly indestructible. Proof. Assume on the contrary that O is strongly indestructible. Then there exist a neighborhood W of F0 in JF and a continuous map H : W —► M such that H(F) = O if F £ W and F(0) = 0. From the continuity of H at F and since H(F) = O it follows that 0 is an indestructible equilibrium point of the vector field F. Obviously, this contradicts the assertion of Proposition 8. 0 The results proved up to now yield:
Perturbation of Invariant Manifolds of Ordinary Differential Equations
231
Theorem. 4 (i) If the index of an equilibrium point is nonzero then it is indestruc tible. (ii) An equilibrium point is persistent if and only if it is hyperbolic. (Hi) An equilibrium point is strongly indestructible if and only if it is nondegenerate. If so, a perturbed equation has a unique equilibrium point close to the initial one. (iv) If an equilibrium point is degenerate and its index is nonzero then it is weakly indestructible.
4. Invariant manifolds of pseudo-hyperbolic linear maps Let T : i? 1 —> BJ1 be a linear isomorphism. Definition. 8 A linear isomorphism T is said to be p-hyperbolic, p > 0, if no eigen value of T has modulus p, i.e., its spectrum Spect(T) has no points on the circle of radius p in the complex plane C. A linear 1-hyperbolic isomorphism T is simply called hyperbolic. For a /^-hyperbolic linear isomorphism there exists a T-invariant splitting of FC1 = E\ (B E2 such that the spectrum of the restriction T\ = T\EX lies outside of the disk of radius />, while that of T2 = Tjj^, lies inside it. It is known that one can define norms in Ei and E2 such that associated norms of T{ , T2 admit the estimates | I T l | < i \T2\ < p. P
(26)
For details see Hirsch and Smale [8]. Conversely, if T admits an invariant splitting T = Ti © T2 with \T^\ = a, \T2\ = 6, ab < 1 then Srpect(T1) lies outside the disk {A e C : |A| < 1/a}, and Spect{T2) lies in the disk {A <E C : |A| < 6}. Thus, T is p-hyperbolic with b < p < 1/a. In what follows we assume that T is /^-hyperbolic and Q < m = dim^i < n. Suppose that norms in E\, E2 are such that (26) holds. Define the norm in FT as follows: set |x| = max{|j/|, \z\} for x 6 i T , x = y © z, y G E1: z € E2. 4-1 Center-Unstable and Stable Manifolds The following result is called Center-Unstable Manifold Theorem. Theorem. 5 Let T be a p-hyperbolic linear isomorphism, p < 1. Assume that f : K1 -+ fl71 is Cl, f = T + g, g = 0 outside the disk K = {x G i T : \x\ < d}, d > 0. There is 6 > 0 such that if
IMI = l l / - r | | < * ,
(27)
232
George Osipenko & Eugene Ershov
where \\g\\ = sup \g(x)\ + sup x€Rn
x€Rn
max\Dg(x)v\ \v\=1
then there exists a so-called center unstable manifold Wfu, dimVVy" = m which is a graph of a C 1 map ip : Ei —> E2, i.e., Wf* = {x e Rn : x = y ©
+ 2e)\xl
(28)
where a < l/p, e is small provided 6 is small, (v) if /(0) = 0 and Df(Q) = T, then Wf1 is tangent to Et at 0. Proof. Choose coordinates (y,z) in JFC1 so that Ei = {(y,0) : y e Rm}, E2 = {(0,z) : z € i T " m } , Rn = Ei x E2. In coordinates (y,z),
T takes the form T(y,z) = (AyrBz)i
(29)
where |^4 _1 | = a < l/p, \B\ = b < p. By the hypotheses b < 1, ab < 1. Let iTi : Ei x E2 —> Ei, ir2 '• Ei x E2 —* E2 be projection operators. The map / can be represented in the form f{y,z) = {Y{y,z),Z{y,z)), where Y = TTif, Z = Tc2f. Since T is an isomorphism, for small 6 the map / is invertible and its inverse / - 1 satisfies
lir 1 -!" 1 !**,
(30)
where e —> 0 as 6 —> 0. Setting P = nif~x, Q = n2f~l, we obtain f-\y,z)
= {P{y,z),
Q(y,z)).
From (27), (29), (30) it follows that \DyZ\<8, \DzP\<e,
\DZZ-B\<8, \DVP-A~l\<£,
\DzZ\
(31)
Perturbation of invariant Manifolds of Ordinary Differential Equations
233
Denote by $ the Banach space of bounded smooth maps
H>{Y(vMv)) = z(vMv)),
V^EU
(32)
and vice versa, W G W is /-invariant provided 9? is a solution of Eq.(32). Let us show that Eq.(32) is equivalent to the equation
(33) then
v{P{yMy))) = Q(yMy))Projecting the equality ff~x{y,ip{y))
= {y,
Y(P(y,v(y)),Q(yMy)))
= y-
Thus, P(y,ip(y)) is the inverse of y(y,(^(y)), i.e.,
{Y{yMy)Tl = P{yMy))Eq.(33) follows now from Eq.(32) and the last equality. In a similar way one can derive Eq.(32) from Eq.(33). To solve Eq.(33) define the operator A : <S> -» # for
A W W = Z(h(y),
(34)
where h(y) = P(y,
|A(p)| < W O I + |Z(*,¥>W) - Bv»(A)| < % | + ||/ - r | | < b\tp\ + S. It follows hereof that IA(^)| <
ft
(35)
if M < A) = <5/(l - 6). Let us estimate the norm of the differential DA((p). Differen tiating (34) with respect to y, we obtain DA(
(36)
234
George Osipenko & Eugene Ershov
where Dh = DyP + DZP Dip. In virtue of (31), from (36) we obtain \DA(
Mhi))
- Z(h2, ip2 (&,))| + |Z(Aa, va(fti)) - Z(A,,
< |I>,Z| |y>i - V2I + \DyZ\ \hx -h2\ where ^1 = P(y,ipi),
(38)
+ \DZZ\ \y2(hx) - ip2{h2)\,
h2 = P(y,
Ifo - fe| < l ^ l
\
\
from (38) and in virtue of (31) we obtain |A(v?x) - A(y>2)| < (6 + 6 + fe + (6 + <5)fte) |y>i -
(40)
Perturbation of Invariant Manifolds of Ordinary Differential Equations
235
derived from Eq.(39) by substitution of u for D
DzPu).
Notice that DA depends on ip since P(t/,
y€E!
M=1
Let Q0 = {w € fi : |u;| < /?i}. Given
+ \D2Z\\DyP\
- DA{u2)\ < k - u ; 2 | + \DZZ\\DZP\ {\ux\ + \u2\) \ut - u>2\
< {6e + (a + e)(6 + 6) + (6 + *)e2/9i) |^i -u2\=
X\wi
- w*|.
Since aft < 1, it follows hereof that x < 1 provided £ is small. Therefore, there is a unique fixed point u of DA in flo, i.e., Eq.(40) in fio has a unique solution w = w^ depending on
a; = DA^u). Denote by w* G Q.0 the solution of Eq.(40) with
- M , . ( w * ) = DAv{u)
- DA^u*)
+ DA,(w*) - M , . ( w * )
then we obtain | w - w - | < T-J— |DA„(u;*)-Z>A^(u;*)|,
(41)
1 -x where
flA^u/J-DA^u;*) = (DyZ(P{
(42)
236
George Osipenko & Eugene Ershov
DAv(u*) - DAV.(LJ*) is contained in the disk Kx = {y G £1 : |y| < \A\d}. this into account, from (31) and (42) we obtain
Taking
|J>A*(w') - DA„.(w*) | < | DvZ(P(
< oidr 1 ! I w * l ) ( i + A)(« + « + «A)+ (* + «) 7(1^1^-^1) x ((a + e) + eft) + (* + (b + *) A) a 2 (|p - p'|) (1 + A),
(43)
where a^, a 2 are the continuity modula of Df, Df~l, 7 is that of u* restricted on Kj. From (41) and (43) it follows that |w„ - w*| -> 0 if |y> - y>*| -> 0. Let us show that u* = Dip*. Define the sequence
(44)
i.e.,
uij = DA^Auj)-
(45)
From (44), (45) it follows |WJ - ZVj I < x |«j - #V?j-i I < X(K- - uij-i I + |wj_! - D(pj-! |).
(46)
Set ctj = \LOJ — Dy>j\, 7j = \u)j — u>j-i|. Since |<,0j — y?*| —> 0 then yj —► 0 as j —► 00. From (46) it follows <*j < x K - i + £j) < ». < X3&o + X J 7i + - + X 2 7j-i + X7jConsider the sequence Pj = X J 7 i + - + X7jWe have ?2j = X 2 J 7 I + - + X J 7j+i + X J_1 7i+2 + ... + X72j < 7 ^ — sup lk + —%— sup 7*. 1 — X i<*<j+i 1 - X *>i+2 From the last inequality it follows that p2j —» 0. Since P2J+1 = x(p2j + 72j+i) then pj —* 0 as j —> 00. Thus, <Xj = \UJ - D
Perturbation of Invariant Manifolds of Ordinary Differential Equations
237
Another method of proving the smoothness of
\f-\x)\ = m*x{\P(y,
Figure 3: The map T(y,z)
= (j/,A2;z) and its perturbation f(y,z)
= (y + <$, A2z).
Example. 5 Center-unstable manifold when a perturbation has no fixed points.
238
George Osipenko k Eugene Ershov
Let the linear map T : R2 —► R2 be of the form T(y,z) = (\1y,\2z)1
(47)
where 0 < A2 < Ai = 1. Clearly, T is p-hyperbolic with p = (1 + A 2 )/2 < 1, E\ = {(y,0)}, £ 2 = {(0,z)} (see Fig.3). Consider the perturbation of the form f(y, z) = (y + 6, A2 z). By Theorem 5 there exists the center-unstable manifold Wj" = Ex in spite of the fact that / has no fixed points for 6 / 0. It is not difficult to see that the center-unstable manifold Wf* is unique in if1. However, as the following example shows, this is not always the case if we consider T in a some neighborhood of the origin, i.e., Wf" is not necessarily locally unique.
Figure 4: Center-unstable locally non-unique invariant manifold. Example. 6 Center-unstable manifold, which is not locally unique. Consider the linear map T denned by (47), where 0 < A2 < Ai < 1. Clearly, T is /o-hyperbolic with p = (Ai + A2)/2 < 1, E\ = {(y,0)}, £ 2 = {(0, z)}. Invariant curves of T are shown in Fig.4. By Theorem 5 every / C 1 close to T has a unique invariant manifold C close to Ei in R2. As evident from Fig.4, T has many invariant manifolds of the form
w = {(yMv)),
y€£i},
where (p(Q) = 0, ify(0) = 0. If we restrict W to a d-neighborhood of 0 then W will be C1 close to the interval (—d, d) of the y-axis. Thus, the center-unstable manifold is not locally unique. This is also true for perturbations of T. The following example shows that the center-unstable manifold can sometimes be locally unique.
Perturbation of Invariant Manifolds of Ordinary Differential Equations
239
Figure 5: Center-unstable locally unique invariant manifold. Example. 7 Center-unstable, locally unique invariant manifold. Consider the map (47), where 0 < A2 < 1 < Aj, which is 1-hyperbolic, JE?i, E2 are the same as in Example 6. Invariant curves of T are shown in Fig.5. As evident from Fig.5, the center-unstable manifold Wf1 = E\ is unique not only in R2 but in a neighborhood of 0 as well, i.e., it is locally unique. In fact, if x £ E\ then the trajectory {Tkx} leaves any neighborhood of E\ as k -* — oo. This holds also for perturbations of T. Thus, if T is /^-hyperbolic with p < 1 then to the part of the spectrum of T outside of the p-disk corresponds the linear subspace E\ which is indestructible under C1 perturbations of T. The linear subspace E2 corresponding to the part of spectrum of T inside the p-disk turns out to offer certain properties of preservation as well. This result is known as Stable Manifold Theorem. Theorem. 6 Let T be a p-hyperbolic linear isomorphism and p < 1. There is a 6 > 0 such that if f : if1 —> BJ1 is a C1 map and = \\f-T\\<6,
(48)
/(0) = 0,
(49)
then there exists a so-called stable manifold Wf, dimWj = n — m, which is a graph of a smooth map tp : E2 —+ E\, i.e., Waf = {xeRn:x
= ip(z) ®z, ze E2)
and satisfies the following properties: (i) Wj is f-invariant, (ii) Wj depends continuously on f in the C1 topology, (in) W} = E2,
(50)
240
George Osipenko & Eugene Ershov (iv) WJ passes through 0 and for x G Wf
\f(x)\<(b + 26)\x\, (v) ifDf{0)
= T then W} is tangent to E2 at 0.
Proof. Choose coordinates (y,z) in R* as in the proof of Theorem 5, i.e., E\ = {(y,0)}, E2 = {(0,z)}. If so, the maps T, / , / _ 1 take the form T(y,z) = (Ay,Bz),
f(y,z)
= (Y(y,z),Z(y,z)),
r\y,z)
=
(P(y,z),Q(y,z)),
where Y, Z, P, Q satisfy (31), and /(0) = / _ 1 ( 0 ) = 0. It is easy to see that if the manifold W = {(*(*), z\
z G E2},
where u) : E2 —* E\, is /-invariant, i.e., fW = W, then ^ satisfies the functional equation if,(Z&(z),z)) = Y(i,(z),z). (51) Just as in the proof of Theorem 5, one can verify that Eq.(51) is equivalent to the equation 1>(z) = P(^(h(z)), h(z))t (52) where h(z) = Z(i/>(z), z) depends on ip. Denote by \J/ the Banach space of maps 0 : E2 —* Ei satisfying the Lipschitz condition: |#*i) - #*2)| <
l\zi-z2\
and vanishing at 0 : ip(0) = 0. The norm in VP is defined by
**0
\Z\
Let #o be a closed subset of # of the form # 0 = {V' £ * : Lip(rf>) < 1}. To solve Eq.(52) define the operator A : $ —> ^ by A(tf) (z) = P(ip{h{z)),
h(z)\
where h is as before. Let us show that A takes *£ into itself and is a contraction provided 6 is small. In virtue of (31), for V> € ^0, we have Lip(h) < \DyZ\ Lip(^) + \dzZ\ < b + 26, and, hence, Lip (A (VO) < \DyP\ Lip{i>) Lip{h) + \DZP\ Lip(h) < (a + 2e) (6 + 26).
Perturbation of invariant ManifoJds of Ordinary Differential Equations
241
Since ab < 1 then Lip(A(tp)) < 1 provided 6 is small. Thus, A takes * 0 into itself. Prove now that A is a contraction. For if>i,if)2 G tfo* we have |A(tfi)M - A(fc)(*)| = \P(Mhi(z)), < \P(tl>i(hi(*))M*))
h(z))
~
- P(rJ>2(Wz)), h2(z))\
P{1n(W*))M*))\
+\P(MW*)), &iW) - P(MW*))i hi(z))\ +\P(Mh2(z)), < \DyP\
hiz))
- P(fh(h2(z)),
LXJW)
MZ)
+ \DyP\\il>x{h2{z)) - MHz))\
h2(z))\
- h2{z)\
+ \D,P\Mz)
- h2{z)\.
(53)
Since \h2(z)\ = \h2(z) - h2(0)\ < Lip(h2)\z\ < (6 + 26) |*|, | M * ) - fe(*)| = \Z{Uz),
z) - Z{Uz),
z)\
< \DyZ\ \^{z) - fe(*)| < * | * ( s ) -
fc(z)|,
(54)
from (53) it follows that
iiAW-AWir=.upi^)w-Aww < [ (a + 2e)£ + (a + e) (6 + 26) ] ||fc - 0 2 ||*.
Since aft < 1, from the last inequality it follows that A contracts in \&o provided 6 is small. Denote by I/J* £ #o the fixed point of A. Clearly, tp* satisfies the Lipschitz condition and Lip(ipm) < 1. Just in the same way as in Theorem 5, one can prove that tp* is smooth and depends continuously on / in the C1 topology. Set
w; = {x e Rn: x = (*l>*(z), z), z e E2}. Properties (i)-(iii) of Wf hold by construction of if>*. The proof of (iv), (v) is similar to that of (iv), (v) in Theorem 5. 0 Remark. 5 It should be remembered that in Theorem 5, /(0) = 0, i.e., the map f preserves the fixed point 0. We show by example that this requirement cannot be dropped. More precisely, if /(0) ^ 0 then there may be no /-invariant manifolds of the form (50) at all. Example. 8 Destruction of a stable manifold under the collapse of a fixed point.
George Osipenko k Eugene Ershov
242 Let the map T : R2 —>■ R2 be of the form
T{y,z) =
{y,\2z),
where 0 < A2 < 1. Clearly, T is />-hyperbolic with p = (1 + A2)/2 < 1, £i = {(y, 0)}, E2 = {(0, z)}. Consider the perturbation / of the form /(y, z) = {y+6, A2z), 6^0. Invariant curves of T are shown in Fig.3. Clearly, / has no stable invariant manifold WJ of the form (50). Notice that / has nevertheless the center-unstable manifold Wf = E1. Remark. 6 From the proof of Theorem 6 it follows that the stable manifold WJ is unique in RJ1. Let us show that WJ is even locally unique in a neighborhood of the origin provi ded 6 is small. On the contrary, assume that in a d-neighborhood of 0 there are two locally invariant manifolds, say
W? = {(*(*),*) : . * € ^ ,
\z\
where ^i(O) = ^ 2 (0) = 0, Lip(rJ>i), Lip(il>2) < 1- Set a =
sup
j—|
z#0, \z\
\z\
.
If 6 + 26 < 1 from (54) it follows \h2(z)\ = \Z(ip2(z\z)\
< (b + 2S)\z\ < d for \z\ < d.
(55)
Similarly to (53) and in view of (55), we obtain a<[{a + 2e)8 + (a + e) (b + 26)]a = Xa . Since ab < 1 then we can assume A < 1. If so, the last inequality holds only for a = 0. Thus, i/>i = ift2 for \z\ < d. 4-2 Center-stable and unstable manifolds Consider now a p-hyperbolic linear isomorphism S with p > 1. Let Fj, F2, dimF\ = m, dimF2 = n — m, be invariant subspaces of S corresponding to the spectrum of S lying outside and inside of />-disk of the complex plane C, respectively. Let Si = S| F l and S2 = S\F2 be restrictions of S on Fx, F2. Consider the inverse map T = S"1. It is easy to verify that T is p\-hyperbolic with px = l/p < 1. As
Perturbation of Invariant Manifolds of Ordinary Differential Equations
243
before, let E\, E2 be invariant subspaces of T lying outside and inside of the pi-disk of C and Ti, T2 be restrictions of T on £ 1 , E2. Clearly, E\ = F2, E2 = F\ and T\ = S^1, T2 = S£" . As before, choose norms in E\, E2 such that \T-1\ = \S2\ = a<-
= p, P\
\T2\ = \S^\ = b
=
-
P From Theorems 5, 6 applied to T we obtain at once the theorems on center-stable and unstable manifolds of S. Theorem. 7 Let S be a p-hyperbolic linear isomorphism and p > 1. There is 6 > 0 such that if f : rV1 —► BJ1, \\f — S\\ < 6 and f coincides with S outside the dneighborhood o/O then there exists a so-called center-stable manifold Wj9, dim Wj" = n — m, which is the graph of a smooth map F\, i.e., Wf = {x £ Rn : x =
|/(*)|<(« + 7)M. where 7 is small provided 8 is small, (v) if /(0) = 0 and Df(0) = S then W]3 is tangent to F2 at 0. Theorem. 8 Let S be a p-hyperbolic linear isomorphism and p > 1. There is 6 > 0 such that if f : FC1 —> BJ1 is a C 1 map and \\f — S\\ < 6, /(0) = 0 then there exists a so-called unstable manifold Wf, dim WJ = m, which is the graph of a smooth map ip : F1 -* F2) i.e., W] = {x € Rn : x = y 0 ip{y), y e Ft] and satisfies the following properties: (i) WY is f-invariant, (ii) Wf depends continuously on f in the C1 topology, (iii)W2 = Fu (iv) W? passes through 0 and for x 6 Wj
irV)i<(<>+7)M, where 7 is small provided S is small, (v) if Df{0) = 5 then Wf is tangent to i^ at 0.
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4-3. Center manifold Suppose now that there are p\, pi > 0, p\ < 1 < p2 such that the spectrum of a linear isomorphism T is decomposed into three parts contained inA r i = { A G C : | A | < pi}, Ko = {A G C : pi < |A| < p 2 }, K2 = {A G C : |A| > p 2 }, respectively. According to this decomposition of the spectrum, we have a T-invariant decomposition R n = E3 © E c © £ u such that the spectra of T3 = T V , T c = T V , and T u = T V are contained in A^, K0, and #2, respectively. As before, one can determine norms in E3, Ec, and Eu such that \T*x\ < pi|x|, |x|pi < \Tcx\ < p 2 |x|, \Tux\ > p2\x\. For x G Rn, x = u®v®w,
ue E$, v G Ec, w € Eu set |x| = max{|u|, \v\, \w\).
Let / : ff1 —> i£n be a map C1 close to T such that /(0) = 0 and f(x) = Tx outside the disk {x G BJ1 : |x| < d}. Since T is pi- and p2-hyperbolic, by the theorems stated earlier in this section there exist the following /-invariant manifolds: 1. stable manifold W) = {x G Rn : x = u © i>30(u) © 0 2 (w), « G £ s } , 2. center-unstable manifold H 7 = { x G i i n : i =
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Theorem. 9 Let T be a linear isomorphism whose spectrum is contained in Kit Ko and K2 with pi < 1 < p2. There is a 6 > 0 such that if f : BJ1 —► BJ1 is a Cl map, ||/ — ^H < &, and f coincides with T outside the d-neighborhood o/0 then there exists a so-called center manifold Wj, dimWj = dimii^, which is the graph of a smooth mapr : Ec -> E' © Eu, i.e., Wj = {xeRn:x
= TJ(V) © v © r 2 (v), v 6 Ec),
(56)
and satisfies the following properties: (i) Wj is f-invariant, (ii) Wj depends continuously on f in the C1 topology, (Hi) Wf = Ec, (iv) if /(0) = 0 then Wj passes through 0 and for x G Wj PiM < 1/(^)1 < P2\x\ (v) if /(0) = 0 and Df{0) = T then W) is tangent to Ec at 0. Proof. Since T is p\- and p2-hyperbolic the conditions of the theorem guarantee the existence of the center-unstable Wj" and center-stable Wjs manifolds. Set Wc} = Wf n Wf.
(57)
Clearly, Wj is /-invariant since Wf and W" offer this property. As Wf1 and Wj" are C1 close to Ec ® Eu and Es ®EC, respectively, and (Ec © Eu) D (Es © Ec) = E% then the intersection in (57) is transversal provided 6 is small. Thus , Wj is a smooth manifold C 1 close to Ec ( see Hirsch [6] ). From this it follows that Wj is represented in the form (56). Properties (ii)-(v) follow from the appropriate properties of Wj", Wj9 by the definition of Wj. 0
5. Perturbation of invariant manifolds through an equili brium In this section we consider the perturbation problem of invariant manifolds passing through an equilibrium point of a differential equation. To do this we apply the results obtained in Section 4 on a perturbation of invariant subspaces of a linear pseudohyperbolic map. Invariant manifolds passing through an equilibrium point occur very often in dynamical systems. Let O be a hyperbolic equilibrium point of Eq.(l). Assume that m(0 < m < n) eigenvalues of DF0(O) have negative real parts and n-m
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have positive real parts. It is well known that in a neighborhood V of 0 in R" there are smooth locally invariant manifolds W, Wu, dimW3 = m, dimW* = n - m through 0 such that WB = {p € V : X(*,p) G V for t > 0 ; and X(t,p) - 0 as * -> +oo}, W" = {p <E V : X(<,p) eV
fort<0
and X{t,p) -► 0 as t -> - o o } .
The existence of stable W3 and unstable Wu manifolds was first proved by Liapunov [13] for analytic differential equations and by Perron [20] for smooth ones. Since the perturbed equation near 0 has the unique hyperbolic equilibrium 0F then Eq.(2) has the locally invariant manifolds W3F, W% through 0F with just the same properties as Ws and Wu. As will be evident from the following, the manifolds WF, W% are uniquely determined and depend continuously on F in the C1 topology, i.e., W3 and Wu are strongly indestructible. In this section we suppose that Eq.(l) has a smooth locally invariant manifold Mo, 0 < m = dim M 0 < n passing through an equilibrium O. Taking into account the representation (13) of Eq.(l) and the fact that the tangent space ToM0 = E0 of M0 at O is invariant for its linear part, we obtain that by a linear change of variables Eq.(13) is reduced to the form y = Ay + Cz + g0(x), i = Bz + h0{x),
(58)
where x t= (y,z), y e E0 = i T , z e 6 = i T ? m " g0, h0 ara C1 and vanish together with their derivatives at O. Thus, E0 is an invariant subspace of the linear system y = Ay + Cz, z = Bz.
(59)
The manifold M 0 in a ^-neighborhood Vd = V0 X V, Vo C E0, V C E of O is the graph of a C1 function
Perturbation of Invariant Manifolds of Ordinary Differential Equations
Definition. 9 A locally invariant exists a X > 0 such that
manifold M 0 of Eq.(l)
is said to be stable if there
ReXj < — A (j — l , . . . , m ) and ReXj > -X (j = m + 1, ...,n), w/iere .Re Aj is the real part of Xj. Conversely,
247
(61)
if
ReXj > X (j = 1, ....,ra) and ReXj < X (j = m -f 1, ....,n), £/ien Mo Z5 called an unstable
manifold.
Clearly, the definition is independent of the choice of local coordinates. T h e o r e m . 10 Let M 0 be a stable (unstable) locally invariant manifold of Eq.(l). There exist d, S > 0 such that for every F £ T, \\F — F 0 | | < 6, preserving the equilibrium O, i.e., F(0) = 0, Eq.(2) in Vd has a smooth locally invariant manifold Mp, dim M F = m, MF = {x <E Rn : x = (y,y>(y)), y G V0}, where ip : Vo —> V, with the following
properties:
(i) OeMF (ii) Mp depends continuously on F in the C1 topology, (Hi) if Mo is locally invariant for F then MF = Mo, (iv) for every p £ MF, X(t,p) —► O as t —> +oo if Mo is a stable manifold and X(t,p) —> O as t —* —oo if Mo is an unstable manifold, (v) ifDF(O) = DFo(O) then MF is tangent to E0 at O. Roughly speaking, Theorem 10 asserts that M 0 is a strongly indestructible locally invariant manifold of Eq.(l) under perturbations preserving the equilibrium 0. Proof. Since an unstable invariant manifold becomes stable by reversing t —> — 2, it is sufficient to prove the theorem for the case of a stable manifold. By the hypotheses, the matrices A0, B have no common eigenvalues. Hence, there will be no loss of generality to assume C = 0 in Eqs.(59), (60). Therefore, solutions of Eq.(59) are of the form Y(t,y)
= eAty,
Z(t,y)
= eBtz.
In view of (61), there exist P, Q > 0 such that \eAt\ < Pe~x\
\e~Bt\ < QeXt (t > 0).
(62)
As shown in Hirsch and Smale [8], there will be no loss of generality to assume P = Q = 1. To achieve this one needs only to introduce in E0 and E special norms.
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Consider the linear map T : Rn —> Rn defined by
T(yjZ) =
(Y(hy),Z(l,z)).
In virtue of (62), T is ^-hyperbolic with p = exp(-A) < 1. Let X(t,x) be a solution of Eq.(60). For x G i T set f{x) = X{\,x). Clearly, / : R" -> ET is C 1 and C 1 close to T provided d and 6 are small. Notice that / depends continuously on F in the C 1 topology by the fundamental theorems of the theory of differential equations. By Theorem 6 there exists a stable /-invariant manifold Waf = {xeRn:x
= (y,
passing through O. Clearly, Wj is invariant for the map X\ : R71 —► RJ1, Xi(x) = X(l,x). Let us show that Wj is invariant with respect to the flow Xt, t £ R. To do this it is sufficient to verify that Wj is invariant with respect to XT for all small positive T. Putting W}(T) = Xr(W})t we obtain
f(w;(r)) = x1(xT(w;)) = x1+T(wf) = xT(x1(wf)) = xT(w°f) = wsf(r). That is, the manifold WJ(T) is /-invariant. By the uniqueness of Wj we have W}(T)
= w;,
i.e., Wsj is invariant with respect to XT. Since r is any small positive number and Eq.(60) is independent of t, Wj is invariant for the flow Xt, t 6 R. To complete the proof it remains to set Mp = Wf C\ Vd. Properties (i)-(v) of Mp follow at once from the properties of Wy. 0 Remark. 7 From the preceding it follows that Mp is locally unique and exists, in general, only when F(0) = 0.
5.2 Center, center-unstable,
and center-stable invariant
Definition. 10 A locally invariant manifold MQ of Eq.(l) stable if there exists A > 0 such that ReXj > — A (j = 1, ...,m) and Re\3
manifolds
is said to be center-un
< —A (j = m + 1, ...,n).
Conversely, if ReXj < A (j = 1, ...,m) and ReXj > A (j = m -f 1, ...,n)
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249
then Mo is called a center-stable manifold. Finally, if there exist A*, A*, A* < 0 < A*, such that A* < ReXj < A* (j = 1, ...,m) and either ReXj < A* or ReXj > A* (j = m + 1 , ...,n) then Mo is called a center manifold. Using the similar arguments as in the proof of Theorem 10, from Theorems 5, 7, 9 it follows Theorem. 11 Let M0 be a center-unstable, center-stable or center locally invariant manifold of Eq.(l). There exist d, S > 0 such that for every F € T, \\F — F0\\ < S, Eq.(2) in Vd has a smooth locally invariant manifold Mp, dim Mp = m, MF = {xeRn:x
= (y,V»(y)), V € V0},
where tj) : Vo —► V, with the following properties: (i) Mp depends continuously on F in the C1 topology, (ii) if Mo is locally invariant for F then Mp — M 0 , (Hi) if F(O) = 0 then Mp passes through 0, (iv) if F(0) = 0 and DF(0) = DF0(O) then MF is tangent to E0 at 0. In particular, Theorem 11 asserts that the center-unstable, center-stable and cen ter manifold M0 is strongly indestructible. Remark. 8 From the preceding it follows that Mp is not necessarily locally unique and exists even when 0 is destroyed under a perturbation. 5.3 Separation condition As above, let Mo be a locally invariant manifold of Eq.(l) through an equilibrium O. Here we will find conditions under which Eq.(2) has an invariant manifold Mp C1 close to Mo. For real /^i,//2 (t*i < ^2) denote by J(/ii, //2) = {A 6 C : //1 < Re\ < fi2} the strip of the complex plane C (the cases fii = —00, \Li = +00 are not excluded). Definition. 11 We say that the condition of separation holds for MQ if there exist \L\i\i-2 (^1 < ^2) such that the spectra of matrices A, B are inside, outside IQ = I(p\,H2), respectively.
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If the condition of separation holds for Mo then the spectrum of B consists of two parts lying in h = / ( - o o , / / i ) and I2 = / ( / i 2 , + o o ) . Needless to say that one of these parts may be empty. Denote by Ei, E0, E2 the invariant subspaces of DF0(O) corresponding to its spectrum lying in / i , I0, I2, respectively. Definition. 12 7/ the condition of separation holds for Mo with fij < 0 < fi2 then Mo is called a normally hyperbolic manifold. Remark. 9 From Theorem 11 it follows that a normally hyperbolic locally invariant manifold is strongly indestructible. Definition. 13 If the condition of separation holds for MQ with // 2 < 0 {fi\ > 0) then O is called a strong sink (strong source) with respect to Mo. A strong sink (source) O is called nondegenerate if 0 is a nondegenerate equilibrium point of Eq.(l), i.e., detDF0{O)^0. The following result was obtained by Osipenko [17]. P r o p o s i t i o n . 9 If O is a nondegenerate strong sink (strong source) with respect to a locally invariant manifold Mo then Mo is strongly indestructible. Proof. For definiteness, let O be a nondegenerate strong sink. Since O is a nondegenerate equilibrium then Eq.(2) near O has a unique equilibrium point OF depending continuously on F. The transformation x = x + Oy reduces Eq.(2) to the form x = F0{x + 0F) + F(x + 0F) - FQ(x + 0F). (63) Clearly, Eq.(63) has an equilibrium at the origin x = 0 and is C 1 close to the linear equation x = DFo{0)x . (64) Since fi\ < H2 < 0, Eq.(64) has the stable and center-unstable manifolds Es, Em of the form Es = Ex © EQ, E™ = E0® E2. By Theorems 10, 11 applied to Eq.(64), Eq.(63) has the locally invariant manifolds WF, Wf C 1 close to E\ E™, respecti vely. Because the intersection Es f*l E™ = E0 is transversal, then Mp = WF C\ WFU is a smooth manifold C 1 close to E0 and, hence, to M 0 . Obviously, MF depends conti nuously on F in the C 1 topology and MF = M0 if M 0 is invariant under Eq.(2) since these properties hold for WF, Wp1. From Proposition 9 and Remark 9 it follows T h e o r e m . 12 If a locally invariant manifold MQ is either normally hyperbolic or O is a nondegenerate strong sink (source) with respect to M0 then M 0 is strongly indestructible.
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It turns out that the condition of Theorem 12 is not only sufficient for M 0 to be strongly indestructible but also necessary. The following result has been recently obtained by Osipenko and Ershov [19]. Theorem. 13 If the locally invariant manifold M 0 of Eq.(l) is strongly indestructible then it is either normally hyperbolic or O is a nondegenerate strong sink (source) with respect to Mo. We finish this section presenting conditions under which a locally invariant ma nifold is weakly indestructible Theorem. 14 Let O be a degenerate equilibrium point of Eq.(l) whose index is non zero. If O is a strong sink (source) with respect to Mo then it is weakly indestructible. Sketch of the proof. As seen from the proof of Proposition 9, the existence of stable and unstable manifolds of a perturbed equation is entirely determined by the existence of its equilibrium point. By Theorem 4, O is weakly indestructible. From this and the previous conclusion it follows that Mo is weakly indestructible. The details of the proof are left to the reader.
6. Perturbation of periodic orbits and invariant manifolds through them In Sections 3,5 we have studied the perturbation problem of equilibria and in variant manifolds through them. Now we are going to make an analogous study for periodic orbits. 6.1 Perturbation of periodic orbits Let 7 be a periodic orbit of Eq.(l) with a least period u > 0, we refer to 7 as w-periodic orbit. Given po £ 7, consider a hyperplane II, dimII = n — 1, transversal to 7 at po. Clearly, the orbit Xt(po) intersects II as t increases at t — u>. By continuous dependence of solutions of a differential equation, the orbit Xt(p) intersects II at a time near u provided p € IT is close to po- Thus, if V is a sufficiently small neighborhood of po in II we have a well-defined map Po : V —» II, associating to p G V the first intersection point of Xt(p), t > 0, with II. The map Po is called "Poincare map" or return map and is known to be a local diffeomorphism provided the vector field F 0 is smooth. The behavior of orbits near 7 is entirely determined by P 0 , e.g., if Po{p) = p then Xt{p) is a periodic orbit; if PQ{P) —► po as n —► ±00 then Xt(p) —> 7 as t —» ±00, and so on. By the fundamental theorems of differential equation theory, the perturbed Eq.(2) has the Poincare map P : V -* U C1 close to P 0 provided F is C 1 close to F0. Conversely, if P is a local diffeomorphism C1 close to P 0 then P is a Poincare map
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of some Eq.(2) C 1 close to Eq.(l). Thus, the perturbation problem for 7 is identical to that for the fixed point p0 of P0. The following notions are similar to those for an equilibrium. Definition. 14 A periodic orbit 7 of Eq.(l) is called hyperbolic if no eigenvalue of DPo{po), Po G 7 has modulus 1. A periodic orbit 7 is said to be nondegenerate if the linear operator DPo(po) — id : n l R ~ —*■ /T 1 - 1 is invertible, i.e., its spectrum does not contain 1, otherwise it is called degenerate. The index 0 / 7 is defined as the index of the vector field $(/>) = Po{p) ~P at a singular point p0 G 7. These notions are independent of the choice of a point po G 7 and a hyperplane II through it. In fact, two Poincare maps Pi, P2 for points />i, p2 G 7 on hyperplanes IIi, II 2 are known to be C1 conjugated, i.e., there exists a local diffeomorphism h : IIi —* II 2 , h(pi) = p2 such that Pi = h~l o P2 o h. From this it follows that the differentials DP\{p\) and DP2(p2) are linearly isomorphic: Z)Pi(pi) = (Dh(Pl))-1DP2(P2)
Dh(Pl).
The following statements are quite similar to those of Theorem 4 and can be obtained likewise. T h e o r e m . 15 Let 7 be a periodic orbit of
Eq.(l).
(i) If the index of 7 is nonzero then 7 is indestructible. (ii) A periodic orbit 7 is persistent if and only if it is hyperbolic. (Hi) A periodic orbit 7 is strongly indestructible if and only if it is nondegenerate. If so, a perturbed equation has a unique periodic orbit close to 7. (iv) If i is degenerate and its index is nonzero then it is weakly indestructible .
6.2 Invariant manifolds through a periodic orbit Suppose that Eq.(l) has a smooth locally invariant manifold d i m M 0 < n, through an ^-periodic orbit 7 = {x G B71 : x = il>{i), 0 ip{t) is a solution of Eq.(l), ^(0) = 0(u>), tp(t) ^ ^(0) for 0 < t < to. to see that the restriction E0 = T^M0 of the tangent bundle TM0 bundle which is invariant under solutions of the linearized equation
C = DFoMt))C
M0, 1 < m = < t < u}, where It is not difficult on 7 is a vector
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253
Notice that the bundle E0 is diffeomorphic either to R71-1 x {unit circle} or to ft"'* x {Mobius band}. In the first case E0 is a trivial vector bundle, and in the second it is a nontrivial one. Let II be a hyperplane transversal to 7 at p0 = t^(0). Clearly, MQ = M0 n II is an m — 1-dimensional smooth manifold which is locally invariant for the Poincare map P0. Consequently, the tangent space E£ = T^M^ = T^MQ C\ U to M£ at p0 is invariant under the differential iXP0(po}The perturbation problem for the locally invariant manifold M0 of Eq.(l) is iden tical to that for the locally invariant manifold M£ of the Poincare map PQ. Let {y, 2), y £ K0*-1, z e if*"m be coordinates in II such that po = (0,0), y G EQ and z is in a subspace transversal to EQ in II. It is easily seen, that P0 and DPo(p0) take the form: p .( y\ v ( Ay + Cz + go{yyz)\ r °'\zJ \Bz + ko(y,z) )>
«ftW:(:)-(S +Clr ).
(65)
where <jb} ^0 are Cl and vanish together with their derivatives at y = 0, 2 = 0. The following notions are similar to the corresponding ones introduced in Section 5 fox the case of an equihbrium, For real ^ , 1*2, 0 < V\ < 1/2, denote by J{vx, v2) = {A € C : vx < jAj < f 2 } the annulus of the complex plane C. Definition. 15 We say that the condition of separation holds at a periodic orbit 7 C Afo (with respect to Mo) if there exist i>i, Vi such that the spectra of matrices A, B in (65) are inside and outside J{v\, v^), respectively. Evidently, the definition is independent of po, II and local coordinates. Definition. 16 / / the condition of separation holds for a periodic orbit 7 C Mo with V\.< 1 < v 1). In addition, if a periodic orbit 7 is nondegenerate then a strong sink (source) is called nondegenerate, otherwise it is called degenerate, Proofs of the following theorems can be achieved by methods used in Section 5 for the case of an equilibrium and are left to the reader.
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T h e o r e m . 16 Let M 0 be a locally invariant manifold through a periodic orbit 7 of Eq.(l) and one of the following conditions hold: (i) Mo is normally hyperbolic at 7; (ii) 7 is a nondegenerate strong sink (source) with respect to M 0 . Then Mo is a strongly indestructible locally invariant manifold. T h e o r e m . 17 Let a periodic orbit 7 of Eq.(l) be a strong sink (source) with respect to a locally invariant manifold Mo through J-Ifj is a degenerate periodic orbit whose index is nonzero then Mo is a weakly indestructible locally invariant manifold.
7. S a c k e r - N e i m a r k and M a n e T h e o r e m s From this point on we will assume Eq.(l) to have a smooth compact invariant manifold Mo, 0 < dim Mo < n, without boundary. As explained in Section 2, a perturbed equation may have many invariant m-dimensional manifolds C1 close to Mo. In the present section we study the simpler case, where a perturbed equation has a unique invariant manifold Mp C1 near to Mo, i.e., the case, where Mo is locally unique. Definition. 18 An invariant manifold M 0 of Eq.(l) there exists a continuous decomposition
is called normally hyperbolic if
TRn\Mo=TMQ@Es®Eu
(66)
of the restriction on Mo of the tangent bundle TRP4 into a direct sum of subbundles TM0, E\ Eu, invariant with respect to the tangent map DXt : TEJ1 —> TK" and constants K, X > 0 such that for all p (E Mo, \DXt°(p)\
of DXt
for t > 0, ^'> for t < 0.
on T M 0 , Es, Eu, respectively; | . |
Invariance of the bundles E3 and Eu means that for every p 6 M 0 , the image of the fiber E'p{E«) under DXt(p) is the fiber E'q(E*) at q = Xt{p) : DXt(p)E°p»
=
E^{py
The bundles Es and Eu are called stable and unstable, respectively.
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Roughly speaking, Mo is normally hyperbolic if the linearized flow contracts (expands) along the normal (to M 0 ) direction and this contraction (expansion) is stronger than a conceivable contraction (expansion) along Mo. That is, the normal (to Mo) behavior of DXt is hyperbolic and dominates its tangent behavior. In a similar way, one can define the normal hyperbolicity of an arbitrary invariant set U C M 0 , namely, U is said to be normally hyperbolic if (66) and (67) hold for all p G U. It is not difficult to see that for the case when U is an equilibrium O (a periodic orbit 7) the normal hyperbolicity of O (7) means that M 0 is normally hyperbolic at 0 (7) (see Definitions 12, 16). Assuming vector bundles Ea, Eu to be trivial and C°° smooth (see Section 2), in a tubular neighborhood N of M 0 one can introduce coordinates (t/, z s , 2U), where y e M 0 , zs e Rk, zu € Rl, m + k + / = n so that N = M 0 x Rk x Rl, M 0 = M0 x 0 x 0, Ea = M 0 x Rk x 0, Eu = M 0 x 0 x Rl, i.e., M 0 is the zero section of N; Ea, Eu are given by zu = 0, zs = 0, respectively. By the invariance of Mo under Xt and of TM0, E\ Eu under DXt, we obtain that the flow Xt = (Yt, Z/, Ztu) generated by Eq.(l) takes the form Yt{y, z9, Zu) = *o(*, y) + Vi(y, «a, 2«), Ztdti z»i z«) = B&I y)z* + zt(yi
z
*i z « ) '
Z?(y, zs, zu) = Bu(t, y)zu + Ztu(y, z s , zu),
(68)
where 1^, Z/, Z" vanish together with their derivatives at zs = 0, zu = 0. Clearly, DX° = DY0(t, y), DXat = B,(t, y), DX? = Bu{t, y) and, in view of (67), we have \B,(t,y)\
for t > 0, for t < 0,
In a small neighborhood of M 0 in R" the flow (68) can be considered as a C1 pertur bation of the linearized flow (Y0(t,y),
Bs(t,y)zs,
Bu(t,y)zu),
for which the stable E9 and unstable Eu bundles are invariant. As before, without loss of generality we can consider Yu Z/, Z™ C 1 close to zero for all (y, z„, zu) e N. To achieve this it is sufficient to apply bump functions. It is well known [7] that over M0 there exist stable and unstable manifolds Ws, Wu, Wa D Wu = M 0 , invariant under the flow (68), C1 close to Ea, Eu, respectively, and such that (i) for all p G Wa, dist(Xt(p), M 0 ) -* 0 as t -► +00, (ii) for all p e Wu, dist{Xt{p), M 0 ) -> 0 as t -» - 0 0 , (iii) if p i Wa n W^u, then dist{Xt(p), M 0 ) -> 00 as t -> ±co.
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The following result obtained independently by Sacker [22] and Neimark [15] shows that such behavior of orbits near M 0 persists under a perturbation. For d > 0 let Vd = {(y,zs,zu)
:y£M0,z9e
Es, zu 6 Eu, max {|*.|, \zu\} < d}
be a ^-neighborhood of M 0 . Set W£ = Wa n Vd and W0U = Wu H Vd. We refer to W£ and WQ as local stable and unstable manifolds over MoTheorem. 18 Assume that Eq.(l) has a smooth compact invariant manifold M0 wit hout boundary . If MQ is normally hyperbolic, then there exist d, 6 > 0 such that for every F G f , | | F — Fo|| < 8} Eq.(2) has in Vd unique stable and unstable manifolds Wp, Wp with the properties: (i) Wp and Wp are locally invariant, (ii) Wp and Wp are C1 close to W09 and WQ, respectively, (Hi) Mp = Wp D Wp is an invariant manifold C1 close to Mo, (iv) WF, Wp, and Mp depend continuously on F in the C1 topology and Mp = M0 if Mo remains invariant for F, (v) if p € Vd and p ^ Mp then Xt(p) leaves Vd as t increases or decreases, i.e. Mp is a maximal invariant set contained in Vd. The proof of this theorem is beyond the scope of the present article. In addition to the original papers of Sacker and Neimark, it can be found, e.g., in Hirsch, Pugh, and Shub [7], Fenichel [3] and Wiggins [26], where various generalizations and applications of Theorem 18 are presented. From Theorem 18 it follows at once that a normally hyperbolic invariant manifold Mo is indestructible (even strongly indestructible) and locally unique. The following definition is due to Mane [14]. Definition. 19 An indestructible locally unique invariant manifold is called persi stent. Thus we obtain Theorem. 19 A compact normally hyperbolic invariant manifold is persistent. It turns out that Theorem 19 admits the converse (Mane [14]). Theorem. 20 Persistent invariant manifolds are normally hyperbolic. Thus, Theorems 19, 20 give necessary and sufficient conditions for an invariant manifold to be persistent. We complete this section by treating simple equations that have nontrivial persi stent invariant manifolds.
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Figure 6: Unperturbed and perturbed normally hyperbolic invariant mani folds. Example. 9 Preservation of the orbit structure on a normally hyperbolic invariant manifold. In R2 consider an equation having a compact invariant manifold Mo, which is the unit circle S = {(x,y) : x2 + y2 = 1}. Assume that the equation has two equilibrium points disposed on S : the saddle point A and the sink B (see Fig.6). Let the linearized equation at A be of the form x = x, y = — (y — 1) and at B of the form x = —x, y = —2(y + 1). It can be easily verified that S is normally hyperbolic: for all p G 5, Ep = {0}, E* is a straight line transversal to S at p, e.g., EaA, EaB are the vertical straight lines. By Theorem 19, 5 is a persistent manifold. Notice that in this case the map H is uniquely determined. In fact, near S every vector field F C1 close to the original one has a unique invariant manifold SF diffeomorphic to S and consisting of two equilibrium points (the saddle Ap near to A and the sink Bp near to B) and two separatrices from Ap to Bp. Hence, the map H must be of the form H(F) = SF. In the above example, the behavior of orbits of a perturbed equation on SF is similar to that of orbits of the initial equation on S. We show by example that this is not the case in general. Example. 10 Destruction of the orbit structure on a normally hyperbolic invariant manifold. As in Example 9, consider a plane equation leaving the unit circle S invariant. Assume that S consists of equilibria and the linearized equation near S in the polar coordinates is of the form tp = 0, p = - ( / > - 1) (see Fig.7). Clearly, S is normally hyperbolic: for p G 5, E% = {O}, E*v is a perpendicular to S at p. Thus, by Theorem 19, 5 is a persistent manifold. However, on SF the perturbed equation may have no equilibria at all.
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Figure 7: Unperturbed and perturbed normally hyperbolic invariant mani folds.
8. Locally non-unique invariant manifolds Let M 0 be a smooth compact invariant manifold of Eq.(l) without boundary. As stated in section 7 , if Mo is normally hyperbolic then it is necessarily locally unique. Here, we study the case, where Mo is not locally unique and, hence, not normally hyperbolic. In order to give conditions ensuring the indestructibility of such a manifold we introduce some notions. For p € M 0 define the linear subspaces TpRn : \DXt(p)v\ -> 0, \DXt(p)v\ {(DX^p))-^
- 0 as t -► +oo},
E; = {ve TpRn : \DXt{p)v\ -+ 0, \DXt{p)v\ liDXfip))-1]
- 0 as t -+ - o o } ,
E; ={ve
where TpW1 is the tangent space to Ft? at p, DX® : TMQ —> TMQ is the restriction of the tangent map DXt : TBJ1 -+ TBJ1. Definition. 20 We say that the transversality condition holds at p £ MQ if TpRn = TpMo + EMp + E;.
(69)
The transversality condition holds on an invariant set A C Mo if (69) holds at every peA. Notice that the sum in (69) is not necessarily direct. From the results of Fenichel [3] it follows that if (69) holds for all p E M 0 and the sum is direct then M 0 is normally hyperbolic. If so, £*, £ J are stable Es and unstable Eu vector bundles at p. Notice also that, generally, dimi?* and dimE^ depend on p. For convenience we will refer to equilibrium points and closed orbits of Eq.(l) on M 0 , which are strong sinks (strong sources) with respect to Mo, simply as strong
Perturbation of Invariant Manifolds of Ordinary Differential Equations
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sinks (strong sources). Similarly, we define nondegenerate strong sources and nondegenerate strong sinks of Eq.(l). 8.1 Strongly indestructible invariant manifolds The following result obtained by Osipenko [17] gives at present the most general conditions for a compact invariant manifold to be strongly indestructible. Theorem. 21 If the transversality condition holds on a compact invariant manifold MQ, except possibly for the nondegenerate strong sinks and nondegenerate strong so urces, then MQ is strongly indestructible. The necessity of conditions of Theorem 21 has recently been studied by Osipenko and Ershov [18]. To state the obtained result let us give some definitions. For p E Mo define the linear subspaces Esp = {ve TpRn : \DXt(p)v\ ^DX^p))-^
K = ive
^Oast-^
+oo},
\ *(P>\ I(^^°(P))_1| -» 0 ™ * ~> -°o}-
T Rn : DX
P
Definition. 21 We say that the relative hyperbolicity condition holds at p 6 Mo if TpRn = TPM0 + Esp + E;.
(70)
// (70) is satisfied at every point of an invariant set A C Mo then we say that the relative hyperbolicity condition holds on A. A
A
As in (69), the sum in (70) is not necessarily direct. Since E* C E* and ££ C E% then the transversality condition implies the relative hyperbolicity condition. Consider Eq.(l) restricted to Mo : x = Fo(ar),
x G M0.
(71)
Let Xf : M0 —► M0 be the flow of Eq.(71). Recall that a point p G M0 is called wandering if there are a neighborhood U of p in MQ and tp > 0 such that for t >tp
xt(U) n u = 0. Otherwise, p is called a nonwandering point. Let Q, C Mo be the set of all nonwandering points of Eq.(71). The set Q is known to be closed and invariant. Theorem. 22 If a compact invariant manifold M0 of Eq.(l) is strongly indestructible then (i) the transversality condition holds at every equilibrium or periodic orbit on M0, except possibly for nondegenerate strong sinks and nondegenerate strong sources, (ii) the relative hyperbolicity condition holds at every nonwandering point of Eq.(l) on M0.
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Figure 8: Locally non-unique strongly indestructible invariant manifold, on which the transversality condition holds. As an illustration of conditions of Theorem 21, let us consider several examples. Example. 11 Locally non-unique strongly indestructible invariant manifold on which the transversality condition holds. Similarly to Example 9, consider a plane equation leaving the unit circle S invariant and with two equilibria on S: the source A and the sink B (Fig.8). Let the linearized equation at A be of the form x = x, y = 2(y — 1) and that at B of the form x = —x, y = — 2(y +1). The transversality condition holds on S since, as easily seen, EaA = E% = {0}, E\, ESB are the vertical straight lines and for p ^ A, B, E%, Eap are straight lines transversal to S at p. By Theorem 21, S is strongly indestructible. Notice that in every neighborhood of S there is an infinite number of smooth invariant manifolds which are tangent to the horizontal direction at A and 5 , and this holds also for perturbations of the initial equation. From this it follows that the map H associating to a perturbed equations its invariant manifold is not uniquely determined.
Example. 12 Destruction of an invariant manifold when the transversality condition does not hold at a wandering point. In R2, consider an equation leaving the unit circle S invariant and with two saddle equilibria A, B (Fig. 9). Let the linearized equation at A be of the form x = x, y = —(y — 1) and at B of the form x = —x, y = y + 1 . Easily seen, A and B are normally hyperbolic: E% = E\ = {0}, E\, Eg are the vertical straight lines. If p £ S and p ^ A, B then Ea = E* = {0}. Hence, the transversality condition is violated at p. Since the arc AB C S is a separatrix joining two saddles one can construct a perturbed equation having nearby 5" no invariant manifold homeomorphic
to S.
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Figure 9: Destruction of an invariant manifold when the transversality con dition does not hold at a wandering point. Example. 13 Destruction of the C1 structure of an invariant manifold when there is a strong nondegenerate sink and the transversality condition does not hold at a wandering point. Consider, in i? 2 , a differential equation leaving the unit circle S invariant and with two equilibrium points A, B G S. Let the linearized equation at A be the same as in Example 12 and at B be of the form x = —x, y = — (y + l)/2. The behavior of orbits is shown in Fig. 10. As before, the transversality condition holds at A. At B we have EB = Eg = {0} and the transversality condition is violated. Since the contraction at B along S is stronger than the contraction in the vertical direction, B is a strong sink and is nondegenerate. lip E S, p ^ A then E£ = E* = {0}. Thus, the transversality condition is violated at p. Perturbing the equation in a neighborhood of p £ 5, p ^ A, B, one can obtain an equation whose saddle point A has the unstable separatrix tangent to the vertical direction at B. Therefore, a perturbed equation nearby 5 has no invariant C1 submanifold diffeomorphic to S. Observe that there
Figure 10: Destruction of the C 1 structure of an invariant manifold.
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George Osipenko & Eugene Ershov
Figure 11: Strongly indestructible invariant manifold when the transversality condition is violated at a strong source. exists nevertheless an invariant C° submanifold homeomorphic to S. Example. 14 Strongly indestructible invariant manifold when the transversality con dition is violated at a strong source. Consider, in R3, an equation having in the plane x = 0 the invariant unit circle S and two equilibrium points A(0, 0, 1), #(0, 0, —1) € S (Fig.ll). Suppose that in some neighborhood of A the plane y = 0 is invariant for the equation and in this plane the equation is of the form x = 2 — 1, i = —x, i.e., the equilibrium A is of center type on y = 0. Let in some neighborhood of A the linearized equation on S at A be of the form y = y. It is clear that A is a nondegenerate strong source and the transversality condition does not hold at A. Assume that the linearized equation at B is of the form x = —2x, y = —y, i = —2(z + 1). The transversality condition is obviously fulfilled on S \ A. For p £ S \ A, E% = {0}, E*v is a plane transversal to S at p. By Theorem 21, 5 is strongly indestructible. Let us show that a perturbed equation has a unique invariant manifold Mp C1 close to 5, i.e., the map H associating to a perturbed equation its invariant manifold is uniquely determined. Obviously, each vector field F C1 close to the original one has two equilibria Ap, Bp nearby A, B. By the Center Manifold Theorem there exists a locally invariant surface op (given by y = v?(x, z)) containing Ap and C1 close to the plane y = 0. Every orbit of F through a point near to ap tends to ap as t —> — oo. On o~p the equilibrium Ap may be of center, focus or center-focus type. It is not difficult to see that in either case there are only two orbits 7^, j F tending to Ap as t —♦ — 00 and C1 close to S (see the Stable Manifold Theorem). Clearly, 7^, 7^ —► Bp as t —► +00 provided a perturbation is small enough. Since the contraction at B in the transversal to S direction is stronger than along S then 7^, 7^ are tangent to each other at Bp. Thus, Mp = Ap U 7^ U 7^ U Bp is the unique invariant manifold of a perturbed equation C 1 close to 5.
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8.2 Weakly indestructible invariant manifolds Sufficient conditions of the weak indestructibility of a compact invariant manifold were obtained by Osipenko [17]. Theorem. 23 Let Mo be a smooth compact invariant manifold of Eq.(l) without boundary. If the transversality condition holds on Mo, except possibly for strong sinks and strong sources, whose indices are different from zero, and there exists at least one degenerate strong sink (source) then MQ is weakly indestructible.
References 1] R.Abraham and J.Robbins, Transversal Mappings and Flows (Benjamin Inc., New York, 1967). 2] N.N.Bogoliubov and Yu.A.Mitropolsky, The Method of Integral Manifolds in Non-linear Mechanics, in Contributions Differential Equations, II (New York, 1963), pp.123-196. 3] N.Fenichel, Persistence and Smoothness of Invariant Manifolds for Flows, Ind. Univ. Math. J. 21 (1971), pp.193-226. 4] D.M.Grobman, The Topological Classification of the Vicinity of a Singular Point in n-Dimensional Space, Math.USSR-Sbornik 56 (1962), pp.77-94 (Russian). 5] P.Hartman, Ordinary Differential Equations (John Wiley, New York, 1964). 6] M.Hirsch, Differential Topology (Springer-Verlag, New York, Heidelberg, Berlin, 1976). 7] M.Hirsch, C.Pugh, and M.Shub, Invariant Manifolds, Lect. Notes in Math. 583 (Springer-Verlag, New York, Heidelberg, Berlin, 1977 ). 8] M.Hirsch and S.Smale, Differential Equations, Dynamical Systems and Linear Algebra (Academic Press, Orlando, 1974). 9] H.Hopf, Ueber die Drehung der Tangenten und Sehen ebener Kurven, Compositio Math. 2 (1935), pp.50-62. [10] A.Kelley, The Stable, Center-stable, Center, Center-unstable and Unstable Ma nifolds, J. Diff. Eqns. 3 (1967), pp.546-570.
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[11] U.Kirchgraber and K.J.Palmer, Geometry in the Neighborhood of Invariant Ma nifolds of the Maps and Flows and Linearization, Pitman Research Notes in Math. Series, 233 ( Longman Scientific k Technical, Published in the United States with John Wiley k Sons Inc., New York, 1990). 12 S.Lang, Analysis, v.l (Addison-Wesley, 1968). 13 A.M.Liapunov, Probleme Generale de la Stabilite du Mouvement (Princeton Univ. Press, Princeton, 1947). 14 R. Mane, Persistent Manifolds are Normally Hyperbolic, Trans. Amer. Math. Soc, 246 (1978), pp.261-284. 15 Yu.I.Neimark, Integral Manifolds of Differential Equations, Izv. Vuzov, Radiophysics 10 (1967), pp.321-334 (Russian). 16 Yu.A.Mitropolsky and O.B.Lykova, Integral Manifolds in Non-linear Mechanics, (Nauka, Moscow, 1973), (Russian). 17 G.S.Osipenko, Perturbation of Invariant Manifolds, I, II, III, IV, Differential Equations 21 (1985), pp.406-412; pp.908-914; 23 (1987), pp.556-561; 24 (1988), pp.647-652. 18 G.S.Osipenko and E.K.Ershov, On the Condition Necessary for the Preserva tion of a Locally Non-unique Invariant Manifold (submitted for publication in Functional Differential Equations, Israel Seminar). 19 G.S.Osipenko and E.K.Ershov, The Necessary Conditions of the Preservation of an Invariant Manifold of an Autonomous System near an Equilibrium Point, J. of Applied Math and Physics (ZAMP) 44 (1993), pp.451-468. 20 0.Perron, Uber Stabilitat und Asymptotisches Verhalten der Integrale von Differentialgleichungssystem, Math. Z. 29 (1928), pp.129-160. 21 V.A.Pliss, The Reduction Principle in the Theory of Stability of Motion, Izv. Acad. Nauk SSSR, Ser. Mat. 28 (1964), pp.1297-1324 (Russian). 22 R.J.Sacker, A Perturbation Theorem for Invariant Riemannian Manifolds, in Proc. Symp. Diff. Eq. and Dyn. Syst. at Univ. Puerto Rico, ed. J. Hale and J. LaSalle (Academic Press, New York, 1967,) pp.43-54. 23 A.M.Samoylenko, Elements of the Mathematical Theory of Oscillations with Mul tiple Frequences. Invariant tori (Nauka, Moscow, 1987), (Russian). 24 H.Whitney, Diferentiable Manifolds, Ann.Math. 37 (1936), pp.645-680.
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[25] H.Whitney, Analytic Extensions of DifFerentiable Functions defined in Closed Sets, Trans Am.Math.Soc. 36 (1934), pp. 63-89. [26] S.Wiggins, Normally Hyperbolic Invariant Manifolds of Dynamical (Springer-Verlag, New York, Heidelberg, Berlin, 1994).
Systems
T H E R E D U C T I O N OF D I S C R E T E D Y N A M I C A L A N D S E M I D Y N A M I C A L SYSTEMS IN METRIC SPACES Andrejs Reinfelds* Institute of Mathematics of Latvian Academy of Sciences and University of Latvia, Turgeneva iela 19, LV-1524 Riga, Latvia; e-mail: [email protected]
Abstract In an arbitrary complete metric space discrete dynamical (semidynamical) systems generated by homeomorphisms (continuous mappings) are considered. Necessary and sufficient conditions for the existence of invariant sets are ob tained. In this context the problem of decoupling and simplifying dynamical and semidynamical systems by means of topological transformations are stud ied. The results obtained allow to reduce the investigation of the given system to an analogous investigation for a much simpler system. In consequence, the classical Grobman-Hartman theorem as well as the principle of reduction for semilinear dynamical systems in Euclidean and Banach spaces are obtained.
1. Introduction The classical Hartman-Grobman theorem [20-23, 28-30] states that a system of autonomous differential equations in the form dx/dt = Ax + f(x, y), dy/dt = By + g(x, y) is dynamically equivalent to the linear system dx/dt = Ax, dy/dt = By, if the spectra of A and B are separated by the imaginary axis in the complex plane, the mappings / and g are uniform Lipschitzian with sufficient small Lipschitz constant "This work has been completed with thefinancialsupport of Latvian Council of Science under Grant 93.809
267
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Andrejs
Reinfelds
and vanish at the origin. If the spectrum of B is located to the left of the spectrum of A and the spectra are separated by a vertical line in the left complex half-plane, then there is a Lipschitzian mapping u such that the given system is dynamically equivalent [34, 40, 44, 50-52, 54-57, 68-69, 79-82] to the system dx/dt = Ax + f(x,
u(x)),
dy/dt = By. Analogous results are correct for differential equations in a Banach space [12—14^ 53, 65-66, 70-71] and for discrete dynamical systems both in Euclidean and Banach spaces [1, 28-30, 38-40, 60, 67]. The decoupling and linearization for semidynamical systems generated by noninvertible mappings in a Banach space has been studied by B. Aulbach and B. M. Garay [6-8]. We give a detailed proof of the fact that if the mapping T is a homeomorphism having a fixed point then it is conjugate to the decoupled homeomorphism S, where S(x,y)
=
(f(x,u{x)),g(v{y),y)).
Relevant results concerning partial decoupling and simplifying of mapping T are given also [75-78]. Note that the reduction theorem in the stability theory is derived from the equivalence theorems. We also prove that the mapping T has a shadowing prop erty. 2. Preliminaries In this section we set out some basic facts needed for later sections and specify the form of the mapping T. Definition 2.1 A metric space is a set X equipped with a function / » : X x X - > R+ such that: («)
p(x, x) = 0 for all x 6 X.
(ii)
p(x, x') > 0 for all x ^ x', x, x' G X .
(Hi)
/?(x, x') = p(x', x).
(iv)
p(x, x") < p(x, x') + p(x', x") (triangle inequality).
Definition 2.2 A sequence {x,} in a metric space X is called Cauchy if for any e > 0 there exists an integer n > 0 such that p(xi, Xj) < e for all i,j > n. Definition 2.3 A metric space X is complete if every Cauchy sequence in X is con vergent. Let X i and X 2 be metric spaces with metrics pi and /) 2 , respectively.
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269
Definition 2.4 A mapping T:Xi —»• X2 is Lipschitz (with constant k) if, for all x,x' € X i , p2(T(x),T{x'))
X which is
Definition 2.7 A one parameter family {T n }, n € Z of continuous mappings with T 1 = T: X —> X is a discrete dynamical system if: (i) (it)
T° = id, where id is identity mapping. TnoTk
= Tn+k.
If a one parameter family of mappings is defined only for nonnegative integers we have a discrete semidynamical system. Let us note that in the case of discrete dynamical systems the mapping T is a home omorphism. Definition 2.8 Two discrete dynamical (semidynamical) systems T1",T2n:X —► X are topologically equivalent, if there exists a homeomorphism H: X —> X such that the diagram
commutes for all n. Definition 2.9 Two mappings Ti,T2:X —> X are topologically conjugate, if there exists a homeomorphism H: X —> X such that the diagram
commutes.
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270
It is easily verified that two discrete dynamical (semidynamical) systems T" and generated by mappings Tx and T2, are topologically equivalent if and only if mappings T\ and Ti are topologically conjugate. Let X and Y be complete metric spaces with metrics p\ and /?2, respectively. The object of this paper is to study continuous mappings T : X x Y — > X x Y o f the type T2n,
T{xiy) = (f(x1y),g(x1y)). We will make the following hypotheses: (HI)
Pl(x,x')
< a/n(/(x,y),/(i',y)) > a > 0.
(H2) ft(/(*.y)./(*,•)) ^ Myrf). (H3)
P2{g{x,y),g{x',y'))
< jp1(x,xr)
+ 6p2{y,y% where a{6 + 2 v ^ < 1-
(H4) Mapping /(-,y):X —» X is surjective. Our aim is to decouple and simplify the given mapping T by means of a topological transformation. Example 2.1 Let us consider the following mapping in Banach space x1 = Ax + F(x,y), yx=By + G{x,y),
(1)
where x G X, y (E Y, A and B are bounded linear mappings, A is invertible, ||J3|| < || A~l || _1 , mappings F : X x Y —► X, G: X x Y —► Y satisfy the Lipschitz conditions \F{x,y) - F(x',y')\ < e(\x - x'\ + \y - y% \G{x,y)-G{x',y')\<e{\x-x'\+\y-y'\). It is easy to verify that this mapping satisfies the hypotheses (HI) - (H4), where a = (HA"1!!"1 - e)- 1 , 0 = 7 = e, 8 = \\B\\ + e. The condition ct{6 + 2y/f^) < 1 reduces to the inequality
llA-j-1-!!*)! 4
1
The mapping given by formula x = Ax + F(x, y) for fixed y is surjective, if ell^l"1!! < 1. Let us note that e:||v4.~^|| < 1/4. 3. Auxiliary lemmas In order to prove the main results we use three lemmas. Let us consider the set of mappings Lip(fc) = {u | u:X -> Y and />2(u(x),tx(s')) <
kPl{x,x')}.
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Lemma 3.1 Let a/3k < 1 and u € Lip(A;). Then the mapping <£>:X —► X, defined by i(x,x') <
api(f(x,u(x))J{x',u(x)))
= ap1(f(x',u(x')),f(x',u(x)))
<
apkp^x.x').
Since a(3k < 1, it follows immediately that pi(x, x') = 0 and x = x'. Next we show that (p is surjective. Let us define now a mapping <&: X —»• X by the equality / ( $ ( x ) , u(x)) = x\. According to ( H i ) and (H4), for every x G X there is a unique $(x) and px{${x)^{x'))
=
< op1(/(*(x),ti(x)),/(*M>«(x)))
ttpi(/(*(*VM)./(*(*V(*)))
< apkPl(x,x').
We obtain that $ is a contraction on X. Therefore for every xi 6 X there is a unique x € X such that / ( x , u(x)) = X\. Thus
/(*',u(x'))) + aPl(f(x',u(x)),
< otp1(f(x1u(x))if(xt,u(x')))
f(x',u{x')))
+ aPkpi(«,&').
We obtain ft(*,*0
< a(l - a W V i ( / ( x , U ( x ) ) , / ( x ' , u ( x / ) ) ) .
(2)
Consequently, pl(ip-1(x)i
< a(l -
aPk)-1p1(x,x').
We get that y? is a homeomorphism. The lemma is proven. Next introduce the operator C acting on Lip(A;) defined by the equality (£u){f(x,u(x))) Lemma 3.2 There exists k>0
=
g(x,u(x)).
such that £(Lip(fc)) C Lip(fc).
Proof. Taking into account (H3) we get * ( ( £ « ) ( / ( » , «(*))), (Cu)(f(xf, = p2(g(x1u{x))Jg(x\u{x')))
u(x'))))
< (7 + £ % i ( x , x ' ) .
According to (2) we get p2((£u)(f(x,
u(x))), (Cu)(f{x',
u{x'))))
D
272
Andrejs Reinfelds
< a(7 + **)(l -#)" 1 /'i(fc«W)»/( a! '«M))If fc > 0 satisfies the inequality 0 < a ( 7 + flfe)(l - a^fc)" 1 < fc, then £(Lip(fc)) C Lip(Jfc). Thus fc > 0 exists, if a{6 + 2 v T O < 1. We choose k =
2a7 1 - aS + v / ( l - a « 5 ) 2 - 4 a 2 ^ 7 * D
Then the lemma is proyed. Next let us consider the set of mappings Lip(0 = {<; | v: Y -> X and pi( U (y),«(y')) < 'Pa(y.y')} and let us introduce the operator K acting on Lip(/) by the equality f{£v{y),y)
=
v(g{v(y),y)).
The operator K is well defined, because the mapping /(•, y): X —* X is surjective and hypothesis ( H i ) is fulfilled. L e m m a 3.3 There exists an I > 0 such that /C(Lip(/)) C Lip(/). P r o o f According to (HI) - (H3), we get P\{Kv(y),Kv{y'))
< api(/(/Cv(y),y),/(/Cv(y'),y))
< *Pi(v(g(v(y),y))A9WW)))
+
ccp,{f{Kv{y%y),f{Kv{y%y'))
< ( a / ( 7 / + £) + a/?)p 2 (y,y'). K 0 < 0/(7/ + <5) + a £ < I, then /C(Lip(/)) C Lip(/). Therefore / > 0 exists, if a(6 + 2y/^y) < 1. We choose 2a/?
/ =
y/(l-a6)2-4a2^'
1 - a<5 + The lemma is proved.
D
Later on we always assume that 2a k =
7
1 - aS + ^ ( 1 - a<5)2 - 4a 2 ^7 and /=
2a/3 1 - a<5 + v / ( l - a « 5 ) 2 - 4 a 2 / ? 7 '
It should be noted that /?fc = 7/, a ( 7 + £fc)(l - a/?*:)"1 = fc, a/ft/ + <$) + a/3 = /. a/?fc = a7/ < 1/2 and A;/ < 1.
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Example 3.2 Let us consider the mapping (1). We get k = l=
2e
UA-i-i - ||B|| - 2e + 7(p-i||-i-||5||)(||A-i||-i-||B||-4£)
| | A - j - » - ||B|| - 2 £ - y/(\\A-i\\-* Y
=
_
- \\B\\)(\\A-*\\-i -
\\Bf^4e) <
L
4. Fixed point We give sufficient condition for the existence of a fixed point. Theorem 4.2 / / (1 — a)(l — 6) — a/?7 > 0, then the mapping T has a unique fixed point T(x0,y0) = (x0,y0). Proof. Wefixyi G Y and define a mapping $ : X —» X by the equality x = / ( $ ( z ) , y i ) . According to ( H i ) and (H4), for every x G X there is a unique $(x) and />!(*(*),SM) < ap1(f($(x),yl),f($(x'),y1)) = aPl(x,x'). It follows that $ is a contraction on X. Therefore for every t/i G Y there is a unique i e X such that x = f(x,yi). Thus the mapping x- Y —* X, where
x(y) = f(x(y)*y) is well defined. Therefore
pi{x(y),x{y')) < api(/(x(y).y)./(x(y0.y)) < *pi{x(y),x(y')) + «^2(y,y/)We get that x € Lip(a/?(1 - a)'1). Next let us consider the equation y = g(x{y),y)We get
p2{9{x(y),y),g{x{y'),y')) < (afa[i - a)" 1 + %2(y,y'). Since a^-y(l - a ) - 1 + 8 < 1, there is a unique y0 G Y such that y0 = g(x{yo), yo)- Let x0 = x(yo)- This completes the proof of Theorem 4.2. □ Example 4.3 Let us consider a mapping of the form (1). The condition (1 — a)(l — 6) > a/?7 reduces to the inequality , ( H A - ! " - 1 ) ( 1 - \\B\\)
||A-l-> - P||
•
Using the relation between geometric and arithmetic mean we obtain
(M-MI-1 - i)(i - ygll) ^ ll^-1!!-1 -11^1 ||A->||-'-||S||
-
4
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274
5. Invariant sets We give a necessary and sufficient condition for the existence of mappings u: X —* Y and v: Y —> X, whose graphs are invariant sets [1-2, 9-11, 15-16, 18-19, 24-27, 30-33, 36, 41-42, 45-47, 59, 61, 63, 72-74, 83-86]. Theorem 5.3 Let the hypotheses (H1)-(H4) hold. For the existence of mappings « : X - » Y and v: Y —> X that satisfy the functional equations w(/(x, u(x))) = g(x, u(x)), f(v{y),y)
(3)
= v(g(v(y),y))
(4)
and the Lipschitz conditions p2{u(x),u(x'))
(5) (6)
Mv(y),v(y'))<Wy,yO,
it is necessary and sufficient that the mapping T has a fixed point T(x0, y0) = (x 0 , J/o)Proof. Sufficiency. The set Mi = {u | u € Lip(fc) and w(xo) = yo} becomes a complete metric space, if the metric is defined by the equality d\ (U, U ) = SUp x
r pi[X,X0)
.
It is obvious that Mi is a metric space and that for every u, u' G Mi we get di(u, u') < 2k. It remains to be proved that Mi is complete. As Y is a complete metric space then for any Cauchy sequence there exists a continuous limit mapping if x ^ XQ. The continuity of the limit mapping for x = xo follows from the estimate p2(u(x),yQ) < d1(u,y0)p1(x,x0)
<
2kp1(x,x0).
Next introduce the operator C on Mi by the equality (Cu)(f(x,u{x)))
=
g(x,u(x)).
Let w, u' 6 M i . We get ft((£ii)(/(*, = p2{g{x,u{x)),g(x',u'(x')))
u(x))), (Cu')(f(x\ < ( 7 + 6k)Pl(x,x')
u'(x')))) +
On the other hand, we obtain Pi(x,x') <
otPl{f(x,u{x)),f{x',u{x)))
6p2(u(x),u'(x)).
The Reduction of Discrete Dynamical Systems in Metric Spaces < aPl(f(xMx))J(x',u'(x')))
+
< ctpi{f{x,u{x)),f{x\u\x')))
275
aPl(f(x',u'(x')),f(x',u(x)))
+ afikfrfaJ)
+
ctPp2{u{x),uXx)).
It follows that cc{\-*pk)-lPl{f{x,u{x)),f{x',u'{x')))
Pi(x,x') <
a0k)-lp2(u(x),u'(x)).
+a0(l If x' = XQ and u' = u, then
Pi(x, x0) < a(l - aj3k)~1p1(f(x, u(x)), f(x0,
u(x0)))
a/3k)~1p1(f(x,u(x)),xo).
= a(l Therefore
p2((Cu)(f(x,«(*))), (£'(x'))) + ( a £ ( 7 + 6k)(l - ctPk)-1 + ^ i ( t / , u > ( l - a/?fc)"Vi(/(x,«(*))> *o) = */*(/(*ft*(*))i/(*'»«'(*'))) + M( u >w')/ 5 i(/( a : » w ( : r ))^o), where , li = a(/3k + S)(l - apk)-1
1 + a6 - J(l - a6)2 - 4a 2 fa V = ; ' < 1. 2 2 1 + aS + ^ ( 1 - a6) - 4a /? 7
In addition, let us note that f(x,u(x))
^ x0 if x ^ XQ. It follows that
(£u)(x0) = (Cu)(f(x0,u(x0)))
= g(x0,u(x0))
= y0.
It follows that £ ( M i ) C M i . Let xi = f(x,u(x))
=
f(x',u'(x')).
Therefore di(£u,£u)
< p,di(u,u').
We obtain that £ is a contraction on M i . It follows that in Mi there is a unique mapping u satisfying the functional equation (3) and the Lipschitz condition (5). The first part of sufficiency is proven. Let us prove the existence of the invariant set given by the mapping v: Y —► X. The set M 2 = {v | v e Lip(/) and v(y0) = x0} is a complete metric space, if the metric is denned by the equality
r
.
275
Andrejs Reinfelds
Let us introduce the operator /C by the equality f(JCv{y),y) =
v{g(v(y),y)).
Using the definition of /C, we obtain JCv(y0) = x0. It foUows from Lemma 3.3 that Kv e Lip(Z). We get that /C(M2) C M 2 . Let v, v' G M 2 . It follows that Pl{Kv{y),Kv'(y))
< otpi(f(Kv{y),y),f{K>v'(y),y))
= api{v(g(v(y),y)), v'(g(v'{y), y)))
< {<xp2(g{v{y),y), yo) + o-yWl/. yo))Mv>v')Let us estimate
P2{g{v{y),y),y0) = p2(g(v{y),y),g{v{y0),yo)) < {il + S)p2(y,yo). It follows d2{Kv{y),K,v'{y'))
<\d2{v,v'),
where A = aS + 2a 7 / = 1 - y/(l - a6)2 - 4 a 2 ^ 7 < 1. We obtain that K, is a contraction on M 2 . It follows that in M 2 there is a unique mapping v which satisfies the functional equation (4) and the estimate (6). Necessity. In virtue of the inequality kl < 1, a simple contraction argument implies that the system of equations v(u(x)) = x and u(v(y)) = y has a unique solution (xo,yo). We obtain u(v(u(x0))) = u(x0) and v(u(v(y0))) = v(y0). Prom the uniqueness of the solution it follows that U(XQ) = yo and v(yo) = xo- Finally, we get /(zo,2/o) = f{v{yo),yo) = v(g(v(y0),yo)) = v(g(x0,u(x0))) Consequently, f(x0,yo) of the theorem.
=
v(u(f(x0yy0))).
= x0. Analogously, g{x0,yo) = y0. This completes the proof D
Let us note that if aS + 1 < 2a, then 1 - a6 - J(l - aS)2 - 4 a 2 £ 7
\ _ a6
2a
2a
In the case a8 + 1 > 2a we get
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277
Lemma 5.4 If fik + 6 < 1 and a ( l + 7/) < 1, then (1 - a)(l - 6) > afa, conversely, if (1 - a)(l - 8) > apf, then /3k + 6 < 1 and a ( l + 7/) < 1.
and
Proof. Let us assume that pk + 6 < 1 and a ( l + 7/) < 1. We have 1 + aS-
J{1 - aS)2 - 4<*2/?7 —^ < 1 2a
and + 1 - aS - ^ ( 1 - a£) 2 - 4a2/?7
lot
< L
2 It follows that
|1 + a£ - 2a| < ^ ( 1 - a£) 2 - 4 a 2 £ 7 . Therefore, (1 - a)(l - 6) > a/3 7 . If (1 - a ) ( l - 6) > a/87, then Y ( l - ad))2 - 4a 2 £7 > ^ ( 1 - a£) 2 - 4a(l - a)(l - 6) = |1 + a<5 - 2a|. Now let a£ + 1 < 2a. Then a ( l + 7/) = a +
1 - a<5 - ^ ( 1 - aS)2 - 4a2/?7 2
1 - a6 + 1 + ct6 - 2a < a + = 1. Let oc8 -\-l> 2a. Here we have '
p
1 - a £ - l - a < ! > + 2a
0k + 6<
.
„ + 6=1.
2a Theorem 5.4 Let the hypotheses (H1)-(H4) hold, and let fik + 6 < 1. For the existence of a mapping u:X —* Y that satisfies the functional equation (3) and the Lipschitz condition (5) it is necessary and sufficient that there exists a mapping UQ 6 Lip(fc) such that supP2(u0(f(x, u0(x))),g(x,u0(x)))
< +00.
Proof. Sufficiency. The set Ms = {u I u e Lip(fc) and sup p2(u(x),u0(x))
< +00}
(7)
Andrejs Reinfelds
278 is a complete metric space, if the metric is defined by the equality d3(u,u') =
supp2{u(x))u'(x)). x
Let us prove that £ is a contraction. Let x1 = f{x,u{x))
=
f{x',u'{x')).
We have P2((Cu)(x1),(Cu')(x1))
=
pMx,u(x)),g(x\u'{x')))
< (7 + 6k)pi{x, x') + 6p2{u(x), u'{x)). On the other hand Pi(x,x') < a/»i(/(i,u(i)),/(ic',uW)) = ap1(f{x\u'(x%f{x,,u(x))) < OLPP2{U'(X),U(X)) +
a^kp1(xJx').
Therefore a(3k)-1pl(u'{x)yu(^)-
Pi(x,x') < o0(l We get p2((Cu)(x1),(Cu')(x1))
< (048(7 + M)(l - a^fc)"1 + Qpa(tt(»)X(«)).
Hence d3(Cu,Cu') < (^ + ^jfc)4(w,«/). We have p2((£uo)(zi),uo(:ri)) = p2(5,(^,w0(x)),wo(/(a;,Wo(x)))). Therefore d 3 (£u 0 ,u 0 ) < sup/> 2 (0(z,uo(x)),uo(/(z,uo(z))))Hence d3(Cu, u0) < d3(£u, Cu0) + (23(£u0, u 0 ) < (6 + (3k)d3(u, u0) + sup p 2 (0(s, Mo(*)), «o(/(z, u0(x))))-
(8)
We obtain that £ is a contraction on M 3 . It follows that in M 3 there is a unique mapping u satisfying functional equation (3). Prom (8) we have d(u, uo) < (1 - S - /3k)-1 sup p2{g{x, u 0 (x)), u0(f(x,
u0(x)))).
X
The theorem is proven. Necessity. It is obvious.
□
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279
Theorem 5.5 Let the hypotheses (Hl)-(H4) hold, and let a ( l + 7 / ) < 1. For the existence of a mapping v: Y —> X that satisfies the functional equation (4) and the Lipschitz condition (6) it is necessary and sufficient that there exists a mapping vo 6 Lip(/) such that sup pi(v0(g(v0(y)yy)), f{v0{y), y)) < +00. (9) Proof. Sufficiency. The set M 4 = {u I u e Lip(/) and sup p2(v(y),v0(y))
< +00}
y
is a complete metric space, if the metric is defined by d4(v,v') =
supp1(v(y),v'(y)). y
Let us prove that K is a contraction. We have Pl(/C<,(y),/CV(y))
=
<
^{f{K,v{y\y\f{Kv\y),y))
ocPl(v(g(v(y),y)),v'(g(v'(y),y)))
< api{v(g(v(y),y)),v'(g(v(y),y)))
+
ctpi(v'(g(v(y),y)),v'(g(v'(y),y))).
Therefore d4{Kv,Kv')
<
a(l+-yl)d4(v1v').
We have pi{£v0(y),v0{y))
< api(/(/Cv 0 (y),y),/(v 0 (y),y))
= o:/E>i(i;oWuo(y),y)),/(vo(y),y)). We get d4(JCv0,v0) <
asupp1(f(v0(y),y),v0(g(v0(y)yy))). y
Therefore d4(Kv,v0)
< d4(KvyK.v0) + d4(JCv0,vQ) < a ( l + -yl)d4(v,vQ) + a sup pi{f{v0(y), y), v0(g{v0{y), y))).
(10)
y
We obtain that K is a contraction on M 4 . It follows that in M4 there is a unique mapping v satisfying the functional equation (4). From (10) we have d(v,v0) < ( a - 1 - {I + ll))'1
suppi(f(vo(y),y)Jv0(g(v0(y),y))). y
The theorem is proven. Necessity. It is obvious.
D
Andrejs Reinfelds
280 Remark 5.1 It is easy to verify the following estimates
Pa(u(/(*,y)),0(*,y)) < P2W{x,y))Mf(xMx)))) +P2(g(xMx)),9(x,y))
< (/3k + 6)p2(u(x),y)
(11)
and pi(v(y),x)
<
api(f{v{y),y),f{x,y))
= aPl{v(g{v{y), y)),/(x,y)) < aPl(f{x,y),v{g{x,
y))) +
a^lPl{v(y)yx).
It follows that Pi(v(y),x) < oc(l-a1l)-1Pl(v(g(x,y))J(x,y)).
(12)
Example 5.4 Let us consider the mapping (1). The condition (7) is fulfilled if sup|G(x,0)| < +oo, x
and (9) is fulfilled if sup|F(0,y)|<+oo. v Lemma 5.5 Let T be a homeomorphism and let there be a mapping v: Y —* X sat isfying (4) and (6). Then the mapping ip: Y —► Y, defined by ip(y) = g(v(y),y), is a homeomorphism. Proof. At first let us show that ip is injective. Otherwise there exists y ^ y' and 9{v{y),y) = 9{v{y'),y'). Hence according to (4) we get f{v(y),y) = f{v(y'),y'). We obtain T(v(y),y) = T(v(y'),y'). It turns out that the converse assertion is not true. Consequently, ip is injective. Next we show that ip is surjective. For every j/i € Y there is (x,y) such that T(xty) = (v(yi),i/i). It follows that f(x,y) = v(g(x,y)). Then according to (HI), (4) and (6) we obtain
pi{v(y),x) < <*pi(f(v(y),y),f(x,y)) = <*Pi{v(g{v{y),y)),v(g(x}y)))
< 0:7/^1 (v(y),x).
Since ccyl < 1, it follows immediately that x = v(y) or g(v(y),y) = y\. We obtain that ip is bijective. Let us prove the continuity of ip'1. We have T(v(y)}y) = {v(ip(y)),ip(y)). Hence T - 1 (u(y),y) = {v(tp~1(y))^~1(y)). ^ follows that ip'1 is continuous. The lemma is proven. □ Corollary 5.1 Let T be a homeomorphism and let the mappings u : X —* Y and v: Y —► X satisfy (3)-(6). Then the mapping 5 : X x Y — > X x Y defined by the equality S(x,y) = (f{x,u(x)),g(v(y),y)) is a homeomorphism.
The Reduction of Discrete Dynamical Systems in Metric Spaces
281
6. Conjugacy of homeomorphisms. 1 We now consider the case when the mapping T is a homeomorphism having a fixed point. Theorem 6.6 Let the hypotheses (Hl)-(H4) hold and let T be a homeomorphism with a fixed point. Then there exists a homeomorphism ff:XxY-+XxY such that the diagram
commutes, where S(x,y) =
(f(x,u(x)),g(v(y),y)).
Proof. Step 1. Mapping p. Denote by M 5 the following complete metric space
{
Oi \ J}( X It)
p I p: X x Y —> X is continuous and sup
'
x,y
X)
\
p2{u(x),y)
< +oo
equipped with the metric ds{p,P ) = sup *,y
. P2{u{x),y)
Let us consider the mapping p — i ► £p, p G M 5 defined by the equality f(£p(x,y),u(p(x,y)))=p(T(x,y)). According to ( H I ) , (H4) and the continuity of T there exists a unique continuous Cp. It remains to prove that £ is a contraction in M 5 . First we obtain Pl(£p(x,y),Cp'(x,y))
< apx{f(Cp(x,y),u{p{x,y))),
f(Cp\x,
y),u(p'(x,y))))
+a/>i(/(£y(z, y), u{p'{x, y))), f(Cp'{x, y), u{p{x, y)))) ^apMTix^p'iTix^
+
apkpripix^p'iXiy)) +
< (a(j3k + 8) + apk)d5(p,p)p2{u{x),y)
a{3kd5(p,p)p2(u(x),y) =
\d5(p,p)p2(u(x),y).
Next we get Pl(Cidx(x),x)
< api{f{x,y),f(x,u(x)))
<
app2(u(x),y).
282
Andrejs
Reinfelds
Since
4(£p,£p')<M>b,A d5(Cp,idx) < d5{Cp,Cidx) + d5{Cidx,idx) < Xd5(p,idx) + a/3 it follows that Cp G M 5 and therefore the functional equation f(p(x1y),u(p(xJy)))=p{T{x,y)) has a unique solution p G M 5 . S t e p 2. Mapping TT. Consider the complete metric space
{
/9 2 (7r(z,y),y) ir \ T: X x Y —»■ Y is continuous and sup . . .—r— < +00 > x,y pi{v(y),x) J
equipped with the metric
,
d .6 (7T, ) = SUp / 7T ,s
——. :{x,y))• p2{'K{x,y),'K
*.«
Pi{v[y),x)
Let us consider the mapping ir 1—> Cn, TT G M 6 defined by the equality Cir(T(x,y))
-
g(v{Tr(x,y)),n{x,y)).
Since T is a homeomorphism, we get that Cn is continuous. We have p2(CTr(T(x,y)),Cir'(T(x,y))) = P2{g{v{-rr(x, y)), 7r(a?, y)),j(u(jr'(i, y)), x'fa, y))) < (7/ + <5)p2(7r(x,y),7r'(x,y)) < (7/ + 5)de(x, ^)pi(v(y)t x) 1
< a( 7 / + 6)(i - enrO" *^, O/nMrt*. 2/))> /(*, y)) = ^ ( T T , 7r>i(u(^(ar, y)), / ( i , y)). Let us note that P2{g{v{y),y),g{x,y))
<
1Pl{v{y),x)
< 07(1 - a 7 0 " V i ( v ( ^ ( x , y)), / ( x , y)). It follows that d6(Cidy,idy)
< 0:7(1 - cry/)" 1 .
Since d6(Cir,Crc') < p,d6(ir, ir') and d6(Cir,idy)
< d6(C7c,Cidy) +
d6(Cidy,idy)
< nd6{Tr,idy) + 0:7(1 - cry/) - 1 ,
The Reduction of Discrete Dynamical Systems in Metric Spaces
283
it follows that the functional equation *(T(x,y))
= g(v{Tr(x,y)),Tr{x,y))
has a unique solution IT £ M 6 . Step 3. Mapping q. Consider the closed subset M
s ( 0 = {q\qeM5,
d5{q,idx) < I and Pi{q(x,z),q{x,z'))
<
lp2(z,z')}
of the complete metric space M 5 . Let us consider the mapping q i-> Cq, q € Ms(/) defined by the equality f(Cq(x,z),z)
=
q(f(x,u{x)),g(q(xyz),z)).
Note that Cq is well-defined and continuous. We have px{Cq{x,z),x)
<
ctpi{f{Cq{x,z),z),f(x,z))
< dpi{q{f{x,u(x)),g(q(x,z),z)),f(x,u(x)))
+ ^(/(z,u(x)),
f(x,z))
< ctds(q,idx)p2(u(f(x,u(x))),g(q(x,z),z))
+ a/3p2{u(x), z).
P2{u{f{x,u{x))),g(q(x,z),z))
u{x)),g(q{x,z),z))
Since < lP2(q(x,z),x)
+ 6p2(u(x),z)
= p2{g{xy
< (7^5(9,idx) +
6)p2(u(x),z),
we obtain d5(Cq,idx) < a(jd5(q, idx) + 6)d5(q, idx) + afi < al{^l + 8) + a/3 = I. In addition p1(Cq{x,z),Cq(x,z')) <
<*pi{q{f(x,u(x)),g(q(xtz),z)),q(f(x,u(x)),g{q(x,z'),z')))
+ct0p2(zy z') < al(jl + 6)p2{z, z') + a/3p2(z, z') = lp2{z, z'). It follows that Cq G M 5 (f). We obtain Pl(Cq{x,z),Cq\x,z))
<
api{f(Cq(x,z),z)J(Cq'{x,z),z))
= api(q{f{x, u(x)),g(q(x, z),z)), q'(f(x, u(x))yg(q'(x, z), z))) < ad5{q,q')p2(u(f(x,u(x))),g(q(x,z),z)) < {a{ld5(q,idx)
+ ajlpt(q(x, z), q'(x, z))
+ 6) + a7/)d 5 (g,g> 2 (u(x),z) <
\d5(q,q')p2(u(x),z).
It follows that there is a unique solution q 6 M 5 (/) of the functional equation f(q(x,z),z)
=
q(f{x,u{x)),g(q{x,z),z)).
284
Andrejs Reinfelds
Step 4. Mapping 9. Consider the complete metric space
{
P2(9(x,y), y) 9 1I 6: X x Y —* Y is continuous and sup —j—. r—i-rc < +co > *,» Pi(q{x,y)My)) J
equipped with the metric , ,A *
d7(9,9
P2(9(x,y),9>(x,y))
) = sup
—;
r—T-TT-.
Let us consider the mapping 9 ^ C9,9 € M 7 defined by the equality C9(S(x,y)) =
g(q(x,9(x,y)),9(x,y)).
Since 5 is a homeomorphism then C9 must be continuous. We have
pi{qfay)My)) ^ «M/(tf(*»y)»»)»/(v(v)»v)) = ocpi(q(f{x, u(x)),g{q(x, y),y)), v($(v(y)> y))) < o/>i(g(5(x,y)),u(«7(u(y),y))) + a7//)i(3(aj,y),u(y)). It follows that
MflfosO.^y)) < a(i -an'0~ViM%^))>v(flrMsO>y)))Since p2(C9(S(xiy))i£i,(S{xiy))) =
P2{g(q{x,9{x,y)),9{x,y)),g{q(x,9'{x,y)),9'(x,y))) <(1nS)d7(9,9,)Pl(q(x,y)Jv(y))
< ctfrl + fl(l - o r y / ) - 1 ^ ^ , 9,)p1(q(S(xi y)), vfaMy), y))) we obtain
d7(z:0,z:0')
<
!Pi{q{x,y),v(y))
an/l)-lPl(q(S(x,y)),v{g(v{y),y))).
Then d7{Cidy,idy)
< 07(1 - a 7 /)- 1 d 7 (£0,id y )
< d7{C9,Cidy) + d7(Cidy,idy)
< fid7{9,idy) + Q 7 ( 1 - cry/)" 1 .
It follows that there is a unique solution 0 G M 7 of the functional equation 6(S(x,y)) =
g(q(x,9(x,y)),9(x,y)).
The Reduction of Discrete Dynamical Systems in Metric Spaces
285
Step 5. Mapping P. Let us consider the functional equation P(S(x,y))
=
f(P(x,y),u(P(x,y)))
in the complete metric space M 8 = \P | P : X x Y -> X is continuous and sup ^ f e ^ l l f L < I *-y P2[0{x,y)yu{x))
+0
ol J
where the metric is defined by the equality a&{r, f ) - sup *,»
— — /> 2 (0(z,y),u(z))
.
It is easily verified that this functional equation has the solution P(x,y) = x. Let us prove the uniqueness of the solution in Mg. Otherwise there exists (x,y) and x ^ P(x,y). We get pi(*(5(ar f y)), «(/(*, u(*)))) = P2{g{q(x, 0(x, y)), 9{x, y)),g{x, u(x))) < lPl(q{x, 9(x, y)),x) + 8p2(6(x, y), u(x)) < fd5(q, idx)p2(0(x, y), u(x)) + 6p2(9(x, y), u(x)) < ( 7 f + % 2 ( 0 ( s , y ) , «(*)). It follows * ( P ( « , y ) t * ) < c V l (/(P(*,y),ti(i , (*.y))),/(*,«(^(*iy)))) < aPl(f(P(x,
y), u(P(x, if))), f{x, u{x))) + a/3kPl(P(x, y), x)
= ap1{P(S{x, y)), / ( x , w(x))) + a(3kd8(P, idx)p2(6(x, y), w(x)) < 0(7/ + £ + ^fc)4(P, idx)p2{0{x, y), u(x)). Hence d8{P,idx)
<\d8(P,idx).
It follows that P(x,y) = x. The mapping P', where P'{x,y) = p(3(x,0(x,y)),0(x,y)), also satisfies the func tional equation. P'(S(x, y)) = p(q(f(x, u(x)), 9(S{x, y))), 0(S(x, y))) = p{f(q(x, 0(x, y)), 0(x, y)), g(q(x, 9{x, y)), 0(x, y))) =
p(T(q(x,9(x,y)),0(x,y)))
= /«
f(P'(x,y)MP'(^y)))-
Andrejs Reinfelds
286 Let us prove that P' € Mg. Indeed, pi{P\x,y\x)
=
pi(p{q{x,e(x,y)),6{x,y)),x)
< Pl{p{q{x, 6(x,y)),${x,y)),q(x,9{x,y))) < ds{p,idx)p2{9(x,y),u{q(x,9{x,y))))
+
+ Pl{q{x,9{x,y)), x) d5(q,idx)p2(9(x,y),u(x)).
Let us note that P2{0{x,y),u{q(x,6(x,y)))) < p2{9(x,y),u(x)) < p2(9(x, y), u(x)) +
+
kpx{q(x,9(x,y)),x) kd5(q,idx)p2(9(x,y),u(x)).
Hence d8{P',idx)
< d5(p,idx) + d5(q,idx) +
kd5(p,idx)d5(q,idx).
Consequently, we have P'(x, y) = p(g(z, 9{x, y)), 0(z, y)) = x. Step 6. Mapping II. Let us consider the functional equation U(S(x,y)) =
g(v(U(x,y)),U(^y))
in the metric space M7. This functional equation has a solution II(x,y) = y. Let us prove uniqueness of this solution in M7. Otherwise there exists (x, y) and y ^ II(x, y). We get p2(tt(S(x,y)),g(v(y),y))
=
p2(g(v(IL(x,y)),Il{x,y)),g(v(y),y))
< (7/ + <5)
idy)Pl(q(S(x,
y),v(y)) y)), v(g(v(y), y))).
d7(U,idy) < a(7Z + <5)(1 - a 7 /)- 1 d 7 (n,^ y ) = ^ 7 ( n , ^ y ) .
It follows that TL(x,y) = y. The mapping n', where n'(x,y) = ir(q(x,9(x,y)),9(x,y)), tional equation.
also satisfies the func
n'(5(x,y)) = *(q(f(xM*))AS&y)))AS(x,y))) = *(f{q(x, 9(x, y)),9(x, y)),g(q{x, 9(x, y)),9(x, y))) = 9{v(*(q(x, 9(x, y)), 9(x, y))), 7t(q(x, 9{x, y)), B(x, y))) = ^(n'(x,y)),n'(x,y)). Besides n ' € M 7 . Indeed p2(W(x,y),y)<
p2(*{q{x,9{x,y)),9{x,y)),9(x,y))
+
p2(9(x,y),y)
The Reduction of Discrete Dynamical Systems in Metric Spaces < d6{ir,idy)p1(q(x,$(x,y)),v(0(x,y)))
+
d7(0,idy)p2(q(x,y)yv(y)).
Let us note that Pi(g(x,0(x,y)),v(0(x,y))) < pi(q(xyO(x,y)),q(x,y)) < lp2(6(x,y),y)
+ p1{q(x,y),v{y)) + pi{q{x,y),v{y))
<(2ld7(d,idy)
+
+ +
p1(v{y),v(d{x,y))) lp2{0(x,y),y)
l)Pl(q(x,y),v(y)).
Hence d7(W, idy) < d6(TT, idy) + d7(0, idy) + 2Id6(-K, idy)d7(9, idy). Consequently, we have n'(x,y) = 7r(g(x,0(x,y)),0(x,y)) = y. Step 7. Mapping Q. Let us consider the functional equation Q(T(x>y),g(Q(x,y,z),z))
=
f(Q{x,y,z),z)
in the complete metric space M 9 = {Q | Q: X x Y x Y -> X is continuous, Pi{Q{^,y,z),Q(x,y,z'))
< lp2{z,z') and
p1(Q(x,y,z),x) x,y,z max(p 2 (u(x),y), p2(z, y))
^
with the metric d9[Q, Q) = sup x,y,z max {p2{u(x), y), /92(z, ?/)) and in its closed subset M 9 (/) = {Q | Q E M 9 and d9{Q,idx) < I}. Let us consider the mapping Q H* £(£, <2 G Mg defined by the equality f{£Q(x,y,z),z)
=
Q{T{x,y),g{Q(x,y,z),z)).
CQ is well-defined and continuous. Let us note that p2(g(Q(x,y,z),
z),g{x,y))
< -ypi{Q(x, y, 2),x) + fy2(y, 2:)
< jd9(Q, idx) max (p 2 (u(x), y),p2(y, z)) +
<
ap1(f(CQ(x,y,z),z)J(x,z))
287
Andrejs Reinfelds
288
< aPl(Q(T{x,y),g{QfaV,*),*))> /(*>v)) +
^K2/)))
Hence d 9 (£Q, i 4 ) < a(7 max (/, d9(Q, idx)) + <5)d9(<2, idx) + a/3. If g e M 9 (/), then <*9(£Q, 1*4) < «/( 7 / + *) + 00 = ?. £ Q satisfies the Lipschitz conditions />i(£Q(x,y,z),£Q(x,y,z')) < aPl{f(£Q(x,y,z),z),/(£Q(x,y,/),z'))
+ a/?p 2 (z,z')
< (0/(7/ + (5) + a/?)/>2(z, z') = //>2(z, z'). Let Q G M9(Z) and Q' € M 9 . We obtain Pl{CQ{x,y,z),£Q'{x,y,z))
< api(Q(T(x,y),y(Q(x,y,
<
ap1(f(£Q(x,y,z),z),f(£Q'(x,y,z),z)) z), z)), Q'(T(or, j/),^(Q(ar, y,z), z)))
+a7//)i(Q(x, y, z), Q'(x, y, z)). We have d9(£Q, CQ') < (0(7/ + (5) + oryQifeW,
= 0(/OO, y)> «(P(*I iO))i0(«(pfo y)> *)>z))
= ffW/(»,y).^(»,y)),ffhW*,ri,*),*)) - g'(r(x,y)^(g'(x,y,z),z)) and the Lipschitz conditions
PiW(z,y> *)> Q'0>y>z')) = Pi(«00,y), *), g(p(s,y), *')) < W*> *')■ Let us note that Pi(Q'(x, y, z), x) < pi(q(p(x,y), *),p(x,y)) + pi(p(z, y), x) < d5(9, idx)p2{u(p(x, y)), z) + d5(p, uQ/9 2 (u(x),y) and
/>2(u(p(x, y)), z) < p2(y,*) + /^("M. y) + M(p, *4)/»2(uM, y).
The Reduction
of Discrete Dynamical Systems in Metric Spaces
289
We obtain d9(Q',idx)
< 2d5(q,idx)
+ kd5(p,idx)d5(q,idx)
+
d5(p,idx).
Consequently, we have Q{x,y,z)
=
q{p{x,y),z).
It is easily verified that Q{x,y,y)
= x.
Therefore q(p(x,y),y)
= x.
S t e p 8. M a p p i n g 0 . Let us consider the functional equation 0 ( T ( x , y ) ) = <7(Q(x,y,0(x,y)),©(x,y)) in the metric space Me. It is easily verified that the functional equation above has the solution 0 ( x , y ) = y. Let us prove uniqueness of this solution in M6- Otherwise there exists (x,y) and y ^ 0 ( x , y ) . We have /> 2 (0(T(x, y)), g(x, y)) = p2(g{Q(x, y, 0 ( x , y)), 0 ( x , y)),g(Q(x,y,y),
y))
< ( 7 / + %2(©(x,y),y). Hence d6(e,idy)
= a ( 7 / + <5)(l - a 7 / ) - V 6 ( 0 , i r f y ) =
pd6(
It follows that 0 ( x , y ) = y. The mapping 0 ' , where 0 ' ( x , y ) = 0(p(x,y),7r(x,y)), also satisfies the functional equation 0 ' ( T ( x , y)) = d(p(T(x, y)), TT(T(X, y))) = 9{S(p(x, y), TT(X, y))) = 9{
+ Pi{v(y),x)
< (2lde{7T,idy) +
+ +
p1{v(ir(x,y)),x) lp2(ir(x,y),y)
l)pi(v(y),x).
Therefore P2(0(p(z,y),7r(z>y)),2/) < p2(6(p(x, y), TT(X, y)), TT(X, y)) + p2(n(x, y), y) < <27(0,idy)pi(q{p(x, y),TT(X,y)),U(TT(X,y))) +
d 6 (7r,id y )p 1 (v(y),x).
290
Andrejs Reinfelds
This implies that 4(0',idy) < defrtidy) + 2lde(ir,idy)d7{$,idv) +
d7(0,idy).
We have
0(&,y) = Wi.ri^Ky)) = yWe obtain that the mappings tf,T:XxY -» X x Y defined by H{x,y) = (p(x,y),7r(x,y)) and T(x,y) = (g(x,0(x,y)),0(x,y)) are inverse to each other and if is a homeomorphism establishing a conjugacy of the mappings T and 5. The the n orem is proven. Example 6.5 Assume in addition that the mapping (1) is a homeomorphism hav ing a fixed point. Using Theorem 6.6 we obtain that the homeomorphism (1) is topologically conjugate to x1 = Ax + F(x,u(x)), y1 = By-^G(v(y),y). 7. Conjugacy of noninvertible mapping We consider the case when the mapping T has an invariant set. Theorem 7.7 Let the hypotheses (Hl)-(H4) hold and let there be a mapping u: X —» Y that satisfies (3) and (5). Then there exists a continuous mapping j : X x Y —*■ X, which is Lipschitzian with respect to the second variable, and a homeomorphism . f f r X x Y — » X x Y such that the diagram
commutes, where R(x,y) =
(f(x,u(x)),g(q(x,y),y)).
Proof. Step 1. Mapping p. See Step 1 of Theorem 6.6. Step 2. Mapping q. See Step 3 of Theorem 6.6. Step 3. Mapping P. Let us consider the functional equation P(R(x,y))
=
f(P(xyy),u(P(x,y)))-
It is easily verified that this functional equation has the solution P{x,y) = x. Let us prove the uniqueness of this solution in M 5 . Otherwise there exists (x,y) and x ^ P(x,y). We get Pi(P(x,y),x)
<
aPl(f(P(x,y),u(P(x,y)))1f(xMP(x,y))))
The Reduction of Discrete Dynamical Systems in Metric Spaces < aPl{f{P{x,y),
u(P(s,y))),/(a;, u(*))) +
< aPl(P(R(x,y)),
291
aj3kPl(P(x,y),x)
/(a,u(*))) + a/?fcd 5 (P,t4,Mt*(*),y)
< a( 7 7 + <5 +
/3k)d5{P,idx)p2(u{x),y).
Hence It follows that P(x,y) = x. The mapping P ' , where P'(rr,y) = p(g(x,y),y), l also satisfies the functional equa tion. -P'(#(z,y)) = p{q(R(xyy)),g(q{x,y)1y))
= p{f{q{x, y), v),g(q{x>y), y)) = f{p{q{x, y),y), u(p(q(x, y), y))) = /(P'(x,y),u(P / (x,y))). Let us prove that P' £ M 5 . Indeed, Pi{P'(x,y),x)
=
Pi{p{q{x,y),y),x)
< px{p{q(x,y), y), g(x,y)) + pi(g(a;,y),x) < d5(p, idx)p2(u(q(x,y)),y)
+
d5(q,idx)p2(u(x),y).
Let us note that p2{u(q(x,y)),y)
< p2(u(x),y) + kPi{q(x,y),
x)
< p2{u(x),y) + fcd5(g, idx)p2(u(x),y). Hence d5(P',idx)
< d5(p,idx) + d5(q,idx) +
kds(p,idx)d5(q,idx).
Consequently, we have P / (a;,y) = p{q{x,y),y) = x. Step 4. Mapping Q. Let us consider the functional equation /(Q(x,y,z),z) =
Q{T{x,y),g{Q(x,y,z),z))
in the metric space M 9 . Let us consider the mapping Q t-* CQ, Q € M 9 defined by the equality f(CQ(x,y,z),z) = Q{T(x,y),g(Q(x,y,z),z)). CQ is well-defined and continuous. We have
Pi{£Q{xiy,z)i£Q(xyy,z')) < aPl{f{CQ(x,
y, z), z), /(£Q(x, y, z'), z')) + a f e ( z , z')
= aPl (Q(T(x, y),g(Q(x, y, z), z)), Q(T(x, y), $(Q(x, y, z'), z'))) + « t e ( ^ *')
Andrejs Reinfelds
292 < (al(
< 1Px{Q{x,y,z),x)
+ 6p2(y,z)
< -yd9(Q,ids) max(p2{u{x),y),p2{y, z)) + 6p2(yi z). Therefore Pi{CQ{x,y,z),x)
<
api(f(£Q{x,y,z),z)J(x,z))
< aPl{Q{T{x,y),g{Q(x,y,z),z)), < ad9(Q, idx)max{p2{u(f{x,
/(x,y)) + aPl(f(x,y),
y)),g(x,y)),
/(x,z))
p2(g{Q{x, y, z), z),g(xy y)))
+ 0 ^ 2 ( 1 / , *)•
It follows that d9(CQ, idx) < a(j max (/, d9(Q, idx)) + 6)d9{Q, idx) + a/?. UQe
M9(Z), then <*9(£Q, «*,) < aKll + 6) + a(3 = l
Consequently CQ G M9(l). Let Q G M9(Z) and Q' G M 9 . We have Pi(CQ(x,y}z),CQ'(x,y,z))
< aPl(f(CQ(x,y,z),z),
f(CQ'(x,y,
z),z))
< aPl(Q(T(x,y),g(Q(x,y,z),z)),Q'(r(:n,y), $(Q(s,y,z),z))) +a7/ioi(Q(a;, y, z), <2'(x, y, z)). Hence <29(£Q, £Q') < (a( 7 / + £) + ail)d9(Q, Q') = \d9{Q, Q'). It follows that there is a unique solution Q G M 9 (/) of the functional equation /(<2(x,y,z),z) =
Q{T{x,y),g{Q{x,y,z),z)).
In addition this solution is unique also in M 9 . The mapping Q', where Q'(x,y,z) = g(p(x,y),z) also satisfies the functional equation /(Q'(x, y, z)}z) = /(?(p(x, y), z), z)
= q{R{p{x, y), *)) = q{p{T{x, y)),g{q(p{x, y),z), z)) =
Q\T(x,y),g{Q'(x,y,z),z)),
and the Lipschitz conditions Pi(Q\x,y,z),
Q'(x,y, z')) = />I(?(P(Z, y), z),q(p(x,y),z'))
< lp2(z, z').
Besides Pi{Q'{x, y, z),x) < pi(q(p{x, y), z),p(x, y)) + pi{p{x, y), x)
T i e Reduction of Discrete Dynamical Systems in Metric Spaces < ds(q,idx)p2(u(p(x,y)),
z) +
293
d5(p,idx)p2{u(x),y)
p2(u(p(x,y)),z)
< p2{y, z) + p2(y, u{x)) +
MQ'iM*)
< 2d5(q,idx) + kd5(q,idx)d5(p,idx)
We have that Q' G M 9 (/) and, consequently, Q(x,y,z) verified that Q(x,y,y) = x. Therefore
kds(p,idx)p2{u(x),y) +
ds(p,idx).
= q(p(x,y),z).
It is easily
q{p(x,y),y) = x. We obtain that the mappings # , r : X x Y -► X x Y denned by H{x,y) = (p( , y), y) and T(x, y) = (q(x, y), y) are inverse to each other and that if is a homeo morphism establishing a conjugacy of the mappings T and R. The theorem is proven. x
D
Example 7.6 In the general case of noninvertible mappings using Theorem 7.7 we have that (1) is globally conjugate to x1 — Ax + F(x,u(x)), y1 =By + G(q(x,y),y).
8. Conjugacy of homeomorphisms. 2 We consider the case, when T is a homeomorphism without fixed points. Theorem 8.8 Let the hypotheses (Hl)-(H4) hold, and let there be a mapping v. Y —»■ X that satisfies (4) and (6). If T is a homeomorphism, then there exists a continuous mapping ^ : X x Y - » Y , which is Lipschitzian with respect to the first variable, and a homeomorphism i7: X x Y — > X x Y such that the diagram
commutes, where N(x,y) = (f{x,9(x,y)),g(v(y),y)). Proof. In four steps, we will show that there exists a homeomorphism H that establishes a topological conjugacy of T and N. Step 1. Mapping -K: See Step 2 of Theorem 6.6. Step 2. Mapping 6: Consider the closed subset M6(ifc) = {d | 9 € M6,d6(8,idy)
< k and P2(0(x,y),e(x',y))
<
kp1(x,x')}
Andrejs Reinfelds
294
of the complete metric space M 6 . Let us consider the mapping 9 *-* £9, 9 G M6(k) defined by the equality £9(f(x, 0(x, y)),g(v(y), y)) = g(x, 9(x, y)). So T is a homeomorphism and, by using Lemmas 3.1 and 5.5, we obtain that £9 is well defined and continuous. We now have p2(£9(f(x, 9(x, y)),g(v(y), y)),g(v(y), y)) = P2{g{x,9(x,y)),g{v{y),y))
< (7 +
Sk)Pl(v(y),x).
Since M v (z/)>x) < api{f(v{y),y),
/(x,0(x, y))) +
a0d6(9,idy)p1{v{y),x)
and, consequently, Pi(v(y), x) < a(l - apk)-lPl{v{g{v(y),
y)), / ( x , 0(x, y))).
It follows that d6{£9,idy) < a ( 7 + 6k)(l - afik)-1 = k. In addition, p2(£9(f(x, 9(x, y)),g(v(y), y)), £9(f(x\ = p2{g{x,0(x,y)),g(x\
9{x\ y)),g(v(y),
0(x', y))) < (7 + 6k)Pl(x,
y)))
x').
Let us note that Pl(x,x')
<
< aPl(f(x>9(x,
aPl(f(x,9(x,y))J(x',9(x,y))) /(x',9{x',y))) + afikPl(x,
x').
Hence, we have * ( « , x') < a(l - aj3k)-1p1(f(x,9(x,
y)), / ( * ' , 9{x', y))).
Therefore p2(£9(f(x, 9(x, y)), g(v(y), y)),£9(f(x\
9(x',
y)),g(v(y)}y)))
< a ( 7 + **)(1 - a W - V i ( / ( x , ^ ( x , y ) ) , / ( x ' ^ ( x ' , y ) ) ) . It follows that £9 € M6(k). We obtain p2(£9(f(x, 9(x, y)),9(v(y), y)),£9\f(x, < p2(g(x,9(x,y)),g(x,$'(x,y))) <(6 +
8(x, y)), g(v(y), y)))
+ pkp2{9{x,y), 0'(x,y))
(3k)d6(9,9')Pl(v(y),x)
The Reduction of Discrete Dynamical Systems in Metric Spaces < iidQ{9,B')Pl{v{g{v{y\y)),
295
f(x,9(x,y))).
It follows that d6{C9,C9')
=
g(x,9(x,y)),
where N{x,y) = {f{x,9{x,y)),g{v{y),y)). Step 3. Mapping II: Let us consider the functional equation Il(N(x,y))
=
g(v(Tl(x,y)),Il(x,y)).
It is easily verified that this functional equation has a solution II(x,y) = y. Let us prove the uniqueness of the solution in M6- Otherwise, there exists (x, y) and y ^ II(x,y). We get p2(n.(N{x,y)),g(v{y),y))
=
< (ll + 6)p2(n{x,y),y)
p2(g{v(IL{x,y)),Il{x,y)),g{v(y)yy)) < (1l +
< iideill, idy)p1(v(g(v(y),y)),
6)d6{Ii,idy)p1(v(y),x) / ( x , 0(x, y))).
Hence, d6(n.,idy) < p,d6{Ii,idy). It follows that n(x,y) = y. The mapping II', where II'(x, y) = 7r(x, 9(x, y)), also satisfies this functional equa tion. n W r , < / ) ) = 7r(/(x,0(x,y)),0(iV(x,y))) = g(v(ir(x, 0(x, y))), TT(X, 0(x, y))) = g{v(U\x, y)), II'(x, y)). Let us prove that II' G M6- Indeed, ft(n'(iV(x,!i)),*(y),ri) < piWfa
=
p2(ir(f(x,9(x,y)),9(N(x,y))),g(v(y),y))
9(x,y)),$(N(x,y))),9(N(x,y)))
+
p2(9(N(x,y)),g(v(y),y))
< 4(TT,id y ) Pl (v(9(N(x, y))), / ( * , 0(x, y))) + 4 ( # , idy)pi(v(g(v(y),
y)), / ( x , 0(x, y))).
Let us note that
ftM*(JV(*,y))),/(M(*,lO)) < *(«M»Mi y)), /(».*(»,»))) + W ( t f (*. y))r9(v(y), y))
296
Andrejs < (1 + lde{9,idy))Pl{v(g(v(y),y)),
Reinfelds
f(x,0(x,y))).
Hence, d6(I[',idy)
< d6(6,idy)
+ d6(ir,idy) +
ld6{9,idy)d6(Tr,idy).
Consequently, we have II'(x,y) = 7r(x,0(x,y)) = y. S t e p 4. M a p p i n g 0 : Let us consider the complete metric space Mio = { 0 | 0 : X x Y x X -> Y is continuous, p2{Q{x,y,z),y) sup \ f rr < +oo and x,y,z max (/?i(u(t/), x), pi{x, z)) p2{Q(x,y,z),G(x,y,z'))
<
kp^{z,z')}
equipped with the metric A (c\ A'\ mm P2{Q{x,y,z),ef{x,y,z)) a i o ( 0 , 0 ) = sup —-——— —, x,y,z max [pi{v[y), x), pi{x, z)) and the closed subset of this space M10(fc) = { 0 | 0 € Mio and dw(Q,idy)
< k}.
Let us consider the mapping 0 H-> £ 0 defined by the functional equation £ 0 ( T ( x , y), / ( z , 0 ( x , |f, z))) = g{z, 0 ( x , y, z)). Analogous to step 2, £ 0 is well-defined and continuous. We have P2{C
y), / ( z , 0 ( x , y,»))), CQ(T(x, y), / ( * ' , 0 ( x , y, z'))))
= />2(y(z, 0 ( x , y, z)), p(z', 0 ( x , y, z'))) < (7 + £ % i ( z , z'). Furthermore, Pl(z,z')
< a / n ( / ( z , 0 ( x , y , z ) ) , / ( z ' , 0 ( x , y , *'))) + <x(3kPl(z, z').
It follows that Pl(z,z')
< a ( l - a(3k)-lPl(f(z,
0 ( x , y , z)), / ( z ' , 0 ( x , y, z'))).
Thus £ 0 satisfies the Lipschitz condition with constant k. Let us note that Pi(x,z) < <xpi(f(x,y),
< a/)i(/(x,0(x,y,z)),/(z,0(x,y,z)))
/ ( z , 0 ( x , y , z))) + a/?d l o (0, zdy) max{ P l (v(y), x), ^ ( x , z))
and />i(«(y),a:) < a p ! ( / ( v ( y ) , y ) , / ( x , y ) )
The Reduction of Discrete Dynamical Systems in Metric Spaces < apx{v(g{x,y)),/(x,y))
297
+ a 7 / / 9 1 (v(y),x).
We obtain max(/} 1 (v(y),x),/) 1 (x,2)) < a max (/>i(/(x, y), / ( z , 0(x, y, 2))), pi(v(^(x, y)), / ( x , y))) +o;^max(dio(0,zd y ),A;)max( / o 1 (i;(y),x), i Oi(x,z)). Hence, max(/)i(v(y),x), / 9 1 (x,2)) < Q?max(/? 1 (v(g(x > y)),/(x,y)),p 1 (/(x,y),/(z,©(x,y,z)))) 1 — a/3 max (
= />2(^,0(a;,y,2)),5r(x,y)) < 7Pi(x,z) + W 1O (0,id y ) max (/Oi(u(y),x),/>i(x,2)) <
<*(7 + ^ i o ( 0 , idy)) max (^(t;(g(x, y)), /(x, y)), /?i(/(x,y), / ( z , 0(x, y, z)))) 1 — a/? max (rfio(0, idy), k)
K0GMio(fc),then
Let 0 € Mi0(A:) and 0 ' € Mi 0 . We have
p 2 (£0(T(x, y), / ( z , 0(x, y, z))), £0'(T(x, y), / ( z , 0(x, y, z)))) < /02(ff(^, 0(z,y,z)),0(z,©V.2/> z ))) +
Pkp2{Q{x,y,z),0'(x,y,z))
<*(£ + /?A;) J I Q ( 0 , 0') max (/?i(t;(g(x, y)), /(x, y)), pi(/(x, y), / ( z , 0(x, y, z)))) 1 - a/?fc Hence, d1o(£0,£0')<Mo(©,O')It follows that there is a unique solution 0 € Mio(fc) of the functional equation 0(T(x, y), / ( z , 0(x, y, z))) = p(z, 0(x, y, z)). In addition this solution is unique also in MioThe mapping 0', where 0'(x,y,z) = 0(z,7r(x,y)), also satisfies the functional equation. 0'CT(x,y),/(z,0'(x,y,z))) = 0(/(z,0(z,7r(x,y))),7r(r(x,y))) = 0{f{z, #(z, *"(*, y))),g(v(ir(x, y)), ?r(x, y)))
Andrejs Reinfelds
298 = g(z, 6{z, 7r(ar, y))) = g{z, 0'(x, y, 2)) and the Lipschitz condition P2{Q'{x,y,z),e'(x,y,z')) = p2(6{z, TT(X, y)), fl^, TT(X, y)))
P2^{xyy),y)
+ d10(ir,zdy)/>i(u(y),x)
< dlo(0, idy)(/ni(x, 2) + /»i(u(y), x) +Wio(T,*dw)/t»i(u(y),x)) +
dio(ir,idy)p1{v(y),x).
Therefore, dw(Q',idy)
< 2d10(6,idy) + ldiO(0,idy)dlo(Tr,idy) + d10(7r,idy).
It follows that 0 ' € M10, and therefore Q(x,y,z)
= 0(z,7r(x,y)). It is obvious that
0(x,y,x) = y. Therefore, 0(x,?r(x,y)) = y. We obtain that the mappings H, T:X x Y —> X x Y defined by if(x,y) = (x,7r(x,y)) and T(x,y) = (x,0(x,y)) are mutually inverse and that if is a homeomorphism estabhshing a conjugacy of the mappings T and N. The theorem is proven. □ Theorem 8.9 Let the hypotheses (H1)-(H4) hold, and let there be a homeomorphism go'.Y —> Y such that satisfies the Lipschitz condition with constant less than 1. If T is a homeomorphism, fik + 8 < 1 and sup/>2(#(a:,y),0o(y)) < +00, x,y
then there exists a homeomorphism if: X x Y — > X x Y si/c/i i/iat i/ie diagram
commutes, where Ro(x,y) = (/(x,u(x)),# 0 (y))-
(13)
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Proof. S t e p 1. Let us note that the conditions of Theorem 5.4 are fulfilled. Indeed we may choose u(x) = y 0 . According to (13), the inequality (7) holds. By applying Theorem 7.7 we obtain that there exists a continuous mapping g:X x Y —* X , Lipschitzian with respect to the second variable, such that the homeomorphism T is topologically conjugate to the homeomorphism R. S t e p 2. M a p p i n g TTO: Consider the set of mappings M i l = {no | 7r 0 :X x Y —» Y is continuous and supp 2 (no(x,y),y) < + o o } . x,y
With the supremum metric dii(7To,7To) =
supp2{Tr0{x,y),Tr'0(x,y)), x,y
M n is a complete metric space. Let us consider the mapping TT0 H-* £7r0, 7r0 € M n defined by the equality £TTO{RO(X,
y)) = g{q{x, ir0(x, y)), 7T0(x, y)).
Since Ro is a homeomorphism, Cno is continuous. Consider the estimate p2(CTr0(Ro(x, y)), £7To(i2o(a;, y))) = P2{g(q(x, 7r0(x, y)), 7T0(xy y)),g{q(x, < (-yl + 6)p2(iro{x,y),ir'0(x,y))
ir'0(x, y)), TT'0(X, y)))
< (il+
6)dn(iro1Tc,0).
Prom this, we arrive at dll(£7T0,£7ro) < (7^ + <5)^ll(7TO,7rJ). A simple consequence of condition (13) is the estimate dn{Cidy,idy)
<
x,y
supp2{g{x,y),g0(y)).
Consequently, du{£ir0,idy)
< (7/ + 6)dn(ico,idy)
+
supp2(g(x,y),g0(y)). x,y
Since 7/ + 6 is less then 1, the mapping £ is a contraction in M n and therefore there is a unique solution ir0 G M n of the functional equation ir0(Ro{x,y))
=
g(q(xiTr0(x,y))iT0(x1y)).
S t e p 3 . M a p p i n g 60: Analogous to step 2, we may prove the uniqueness of 90 6 M n satisfying the functional equation
Bo(R(*,y)) = 9o(0o(x,y)).
Andrejs Reinfelds
300 Step 4. Mapping IIo: Let us consider the functional equation U0(R(x, y)) = g{q(x,Ilo(x, y)), n 0 (x, y)).
It is easily verified that this functional equation has a solution n 0 (x,y) = y. Let us prove uniqueness of the solution in M n . Otherwise there is (x,y) and y ^ TL0(x,y). We get p2{Tlo{R{x, y)),g(q(x, y), y)) = P2{g{q{x, n 0 (x, y)), n 0 (x, y)), g{q(x, y), y)) <(7/ Hence,
+ %2(n 0 (x,y),y).
cf n (n 0 , e
It follows that n 0 (x,y) = y. The mapping IIQ, where IIo(x,y) = 7r o (x,0 o (x,y)), also satisfies this functional equation. n'0(R(x,y)) = Tr0(f(x,u{x)),6o(R(x,y))) =
7r0(f(x,u(x)),go(e0(x,y)))
= 9{q{x, M ^ d0(x, y))), TT0(X, 0 O (X, y))) = g(q(x, W0(x, y)), IIo(x, y)). Let us prove that IIQ € M n . Indeed, p2{W0{R{x,y)),g(q(x,y),y)) = P2(7ro(/(x, u(x)), Q0(R{x, y))), 0(?(x, y), y)) < ^(7r 0 (/(x, tz(x)), 0o(fl(x, y))), $0(R(x, y))) + p2(e0(R(x, y)),g(q(xy y), y)) <
dn(0o,idy).
Consequently, we have n
o(z,y) = 7ro(x,0o(x,y)) = y.
Step 5. Mapping 0 O : Analogous to Step 4, we may prove that the functional equation 0o(i2 o (x,y))= 5 f O (0 o (x,y)) has a unique solution 0 O € M n , and 0o(x,y) = 0o(x,7ro(x,y)) = y. We obtain that the mappings i7o,r 0 :X x Y -► X x Y defined by H0(x,y) = (x, w0(x, y)) and r 0 (x, y) = (x, 90(x, y)) are mutually inverse and that H0 is a homeomorphism establishing the topological conjugacy of the homeomorphisms RQ and R. The theorem is proven. Q
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301
Theorem 8.10 Let the hypotheses (Hl)-(H4) hold, and let there be a homeomorphism /o: X —* X such that f$ satisfies the Lipschitz conditions with constant less than 1. If T is a homeomorphism, a(l + 7/) < 1 and sup/9i(/(x,y),/ 0 (x)) < +00,
(14)
x,y
then there exists a homeomorphism if: X x Y — > X x Y such that the diagram T XxY
XxY
H
H No
XxY commutes, where No(x,y) =
XxY
{fo{x),g(v(y),y)).
Proof. Step 1. Let us note that the conditions of Theorem 5.5 are fulfilled. Indeed we may choose v(y) = xo- According to (14), the inequality (9) holds. By applying Theorem 8.8 we obtain that there exists a continuous mapping 0: X x Y —> Y, Lipschitzian with respect to the first variable, such that the homeomorphism T is topologically conjugate to the homeomorphism N. Step 2. Mapping po\ Consider the set of mappings M12 = {po I J>o:X x Y —> X is continuous and supPi{po(x,y),x) < +00}. With the supremum metric duipo, Po) = sup/>i(po(z,y),pi(z,y)) x,y
M 1 2 is the complete metric space. Let us consider the mapping po ^ £po, Po G M12 defined by the equality Po{N0(x,y)) =
f(Cpo(x,y),0(po(x,y),y)).
Consider the estimate pi{Cp0{x,y),Cp'Q{x,y)) < ap^fiCpoix,
y), 6(po{x, y), y)), f{Cp'0{x, y), 0(po{x, y), y)))
+
a^kp1{pti(x1y)1p'0(x,y)).
< a(l + pk)dl2{po,p'o)-
302
Andrejs
Reinfelds
A simple consequence of condition (14) is the estimate d12(Cidx,idx)
<
asupPl(f(x,y),f0(x)). x,y
Consequently, di2(Cp0,idx)
< <x(l + Pk)du(p0,idx)
+ asup
Pl(f{x,y),
f0(x)).
x,y
Since a(l + f3k) is less then 1, the mapping £ is a contraction in M12 and therefore there is a unique solution po 6 M 1 2 of the functional equation Po{NQ{x,y)) =
f{po(x,y),0{po{x,y),y)).
S t e p 3 . Mapping qo: Analogous to Step 2, we may prove the uniqueness of qo G M12 satisfying the functional equation q0{N(x,y))
=
fo{qo(x,y)).
S t e p 4. Mapping Po: Let us consider the functional equation P0(N(x,y))
= f(Po(x,y),0(Po(x,y),
y)).
It is easily verified that this functional equation has a solution Po(x,y) = x. Let us prove uniqueness of the solution in M i 2 . Otherwise, there exists (x,y) and x ^ Po{x,y). We get Pl{P0{x,y),x)
<
aPl(f{Po{x,y),e(Po(x,y),y)),f{x,0{Po(x,y),y)))
< aPl{P0(N{x,y)),f(x,6(x,y)))
+
apkPl{P0(x,y),x).
Hence, dulPoyids)
< a{l +
pk)d12{P0,idx).
It follows that Po(x,y) = x. The mapping PQ, where Po(x,y) = Po{qo(x,y),y), equation. PQ(N(X,
also satisfies the functional
y)) = Po(qo(N{x, y)), g(v(y), y)) = pb(/ 0 (ft,(*,
= f(Po(qo(x,y),y)J(po(qo(x,y),y),y))
= /(i?(*,y)f
tf(/?(*,y)lF)).
Let us prove that PQ € M12. Indeed, ft(^(*,y),*) < Pi(po{to(z,y),y),qo{x,y))
<
Pi(Po(qo(x,y),y),x)
+ Pl{q0(x,y),x)
< d12(p0,idx)
+
d12(q0,idx).
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303
Consequently, we have
Pofay) = po{qo{x,y),y) = x. Step 5. Mapping Q0: Analogous to Step 4, we may prove that the functional equation Qo{No{x,y)) = f0{Qo(x,y)) has a unique solution Q0 E Mi 2, and Qo(x,y) = qo{po{x,y)yy) = x. We obtain that the mappings i f 0 , r 0 : X x Y —► X x Y defined by H0(x,y) = {Po{x,y),y) and T0(x,y) = (qo(x,y),y) are mutually inverse and that H0 is a homeo morphism establishing the topological conjugacy of the homeomorphisms N0 and N. The theorem is proven. □ Example 8.7 Let us consider the mapping (1). The condition (13) is fulfilled if sup|G(s,y)-G(0,y)|<+oo x,y
and (14) is fulfilled if sup|F(x,y)-F(x,0)|<+oo. x,y su
Let the mapping (1) be a homeomorphism and let B be invertible, Vx,y\G(x,y)\ < +00 and 1-111-1
e<<
-\\B\
if||A -111-1
4
(IIA-l^-lXl-ll* \A-^-\\B\
,
||JB||
< 1,
+ IIB||<2
if||A- 1 ||- 1 + | | 5 | | > 2 .
Using Theorem 8.9 we obtain that homeomorphism (1) is topologically conjugate to x1 = Ax + F(x,u(i)), yl = By. Now suppose that ||A _1 || < 1, sup
|.F(a:,y)| < +00 and
r (HA-ir 1 - i)(i - \\B\ , if IIA-1-'+ ||B|| < 2 IIA-H-' - ||B|| £ < <
A-MI-'-IIBI
if ||A-'||-' + ||B|| > 2.
By Theorem 8.10 homeomorphism (1) is topologically conjugate to yl = By + G{v{y),y).
Andrejs Reinfelds
304 9. Application to the stability theory
We prove that the asymptotic behavior of a semidynamical system generated by a continuous mapping T : X x Y - » X x Y , where T{x,y) =
(f(x,y),g{x,y))
is determined by the reduced semidynamical system [4-5, 15, 37, 43, 58, 62, 64] generated by -a continuous mapping
f(x,u{x)).
Theorem 9.11 Let the hypotheses (Hl)-(H4) hold and let the mapping T have a fixed point T(x0,y0) = (x0ly0). Then for any (x,y) € X x Y there exists a £ <E X such that as n —> +oo, Pi{xn,x0) < h{k/3 + 8)n(p2{y,yo) + kPl{x,x0))
+ pi{C,*o),
P2{yn,yo) < (1 + Wi)(*j9 + t)n{p2{y,yo) + *Pi(*,*o)) + **(F.*o) and pi{t,xo) < hP2(y,yo) + (l +
M1)pi{x1x0),
where Tn(x1y) = (r(x,y)ign(x,y))
=
(xniyn)
is n-th iterate ofT and I
al3
Proof. According to Theorem 5.3, there exists a mapping u : X —> Y satisfying (3), (5) and u(x0) = yo- Prom (11) using mathematical induction, we obtain ftWf(M)),5n(^))
< (W + 6)nP2{yM*))
or />2(yn,W(xn)) < (kp +
6)np2(y,u(x)).
Put £ = p(x, y), where p is the mapping defined in Theorem 7.7. Then
? = f(ZA0) = f(p(x,y)Ap(x,y))) = p(f(x,y),g(x,y))
= p{xx ,y 1 ).
Using mathematical induction, we obtain £n = p(xn,yn).
Let us note that
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305
Therefore Pi(xn,C) = < d5(p,idx)p2{yn,u{xn))
Pi(xn,p(xn,yn)) < /i(fc/3 + 5 ) > ( y , u ( x ) ) .
We get Pi(xn,x0)
< Pi(xn,C)
+ +
Pi(C,x0) p1(Cx0)
and P2(yn,yo) < P2{yn,u{xn))
+
p2(u(xn),y0)
< (1 + kh)(kp + 6)np2{y, u{x)) + kPl(C,
x0).
Let us note that p2(y,u(x))
< p2{y,yo) + p2(u(x),u(x0))
< p2{y,y0) + fc/>i(x,x0)
and PiK,aJo) i(x,x0) < hp2(y,yQ) + (1 + H x )pi(a;,x a ). Thus the theorem is proved.
□
Corollary 9.2 If j3k + 6 < 1 and xo is a stable fixed point of f, so (xo,yo) is a stable fixed point ofT. If j3k + 6 < 1 and xo is an asymptotically stable fixed point of if, so {xo,yo) i* an asymptotically stable fixed point ofT. Example 9.8 Let us consider the mapping (1) and let According to Theorem 9.11 we get the estimates \xa\
JF(0,0)
= 0, (2(0,0) = 0.
+ 6)n(\y\ + k\x\) + \ n
\yn\ < (1 + Wi)(*/J + S)n{\y\ + k\x\) + k\C\t where and \t\ < h\y\ + (I + kh)\xl 10. Shadowing lemma We shall prove that the mapping T has a shadowing property [1, 3, 17, 35, 48-49]. Definition 10.10 A sequence {xn,yn}nez
is an orbit of T if
( x " + 1 , ^ + 1 ) = T(x",y") for all n 6 Z.
Andrejs Reinfelds
306
Definition 10.11 A sequence {Cn>*/n}n€Z is a A-pseudo-orbit of T if
max{/)1(/(C",^),Cn+1),P2U(Cn,'?n)^n+1)} < A for all n 6 Z. Definition 10.12 A mapping T is said to have the shadowing property if for every e > 0 there exists A > 0 such that any A-pseudo-orbit {Cn>»7n}nez is e-traced by some genuine orbit {£n,yn}nez> i-e. max{p1(xn,(n),p2(yn,T,n)}<e for all n € Z. Theorem 10.12 Let the hypotheses (H1)-(H4) hold and let (1 - a)(l - 6) > apf. Then the mapping T has the shadowing property. Proof. Let { C i ^ l n e z be a A-pseudo-orbit and let q ( l - * + /?) (l-a){l-6)-a^ and "
=
1 — a + cry (1 - a)(l - 6) - a/? 7 '
We next introduce the complete metric spaces E and F by E = {x | x = {xn}nez,
sup/*(*", C") < *} n
and
F = {y | y = {yn}„€z, suPP2(yn,vn)
< u}
n
with metrics r!(x,X!) = sup{/0i(x n ,x?)} n
and ^ ( y , y i ) = sup{/?2(i/n,2/1n)} n
respectively. Let us note that if (1—a)(l—6) > a/?7, then /?&+£ < 1 and 0(1+7/) < 1Let us consider the sequence of functional equations pn+1(g(pn(wn),wn))
=
f(pn(wn),wn)
in the complete metric space N = {p I p : F -* E and n{p{w),p{w1))
< Zr2(w,Wi)}
The Reduction of Discrete Dynamical Systems in Metric Spaces
307
equipped with the metrics # ( P , P i ) = supri(p(w),pi(w)). w
Let us consider the mapping p H* £ p , p € N defined by the equality
Pn+1(g(pn(wn),wn))
=
f(£Pn(wn),w").
We have
MA>nK), W K ) ) < a/h(W(iB B )/),/WK),t» n )) < aprip^^gip^w^w^p^igip^w-)^)))
+
aj3p2{w\wnx)
+ (7J + S)p2(wniw^))
+
a0pa(vr9vf)
< a ( l + 70^(P»Pi) + W r f + *) + <*P)p2(wn, u>?) < a ( l + 7')-D(P»Pi) + ^ ( w , w i ) . It follows that D(£p,£p1)
< aPl(pn+1(g(pn(wn),wn)),C+1)
+ ^i(Cn+1,/(Cn,^)) +
ctfaWiW")
< aD(p, 0 + a A + a/3r2(7?, w) < a/e + aA + ctfii/ = K. It follows that £p G N. The Banach contraction principle completes the first part of proof. Let us consider the sequence of equalities qn+1=g(pn(qn),qn)Let us consider the mapping q ■-»■ £q, q € F defined by the equality Cq1^1 =
g(pn(qnUn)-
We have P2{Cqn+\Cq?1)
=
P2(9(pn(qnUn),9(pn(tiU?))
< (jl + S)p2(qn, rf) < (7/ + ^)r 2 (q, q j . It follows that r2(£q,£qi) <
(ll-\-S)r2(qyq1).
308
Andrejs Reinfelds
Let us note that P2(Cqn+\Vn+1)
< P2(9(pn(qn,,
+
P2(rin+\9(C,Vn))
< lPi(pn(qn),C) + Sp2(qn,vn) + P2(vn+\g(C,n")) < 7#(p,C) + Sr2{q,r)) + A < 7« +
{pn(qnUnUz, where max{a(l-8 + jg)tl-a + a07A
(1 - a)(l - 8) - afo
D
Remark 10.2 The Theorem 10.12 remains valid in the case of a positive orbit and a positive A-pseudo-orbit. Example 10.9 Let us consider the mapping (1). Using Theorem 10.12 we obtain that the mapping (1) has the shadowing property if €<
and
^
(HA-j^-ixi- nan) HA-l-i - ||fl||
maxKIIA- 1 !!-|-!),(l-11*11)} A e 1 A -(||A-1||- -l)(l-|™-£(||A-1||-1-||5||) -
11. References 1. E. Akin, The general topology of dynamical systems, Grad. Stud. Math., vol.1, Amer. Math. Soc, Providence, 1993. 2. D. V. Anosov, Many-dimeniional analog of Hadamard's theorem, Nauchn. Dokl. Vysh. Shk. Ser. Fiz.-Mat. Nauk (1959), no. 1, 3-12 (Russian). 3. D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967). 4. B. Aulbach, A reduction principle for nonautonomous differential equations, Arch. Mat. 39 (1982), no. 3, 217-232. 5. B. Aulbach, Continuous and discrete dynamics near manifolds of equilibria, Lecture Notes in Math., vol. 1058, Springer, Berlin, 1984. 6. B. Aulbach and B. M. Garay, Linearization and decoupling of dynamical and semidynamical systems, in Differential Equations (Plovdiv, 1991), World Sci Publishing, Singapore, 1992, pp. 15-27.
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of Discrete Dynamical Systems in Metric Spaces
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7. B. Aulbach and B. M. Garay, Linearizing the expanding part of noninvertible mappings, J. Appl. Math. Phys. 44 (1993), no. 3, 469-494. 8. B. Aulbach and B. M. Garay, Partial linearization for noninvertible mappings, J. Appl. Math. Phys. 45 (1994), 505-542. 9. N. N. Bogolyubov, On some statistical methods in mathematical physics, AN USSR, L'vov, 1945 (Russian). 10. N. N. Bogolyubov and Ju. A. Mitropol'skii, Asymptotic methods in the theory of nonlinear oscillations, Gordon and Breach, New York, 1962. 11. P. Bohl, Uber die Bewegung eines mechanischen Systems in der Nahe einer Gleichgewichtslage, J. Reine Angew. Math. 127 (1904), no. 3-4, 179-276. 12. M. A. Boudourides, Topological equivalence of monotone nonlinear nonautonomous differential equations, Portugal. Math. 41 (1982), no. 1-4, 287-294. 13. M. A. Boudourides, Hyperbolic Lipschitz homeomorphisms and flows, in EquadiffS2, Proceedings 82, Lecture Notes in Math., vol. 1017, 1983, pp. 101-106. 14. I. U. Bronshtein and V. A. Glavan, The Grobman-Hartman theorem for ex tensions of dynamical systems, Differential Equations 14 (1978), no. 8, 10711073. 15. J. Carr, Applications of centre manifold theory, Springer, Berlin, 1981. 16. E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New-York, 1955. 17. B. A. Coomes, H. Kocak and K. J. Palmer, A shadowing theorem for ordinary differential equations, J. Appl. Math. Phys. 45 (1994). 18. E. Cotton, Sur les solutions asymptotiques des equations differentielles, Ann. Sci. Ecole Norm. Sup. (3) 28 (1911), 473-521. 19. N. Fenishel, Persistence and smoothness of invariant manifolds and flows, In diana Univ. Math. J. 21 (1971/72), no. 3, 193-226. 20. D. M. Grobman, Homeomorphisms of systems of differential equations, Dokl. Akad. Nauk SSSR 128 (1959), no. 5, 880-881 (Russian). 21. D. M. Grobman, Topological classification of neighborhoods of a singularity in n-space, Mat. Sb. (N.S.) 56(98) (1962), no. 1, 77-94 (Russian). 22. D. M. Grobman, Global topological equivalence of systems of differential equa tions, Mat. Sb. (N.S.) 73(115) (1967), no. 4, 600-609 (Russian). 23. D. M. Grobman, Homeomorphy of dynamical systems, Differentsial'nye Uravneniya 5 (1969), no. 8, 1351-1359 (Russian). 24. J. Hadamard, Sur I'iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. Prance 29 (1901), 224-228. 25. J. K. Hale, Integral manifolds of perturbed differential systems, Ann. of Math. (2) 73 (1961), no. 3, 496-531. 26. J. K. Hale, Oscillations in nonlinear systems, McGraw-Hill, New York, 1963. 27. P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2) 5 (1960), no. 2, 220-241. 28. P. Hartman, A lemma in the theory of structural stability of differential equa tions, Proc. Amer. Math. Soc. 11 (1960), no. 4, 610-620. 29. P. Hartman, On the local linearization of differential equations, Proc. Amer.
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30. 31. 32:
33. 34. 35. 36. 37. 38. 39. 40.
41. 42. 43. 44.
45. 46.
47. 48. 49.
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