Peyresq lectures on nonlinear phenomena : [lectures of two summer schools, the INLN Summer Schools on Nonlinear Phenomena, in June 1998 and June 1999]

6H _=-dtV 5n(x)

(2.8)

where the Hamiltonian H is given by (6 is the step function) H=1-jdxdzO(r,-

*)(W>)2 + \gjdxr?

(2.9)

One might find puzzling that H as defined in (2.9) depends on tp, which is a function of x and z, while Tp is a function of x only. We are going to show that, provided we restrict ourselves to harmonic velocity potentials

= 0, then H can be written in terms of n and Tp alone. Purists might object that one should derive the condition V2

tp + 6
- z){dx
(2.10)

and, thanks to the boundary condition (2.2d) 0fo - z){dz S(n - z)(dz
(2.11)

The last terms of the previous two equations add up to zero thanks to the harmonicity of

134 M. L. Bellac

variation of H at constant r\ is thus <5flj = I dx[-v;dxr]

+ v^6ip(x,z

Following our convention (2.4), 6
T)(X))

= ri(x))

= 6ip(x), and schematically

SHI - j - SHI f <$# = = 5^ + - ^ (JT? dip If)

(2.12)

(2.13)

dT/ lyj

Using the relation (see (2.6) and (2.7)) SJp = 6ip + v;ST}

(2.14)

we may rewrite (2.13) as oti SH\

_ lr (*-

\

SH

— \ c \ SH o H

e

These equations allow us to identify the partial derivatives which we need SH\ SH SH\ _SH\ 1 x i x IEW = SXp dtp ii} or)

0L

=

-7=\ =-vx

dxr) + vz = dtv

, , (2.16)

(2.17)

d(p It,

while the second equation in (2.16) and Bernoulli's equation (2.2b) lead to -Srj\-=

2^^2

+91-^^

= -9t
(2.18)

where we have also made use of (2.6) and of the relation g^6(r)(x)

-z)

= S(u - x)S(r,(x) - z)

Thus (2.8) is equivalent to (2.2b) and (2.2c). It is also clear that one should be able to rewrite the Hamiltonian (2.9) by using only variables which are taken on the surface of the fluid. In doing this, one relies on the harmonicity of the potential, on the periodic conditions in x and on the boundary condition (2.2c); one could not obtain this simplification with an arbitrary shape of the bottom of the tank! In order to transform (2.9), we integrate by parts once more 9* [0(V ~ *)(
An Introduction to Zakharov Theory of Weak Turbulence 135 dz [6(r, - z)(
and, taking into account V2y? — 0, we arrive at the final form of H 2

(2.19) = \ JdxJp(x,t)[v:

- Vx~dxr)J +

\g$dxrj

However one should not be misled by the apparent simplicity of (2.19)! A naive interpretation of (2.19) would lead to

This result is wrong by a factor of 1/2, because a dependence with respect to Tp is also hidden in t>7 and wj! However it would be extremely difficult to take functional derivatives on (2.19), and it seems necessary to follow the approach described above. Let us finally write the full Hamiltonian with two horizontal variables, and including surface tension H = \p j d 2 r W,

t) (d£

- VXT? • V l ^ ) + l-gp j d 2 r

2 V

+ \a j d 2 r

(V±V)2

(2.20) where Vx = (dx,dv) is the horizontal gradient, a the surface tension and we have reestablished the volumic mass p. Actually we have used an approximate form of the potential energy V„ associated with surface tension, which will, however, be sufficient for our purposes. The exact expression reads

V, = ^ | d ' r [ > / l + (Vx.7)2-l] 2.3

The pertubative expansion

Our goal is now to expand the Hamiltonian in powers of the small parameter krj, where k is the wave vector. We are going to write Fourier transforms of Tp and r), where of course k is not limited. We have to assume that there exists an effective ultraviolet cutoff on k: the existence of such a cutoff is a very natural one. Indeed, we are working in the ideal fluid approximation, where dissipation is neglected. As shown for example in [7], the dissipation rate can be estimated to be 2uk2, where v is the kinematical viscosity. Thus dissipation occurs at large wave-vectors, or at small scales, and our approximation of an ideal fluid breaks down at large values of k. Let us write the Fourier expansion of the velocity potential

(r, *) = £ > ( * ) e*V*' ?

(2.21)

136 M. L. Bellac

The form of

(2.22)

k

We expand (2.22) in powers of kt) = EjE, V(*i) (l + *H?(*0 + • ■ •) ei -dr

V&

" i-r = E*„* 2 (f(ki) +fciv(fci)r?(fc2)ei£2f+ • • -)e«*

(2.23)

where we have used the Fourier expansion of 77

We can now obtain the Fourier components of Tp =£/d2re-'*->(r>

m

We have been working consistently to order kr}, and in the spirit of a perturbative expansion, we may replace tp by Tp in the last term of (2.24) and invert this equation ¥>(*) =
+ °(fcl?)2

kMM&Wh)

(2-25)

Now let ip(r,z) = dzip(f,z), so that from (2.21) W, z) = Y, V(k)kekze*f

(2.26)

k

Expanding once more exp(kr)) in powers of krj and using (2.25) and (2.26) we obtain tp(k) on the fluid surface ?(£) = k
% , + & [*i - * • *i]?(*i Mb) + 0(kV)2

(2.27)

*1,*2

We may now plug (2.27) into the expression (2.20) of the Hamiltonian and use Parseval's theorem j A2rTp{f) ^(f) = L2 Y^tf)

VK-*)

(2.28)

An Introduction to Zakharov Theory of Weak Turbulence 137

or generalisations in order to obtain the perturbative expansion, which will be detailed in section 3. In view of the factor of L2 in (2.28), it is convenient to work with the Hamiltonian density V. = H/L2; this Hamiltonian density is expanded in powers of krj H = H0 + Hi+H2

+ ---

(2.29)

where Hn is of order (^77)". The preceding computations allow us to get the first two terms in (2.29), but it is clear, that, at least in principle, the same strategy would allow us to obtain terms of arbitrarily high order in n, although, in practice, calculations become more and more cumbersome when n increases. 3

The normal form of the Hamiltonian

3.1

y.o=sum ofharmonic

oscillators

Let us now write the explicit expression of %0 from (2.20), (2.27) and (2.28) . Since all variables are from now on taken on the fluid surface, we suppress the bar over tp and %l>

Ho = \pY,{
fflv-i)

(3-1)

This expression is easily transformed into a sum of harmonic oscillators through a generalisation of (1.6) (3.2)

Jp[X-lT}_-k-iXk}

a'i = with Xk =

\J2(^kT)

^*%*

3

(3-3)

The inverse formulae are y/pVi = Xk(ai

+ a'_i)

yfpn

= - i X ^

- a*_s)

(3.4)

A simple calculation gives 7i0 in the standard form

K0 = X! w * a £ a *

(3-5)

138 M. L. Bellac

In quantum mechanics , o^ and a*- would be replaced by operators o^ and a t obeying canonical comutation relations

Of course, nobody is thinking seriously of quantising water waves, but quan­ tum mechanics is sometimes more convenient in order to check dimensional analysis. The commutation relations (3.6) are equivalent to the equal time commutation relations of the canonically conjugate fields 77 and pip {r,(f,t),Mr'J)} = ih6™(r- f') (3.7) The dispersion law (3.3) gives the group velocity cg(k)

+3l

^=t'U° f)

<38)

The limiting cases are (i) Gravity waves for small values of k: uk = y/gk (ii) Capillary waves for large values of k

(3.9)

uk = J°-pk3'2

(3.10)

For water waves, with the numerical values in MKS units: a = 7.4 x 1 0 - 2 , p = 103 and g = 10, one checks (see Fig. 2) that cg goes through a minimum in a region of k where gravity and capillarity are equally important; in terms of wave-lengths, this minimum is found at A = 4.33 cm, corresponding to a group velocity c9 = 17.9cm.s - 1 . Of course, we have recovered in (3.3), (3.9) and (3.10) results which are derived in an elementary and direct manner from Euler's equations in standard textbooks on fluid mechanics [4]. 3.2

Nonlinear terms: three wave interactions

The expression (2.27) of tpk gives at once the cubic terms of the Hamiltonian (of course, one must not forget the second term in (2.20))

An Introduction to Zakharov Theory of Weak Turbulence 139

(remember that the bars on

(3.12)

+ 1 £*„&.& ( ^ 2 3 «i02a 3 + c.c)* £ i + £ a + g 5 i 0 where we have adopted the shorthand notation ai = a^ etc. Now comes an important point: Hi in (3.12) can be simplified thanks to a canonical transformation which will be given explicitly later on (eq. (3.18)). Without going through all the machinery of canonical transformations, let us give an intuitive argument for this simplification: a^(t) contains a rapidly oscillating factor exp(-io)jfei). Let us get rid of this factor by defining a slowly varying

o s (t) = ^(tje-****

(3.13)

The equation of motion for ag(t) is (we neglect the term proportional to U in (3.12): this will be justified later on) idtdj- = ukan + £ * l i & [ | V i i 2 0 £ i o f e *:,fcl+fc2 Wi

(3.14)

so that l+*2

+V*k2e^-«*-»)tA-kiAlSiiU~k2]

(3.15)

If Aw = Uk — wi — u)^ and ACJ' = u>i — Wk — &2 do not vanish, then the oscillating factor in the integrand will lead to dtA^ ~ 0: then the cubic terms will be unimportant. On the contrary, if the so-called "decay condition" k = k\ + %2

u>k= ojki + Uk2

(3.16)

may be realised, or, in other words, if the dispersion law is of the "decay type", we have a resonance condition. This is the well-known "resonance problem", or "small divisor problem" of classical mechanics, which has been known since Poincare. If the resonance condition is realised, one cannot eliminate the resonant terms through a canonical transformation. We shall thus have to distinguish between two completely different cases

140 M. L. Bellac

1.8" ~ 1.6 1.41.2 C9

capilL

gravity

10.80.60.40.2 o—

101

102

103

104

k (m"1) Figure 2. Dispersion law for deep-water waves

(i) The dispersion law is compatible with (3.16): it is then of the decay type. Physically it means that a wave can decay into two other waves while conserving wave vector and frequency (in the corresponding quan­ tum problem, it means that an elementary excitation can decay into two elementary excitations while conserving energy and momentum). In that case the cubic terms cannot be eliminated by a canonical transformation and they give the leading contribution, being of first order in krj. Note that the terms proportional to U in (3.12) are always non resonant and can always be eliminated: we shall not write them explicitly in what follows. (ii) The dispersion law is of the non-decay type. Then, barring accidental cancellations (see section 3.5), elastic scattering of two waves is always

An Introduction to Zakharov Theory of Weak Turbulence

141

Figure 3. Kinematics of the decay case

The two surfaces u = ka with origins O and Ox intersect along the 0\A curve or a > 1. possible k\ + ki=k~2 + ki

ukx + <^*j = w*s + u>*4

(3-17)

and the leading term will come from four wave interactions and will be of order (kr))2 It is easy to see (Fig. 3) that in the case of a power like dispersion law of ,he kind u/* oc ka, the decay case corresponds to a > 1 and the non-decay case ; o a < l , except in the one dimensional case which is always of the non-decay ype. As a simple example, let us assume a decay law u;* = ck2 (shallow water :apillary waves). Then the angle 9 between k and ki is fixed by cos0 = ki/k. Deep water gravity waves (w* = v ^ ) are of the non-decay type: four wave nteractions (K2) give the leading contribution in perturbation theory. On

142 M. L. Bellac

the other hand deep water capillary waves wk = s/ajpk3/2) are of the decay type and the cubic terms (Hi) are dominant. To summarize: in the decay case (e.g. deep water capillary waves), the cubic terms are dominant and the terms proportional to U in (3.12) may be eliminated through a canonical transformation a^ -+ b%. In the next two sections we shall be interested in the "resonant manifold" (3.16) only, so that, for notational simplicity, we shall keep a^ instead of 6£ in the equations. However, if one wishes to preserve the Hamiltonian structure, it is compulsory to take properly into account the canonical transformation [9]. In the nondecay case (e.g. deep water gravity waves), all cubic terms may be eliminated, and the dominant term is % • 3.3

Nonlinear terms: four wave interactions

In the non-decay case, the resonant manifold is defined by (3.17); this manifold is of codimension 3 in an 8-dimensional space. In order to eliminate the cubic terms, we write the canonical transformation ar -► br K

4 1 ' = £«..«. TU(i^S

K

*-Ai* - 2E*. A ri!!«A*iWs+*, A (3-18)

with a- = 6jj. The transformation is canonical (namely it conserves Poisson brackets) provided p(2) _ p(2) _ p{2) *AA fci,£A £3.£.£i and the elimination of cubic terms gives

It is easy to check that this choice allows one to eliminate the cubic terms. It would seem that one could stop at this point, but it is necessary to be more careful for two reasons (i) One would like to keep in H2 only terms of the form 6J&26364, which have as many powers of 6 as of 6*, and to eliminate terms like 6i&2&3&4 or &J&26364 (ii) One would like to preserve the Hamiltonian structure of the theory.

An Introduction to Zakharov Theory of Weak Turbulence

143

It can be shown (but the calculation is cumbersome [10]) that both conditions may be realised, provided one takes into account terms of order ak21 (namely cubic in b) in (3.18). However (3.18) limited to order ak1 gives correct results provided one restricts oneself to the resonant manifold (3.17). In the non-decay case, a long but straightforward calculation gives the form of N2

7l2

= 2 p

k2)k2,ks,k4 =

Ek 1,k2,k3,k4 ^k1Wkz^ksqk4Lk 1)k2,k3,

kv4ak1+k2 +k3 + k4,0

1klk2[-2k1-2k2+Ik1+k3l+Ikl+k'4l+Ik2 4

+k31 +^k'2+k4t,

(3.20) Then (3.20) is written from (3.4) in terms of harmonic oscillator variables and the final form is obtained through the canonical transformation (3.18) W2 = 4 Ek1k2,k3,k4 Tkik2,k-3,k4bib2bsb4 Vk 1 +k2.k1,k2 Vk3+k4 ks,k4

Tk1-2 ,k3,k4

= Wk1k2,

3,k4 +

(3.21) +p erm.

Wk1+Wk2- W91+92

where V is defined in (3.12) and W is the coefficient of a*, a2a3a4 in the expression of N2. The second term in the last line of (3.21) is reminiscent of second order perturbation theory in quantum mechanics, and it can be actually represented graphically as a Feynman graph (Fig. 4). The full expressions of V, W and T are rather cumbersome and we refer to the literature for their explicit form [3,8,10]. Expression (3.21) is valid on the resonant manifold (3.17) only, but that is all that will be needed in the next section.

(0 1+ w2= 033 + w4

2'a) 2 Figure 4 . Feynman graphs

The Hamiltonian density N2 in (3 . 21) conserves the wave action N = > btbk k

(3.22)

144

M. L. Bellac

because there are as many 6*s as bs in (3.21). However this conservation law is only approximate, since (3.21) is valid on the resonant manifold only. This could be corrected by using a canonical transformation including cubic terms, but even with this improvement, terms of order (kr))3 contributing to Tiz would spoil the conservation law. 3.4 Dimensional analysis and scaling laws Comparing (3.5) and (3.12), we see that the dimension of Va is the same as that of w: [u] = [Va\; as [a] = M 1 /'^" 1 / 2 (see (3.2) or (3.6)), the dimension of Vis [V] = M-^T-W

(3.23)

while the dimension of W is [W] = M- 1

(3.24)

Let us first consider deep water capillary waves: the only dimensionful quan­ tities at our disposal are p, a and k, and dimensional analysis implies foc(-^r)

(3.25)

The same result can of course be obtained from the explicit expression of the three wave interaction in (3.12) and the dimension of Xu in (3.3): Xt <x pl/4CT-l/4fc-l/4

For deep water gravity waves, the dimensionful quantities are g, p and k, so that dimensional analysis gives W oc p~lk3

(3.26)

Note that dimensional analysis is useless in the case of shallow water waves, because there is one more dimensionful quantity, namely the depth h. Then one must use the explicit expressions (see section 3.5). An interesting example is that of shallow water capillary waves, where

V

^ = iipdl,ik2

< 3 - 27 )

The power laws (3.25-27) are called scaling laws; they are valid provided one type of wave is dominant.

An Introduction to Zakharov Theory of Weak Turbulence 145

3.5 Miscellaneous remarks We conclude this section with some remarks.

(1) Shallow water. When the depth h is finite, the expression of wk and of the nonlinear terms are modified. One finds, for example (3.28)

wk = (gk + o' k3) tanh(kh) P

and Lk',)k2,ky

= - (ki • kz + klk2 tanh(kih) tanh(k2h))

(3.29)

One can immediately take the shallow water limit kh << 1 in these expressions. (2) Deep water gravity waves in D = 1 . There are two trivial solutions to (3.17), but there is also a non trivial one kl = all+() z k2 = all

+()2(2

k3 = -acz

k4 = all+(+(2)

2 (3.30)

where a is a dimensionful parameter and 0 < ( < 1. This equation gives the non trivial resonant manifold . It was shown recently that ?12 is zero on this non trivial resonant manifold [ 11]. This means that N2 may be eliminated through a canonical transformation . Should the process go on, it would mean that one dimensional gravity waves in deep water form an integrable system. "Unfortunately" N3 # 0 [12] , and integrability is valid only up to quartic terms. Of course, the four wave interaction is non zero on the trivial manifold, but this does not affect integrability , because R is of the form (in quantum mechanics)

^l = ^wkbkbk + Tk^,^(bkbk ,)(bt2bk2) k ki,k2

(3.31)

and the eigenvectors are obvious in Fock space. (3) Continuum normalisation . In the next two sections , we shall follow Zakharov' s conventions and use a continuum normalisation . Let us recall that L2

(27r)2

f dzk

(3 . 32)

kk The convention for Fourier transforms is

dzr -ik L2 tp ( r = 27r `Pk `p( k ) = f 27r e

(3 . 33)

146

M. L. Bellac

Similarly £2

o(fc) = ^aE

(3.34)

so that the dimension of a{k) is now A f 1 / 2 L 2 T - 1 / 2 . The free Hamiltonian (and not the Hamiltonian density) Ho is now H0 = fd2kuj(k)a*(k)a(k)

(3.35)

while Hi becomes, instead of (3.12) /f1 =

iy - d 2 fc 1 d 2 M 2 M (2 Hfci-fc2-4)[nfci,* 2 ,A3)a*(fc 1 )o(Jfc 2 )a(fc 3 )+c.c.] (3.36)

where *a&,k.*3) = ^ - V S j i A A 2TT

(3.37)

Similarly for the four wave interaction T(ki,h,hM = Q \

A A A

(3.38)

The equations of motion are idta(k,t) = —-=da*(fc)

(3.39)

where the RHS is now a functional derivative. 4

Kinetic equations

Up to now, we have been dealing with a pure dynamical problem. We are now turning to a statistical description, because we are interested in a situation where there is a large number of excited waves. Then we are not going to ask for a detailed information on all amplitudes and phases of the different modes, but we shall limit ourselves to the knowledge of a probability distribution. The main object of interest will be n(k,t), the number density of waves in wave-vector space. The time evolution of n(k, t) is called a kinetic equation.

An Introduction to Zakharov Theory of Weak Turbulence

147

4.1 Derivation of the kinetic equations The decay and non-decay cases turn out to be governed by somewhat different kinetic equations. We shall treat in some detail the latter case , which turns out to be simpler, and will quote only the results for the first case. We recall the equations of motion

i8a(k) - Wka(k) + 671,

at

Sa * (k)

(4.1)

with (see (3.36))) r

'H1 =

2

J

1 + c.c.

L Vi23 ala2a3

j 6(2)(kl - k2 - ks)d2kid2k2d2k3

(4.2)

The non resonant U-term in (3.12) has been neglected; it can be eliminated through a canonical transformation, and in any case it does not play any role on the resonant manifold (3.16). We introduce an averaging procedure consistent with the dynamics: this is possible because there exists an invariant measure (Liouville measure) on phase space. To order zero of perturbation theory, we shall make two basic assumptions for the correlation functions (a(k1)...a(kn)):

(i) Random phase approximation=RPA (ii) Gaussian statistics: all cumulants of order > 3 vanish. These are standard physicits' assumptions, but no detailed justification is given by Zakharov (the justification is postponed to the second volume of the book [31, which is still unpublished ...). The random phase approximations means that

(a(k))(0) = (Ia (k)I

ei° (k))(° = 0

because the phase averages out to zero; the superscript (0) means "order zero of perturbation theory ". Thus we have, for example, (a(k)a(k'))(°) = 0

(a(k)a*(k'))( °) = n(k)6(2) (k - k') (4.3) (a1*a*a3a4 )(°) = n(k1)n( k2) [5(2 )(k1 - k3 )8(2)(k2 - k4) 1 +J(2) (k1 - k4 )b(2) (k2 - k3) I

where we have used in the last equation the assumption of Gaussian statistics, which allows us to express a correlation function of order four in terms of

148 M. L. BMac correlation functions of order two; note that with the conventions of section 3.5, n has dimension of Planck's constant, ML2T~l. It must be emphasized that these equations will be modified by pertubative corrections: for example, the correlation function of order three vanishes to zeroth order, but will get non-zero corrections from non linear interactions. Taking these assumptions into account, we now derive the kinetic equations. The equation of motion for a(k) is idta(k) -u) k a(k) = / \kVki2aia26(k - ki - k2) J - - - l o o <4-4> 2 2 +Vl\2a1a^S(kl - k - Jk2)jd Jfcid *2 In ojder to derive the expression of dtn, we multiply the previous equation by o'(fc), write the equation of motion for a*(k), multiply it by o(jfc), and finally subtract the second equation from the first with the result dtn(k) = Im( f\vkl2JknS(k J

- k, - k2) -

-W^Jyk25{kX

-

-

-k-

1 o

o

\

<4'5)

k2)\&kl
where Jiiz6(k Now, from (4.3), j[2\ = 0, so to compute .A23 to first order second order in V for dtn(k). an equation for Ji23

- kx - k2) = {ala2a3) that dtn(k) = 0 to first order in V. We need in V in order to obtain a non trivial result to Following the same steps as before, we derive

\idt + (w! - u>2 - u)3)IJ123 = /[-5^1*45 J4523<5(Al " *4 ~ £5)

(4.6)

+ V725./l534<$(fc4 - *2 - £>)

+v;3bJl&2im

- h - £5)]d2M2ik5

with Jri234<5(*i + k2-

k3-

k^ = (aj0^0304)

In deriving (4.6), we have already made use of the random phase approxima­ tion. It would not be difficult to write down the exact equation, and, indeed, it is clear that we could derive a hierarchy of exact equations, involving corre­ lation functions of higher and higher order, similar to the BBGKY hierarchy in plasma physics. A useful result can only be obtained by truncating this

An Introduction to Zakharov Theory of Weak Turbulence

149

hierarchy. This will be done thanks to the assumption of Gaussian statistics, which allows us to write J1234 in terms of a product of n(k)s <°>

= Ka3)<°> (a^a4)W + 0)

(aiajWfaa,)™

0)

= T4 T4 [<5(fci - h)S{k2 - k\) + S(kt - kt)8(k2 - hj\ To first non trivial order of perturbation theory, we thus arrive at r

I

ni

r

"|(°>

\idt + (wi - w2 - W3) Ji 2 3 = V'23 |nin 2 + nin 3 - n 2 n 3 1

= .4

(4.7)

This equation is easily integrated

J<»>=£fe""* + A

(4.8)

with Aw = (wi - W2 - w3). The oscillatory term may be neglected and we get j ( l ) _ ^123( n l"2 + »1"3 ~ "2"3) 123 (Ji - w2 - w3 + t'e

, 4 Q> '

where e -¥ 0+. The justification of this ie (Aw -4 Aw + ie) is causality and is identical to that used in Landau's argument for the solution of Vlasov's equation [13]. One can derive intuitively of this prescription by adiabatic switching of the interaction: V -¥ Veet. Plugging this result into equation (4.5) for dtn, we obtain a closed nonlinear equation dtn(k,t)

=7r / |jVfci2|2/fci2<5(£- ki -k2)8(uk-

orm - w2)

-|Vi* 2 | 2 /i*2 *(*i - * - h)6{ui - w* - wa) - | ^ i | 2 / 2 * i 6(k2 - k - £,)*(«, - wfc - wO] d2*x d2fc2

( 4 - 10 )

where /fci2 = ni»2 - nk{rii + n 2 )

(4.11)

Equation (4.10) is reminiscent of a Boltzmann equation, although the analogy, according to Zakharov, is more misleading than useful. It is of the general form dtn(k, t) = 7£[n(J?)]

(4.12)

where I^[n), which is a functional of n, is called the collision term. In the non-decay case, the four wave kinetic equation is derived along similar lines

dtn(k, t) = %J\Tkl23\2fkl23 6(k + ki -k22

2

k3) 2

X(5(wt + wi - w2 - w 3 ) d fci d fc2 d fc3

( 4 13)

150 M. L. Bellac

where T is defined in (3.21) and A123 = n 2 rj 3 (ni + nk) - nink{n2

+ n3)

(4.14)

This equation was first derived in 1962 by Hasselmann [14], but it is only recently [15] that it was shown to be completely equivalent to that derived by Zakharov. 4.2

Conservation laws

In the weak nonlinear approximation, the average energy density is given by E = I u>(jfc)n(£)d2fc

(4.15)

The time variation of E is determined from the kinetic equation; to be specific, let us consider the four-wave case (4.13)

f «/"(k)\T

2 kl23\ fkl23S(k

+ k,-k22

2

x6(uk + u>i - u;2 - LJ3) d kd ki

k3) 2

d fc2 d2fc3

One uses a trick familiar from the study of Boltzmann's equation [16]: thanks to the symmetry properties of T and / uj(k) -¥ -(u>/t + Ui - u2 — W3)

so that dE/dt = 0, provided the integrals converge. In the four wave case, we have in addition to energy conservation the approximate conservation of the "number of waves", or of "wave action" N=fn{k,t)d2k

(4.17)

One can also derive other results analogous to those familiar in the case of Boltzmann's equation; defining for example an "entropy" S{t)=

f\nn{k,t)d2k

(4.18)

one obtains an "H-theorem" in the form dS/di > 0. This result implies that one should find an equilibrium distribution which obeys 7[neq.(£)] = 0. It is not difficult to check that the Rayleigh-Jeans distribution »«,. = ^ -

(4.19)

An Introduction to Zakharov Theory of Weak Turbulence

151

is precisely the equilibrium solution of I\n) = 0. However this equilibrium distribution is not what we are looking for, because we are interested in open systems, where energy is pumped into the system and then dissipated, through viscosity or wave breaking. What we are looking for are stationary non equi­ librium distributions. Let T(fc) represent the interaction with the environment dtn(k) = Ij;[n(k',t)} + T(k)n(k,t)

(4.20)

so that with a stationary non equilibrium distribution we have r(k)n(k) + /s[n(jf')] = 0

(4.21)

The solution to this equation will give precisely the weak turbulence spectrum n(k). Remember that it is called weak turbulence because n(k) is controlled by the weak nonlinearities of the Hamiltonian. Let us now obtain some constraints on our theory. From the "H-theorem" dtS = f d2k(dtn)n~l

= /d^/^njn-1 > 0

(4.22)

so that fr(k)d2k<0

(4.23)

and this last integral is equal to minus the entropy removed from the system: if entropy is created in the system, it should be removed by the environment in a stationary state. On the other hand the collision and decay processes conserve energy and momentum, so that the total energy and momentum given to the system vanish

fr{k)u(k)n(k)d2k = 0 J

f

_

.

2

T{k)kn(k)d k

(4-24)

=0

The first of the preceding equations shows that T(fc) must have alternating signs, and it is possible to show that T(k) < 0 at small scales. Thus we may expect a situation analogous to that of fully developed turbulence, where energy is pumped at large scalens (A; ;$ fcmi„) and dissipated at small scales (k k, km&x, and where there exists an "inertial range"

such that I^[n(k')] = 0 in the inertial range. At the ends of the inertial range, n(k) should match source and sink. The universality of weak turbulence distributions relies on the fact that they are independent of T(fc). Our program for the last section is thus as follows

152 M. L. Bellac

(i) Find the solutions of /* [n] = 0 (ii) Check the convergence of the collision integrals (iii) Match with source and sink (iv) Check the stability of the solutions. In fact, points (iii) and (iv) are extremely involved and will not be dealt with here: we refer to [3s] for details. Point (ii) will be briefly touched, and we shall be mainly interested in point (i). 5

Stationary spectra of weak turbulence

5.1 Dimensional estimates In this section, it will be instructive to keep the space dimension D arbi­ trary. Let dE/dk be the energy density per unit of k, which has dimension ML2~DT~2, and P be the energy flux pumped into the system, which has di­ mension ML2~DT~3. In Kolmogorov's theory of fully developed turbulence, one assumes that the energy spectrum dE/dk in the inertial range depends on P, k and p only; then dimensional analysis for D = 3 gives at once the famous K41 law

^f = cy/'p2/8*-5/3

(5.i)

where C is a dimensionless constant. More sophisticated assumptions may be made in order to "derive" K41 [17], but they lead to little improvement in our understanding of the problem. In our case we have one more dimensionful quantity at our disposal, namely the frequency u(k). We can form the dimensionless quantity *

=

pk(5-D) 37TT puj6{k)

(5-2) v

an the most general result is

^

= pw2(*)fc(°-6>/(£)

(5.3)

If dE/dk does not depend on w(fc), one recovers (5.1) in dimension D = 3. But, in the general case, we need further information in order to proceed. This information is provided by the scale invariance of the interaction and of the dispersion law.

An Introduction to Zakharov Theory of Weak Turbulence 153

In many physically relevant cases, the frequency and the interaction obey scaling laws u(\k) = Xau(k)

V(\k1,\k2,\k3)

= \mV(kllk2,k3)

(5.4)

For example, in the case of deep water capillary waves, a = 3/2 from (3.10) and m = 9/4 from (3.11). Let us assume that a power law for n(k) gives a stationary solution of (4.10) in the inertial range n(u) = Au-s/a

n(k) = Ak-"

(5.5)

We may write V(k,k1,k2) = V0kmf1{^^)

(5.6)

and the collision integral reads I(k) oc fdDkx dDifc2 \V\2n2S^(k 2D

2m 2

2

D

- fci - k2)5(uk -

Wl

1

(5-7)

oc k V*k A k- >k- u=

- u2)

A2V2kD+2m-2'uj-1

Equation (5.7) has, of course, a purely dimensional origin, and is obtained from simple power counting. Now, assuming isotropy, the energy flux dP/dk in k—space obeys ^- = SD u(k)T(k)n(k) = -SD u(k)I(k) (5.8) ak where we have used the stationarity condition (4.21); So is the surface of the sphere in dimension D (So = 7r(2fc)D_1). Thus P(k) = -v[ J

o

w(*')/(*:')(2* , ) l> ~ 1 dfc' k

2 2

= nA V a(m,D.a,S)f = irA2V2a(m, D, a,

k'D-lk'D+2m-2'dk'

(5 9)

'

Jo 2 D+2m 2 -^ s)k ^

where a(m,D,a,s) is a dimensionless integral. Thus, in the inertial range where there is a constant energy transfer from the source to the sink s=m +D

/ P W2 !»(*)=(—jj *"'

(5.10)

154

M. L. Bellac

and the exponent of k in (5.5) is determined. Note that the normalisation of n is also fixed. In the four wave case, an analogous but longer calculation gives 2m

„

f P

\1/2

(5.11)

However, in this case, we have in addition conservation of wave action with a new exponent s' ,

2m

„

n(k) n(k)=

I

1/2

(-9-) =\m>

k~

(5.12)

where Q is the flux of wave action. Energy and wave action fluxes move in opposite directions (Fig. 5).

fi>,

W

N, ' Q

CD 3

2

N2

1

Figure 5. A system with source and sink

In order to derive the exact expression of P, we have to be more careful than in the preceding calculation. Writing from (5.7) the collision integral as l\K ,S) = K

U

J{S)

we obtain the energy flux in the form 2 P(k) = - I Ak'k',/2(£>+m-»)-l J(S)

(5.13)

For s - D + m the denominator vanishes, but so does J(s), since by definition the collision integral should vanish for a stationary distribution. We have an indeterminacy, which is however easily lifted „

1 d J(s') I

We shall explain later on (after (5.30)) how to compute this derivative.

An Introduction to Zakharov Theory of Weak Turbulence

5.2

155

Zakharov transformation

It remains to check explicitly that a power law for n(k) does lead to a van­ ishing collision integral, and we must look more accurately at its explicit calculation. Let us examine the first term in the square bracket of (4.10). We limit ourselves to the case D = 2 h = J d2*! d2Ar2 |\4i 2 | 2 /*i2 S(k - £, - k2) 6(u - w, - wa)

(5.15)

with 6Jk = u. The /^-integral is performed immediately; assuming a dis­ persion law u(k) = cka, with c = 1 in order to avoid inessential details, we have h = fk1dk1des(ka - Ifcf - (Jfc2 + Jfc2 - 2 * ^ cos0) a / 2> ) J r v . = a-2j dWldu;2 {TAn6 l W / * » S(u - w, - wj)

( 5 - 16 )

where we have used UJ, rather than k, as integration variable. In (5.16) kk\ sin 6 = A is twice the area of the triangle built with the vectors k, k\ and £2 A = [2(Jb2Jfc2 + k2k\ + Jfc2*!) - Jfc4 - Jfc? - * £ ] 1 / 2

(5.17)

Then (5.16) becomes

I, = a-2V£ jdwidu* ^ ^ 2 / ° " 1 „*«/« g ^

fu3

S(fJ

_Ui_

W2) ( 5 l g )

where we have taken a scaling law of the form (5.6) for V \Vkl2\2 = V2k2mg(^-)

(5.19)

If we now assume a power law for the distribution: n(w) = w ", v = s/a, we have for /*i2, with u2 = <*> - u>i A12 = n i n 2 - n w ( n i + n 2 ) = ( w i u j ) " ' [ l - (— J

~ (—)

)

(5-20)

Summarizing h = a- 2 V 0 2 /dwtdwj t""-"^"" 1 u 2 m / ° < / f e ) *(w / _ _ VWV

Wl

- a*) (5.21)

156

M. L. Bellac

Let us now look at the second term of (4.10), obtained from (5.21) with the substitution w « uj in the integrand and a minus sign h = -y*d 2 A; 1 d 2 fc 2 |V li2 | 2 /i*2*(£i - k -

fc2)*(wi

lamagrl ^m/a9(^)

= -a-^jd^d^

-u-Ui)

jfo - w - a*)

(5.22)

x(«^)-'(l - («*.)"" - (S)"") The regions of integration are different in (5.21) and (5.22): in the first case 0 < wi < u> and in the latter w < ui\ < oo. In order to ensure identical regions of integration, we use the following transformation w2 wi = —

aiwi w2 = — r

(5.23)

We note that in this transformation U>2

9(lc>l,td 2 )

^-r-^

*(«* - « - * * ) = - ^ *(« - «J - «£)

(5.24)

Furthermore . 2/a

A(fc,fcx,fcj)= ( 4 )

A(fc,

fcj,

fci)

(5.25)

hile

/t« = «„»! -n l ( «„ + n2) = ( ^ ) " " ( l - ( ^ ) ~ " - ( £ ) - " ) (5.26) and

!«-!»-«^-(^.(^)

(a.27)

Plugging (5.24-27) into (5.22) and relabeling the integration variables u[ -► o>i, w2 -» W2, we get the final result for 72 /2 = - a - 2 F 0 2 | d u ; 1 d a ; 2 ' ^ f " > / ' x

/

/

x(^)-"(l-(^)

\-"

/

( a ) J(w -

g

\-"w

-(*)

Wl

- a*)

,2=^2-2^-1 (

5 2 8

)

)(-)

Setting x = -2 ^ i _ + 2i/ + 1

(5.29)

An Introduction to Zakharov Theory of Weak Turbulence

157

and adding the third term in (4.10) obtained from (5.22) by the substitution u>i <-> U2 allows us to cast the collision integral into

/M = a->V0»(^/-J^d^[^^]a/a","^(=^) 1

(53o)

-[i - w - (A) *] t -fer- (^n

The integrand in (5.31) vanishes for v — 1 (Rayleigh-Jeans) and v = ^^ (in general v — m^-)Thus the Rayleigh-Jeans and Kolmogorov distributions are the only stationary power law solutions of the kinetic equations; then the energy flux P (see (5.14) is easily computed from (dl/di/)\l/_s±2.. The existence of other (i.e. non power law) solutions is an open problem. 5.3

Examples and final remarks

All examples will be given in the case D = 2, assuming isotropy of the distri­ butions. (i) Capillary waves in shallow water a =2

m =2

fi(x) = 1

(5.31)

Note that the function f\ is remarkably simple! One finds n(k) = SV^P1/2(^)1/4k-4

(5.32)

(ii) Capillary waves in deep water 3 a=-

9 m=-

One finds n(fc) = C P 1 / 2 ( ^ ) 1 / 4 J f c - 1 7 / "

(5.33)

As the function / i is more complicated than that in (5.31), the constant C ~ 9.85 must be computed numerically. (iii) Deep water gravity waves. The four-wave interaction can be treated along similar lines, but the calculations are still more cumbersome. FVom the values m = 3

a = 2

one finds n(k) = diPp2)1/2^4 1

(energy)

n(fc) = C 2fl /6 (p 2Q )1 /3 A -23/6

( w a v e action)

(5 34)

158 M. L. Bellac

where C\ and C% are calculable constants. (iv) Convergence of the integrals. The assumption of a power law for n(k) may lead to convergence problems when k -*■ 0 or k -► oo. Of course, there are infrared and ultraviolet cut-offs coming from source and sink respectively. However, divergent integrals would be extremely sensitive to the cut-off, and the results for the spectra would be meaningless. It is easily checked that k -*■ oo gives no difficulty, but problems may arise from k -> 0. Fortunately the convergence is improved because of cancellations in suitable combinations of distribution functions. The k -> 0 behaviour is linked to that of the interaction for small values of one of the wave vectors V(k, £,, ha) ~ V02k?>k2m~m>

*, « k

(5.35)

From the explicit expression of V (see 3.11), one finds mi = 7/2. The condi­ tion for the convergence may be shown to be 2mx>2m

+ 2- 4a

(5.36)

This condition is realised for deep water capillary waves. (v) Stability of the results. In Kolmogorov's theory of fully developed tur­ bulence, one assumes isotropy of the energy spectrum dE/dk, even if energy pumping is anisotropic. The opposite situation may happen in weak turbu­ lence: even if energy pumping is isotropic (i.e. T(k) depends only on k), a linear stability analysis shows that formation of structures may occur spon­ taneously. For example, the isotropic spectrum (5.33) of deep water capillary waves is unstable, and there is a tendency towards an anisotropic spectrum

The spectrum is isotropic at large scales, but becomes anisotropic at small scales: even an isotropic pumping will lead to an anisotropic spectrum. By contrast, the isotropic spectrum is stable for deep water gravity waves, as well as for shallow water capillary waves. Acknowledgments I would like to thank Frederic Dias, Francoise Guerin and Alain Pumir for their help while preparing these notes. References [1] H. Casimir: Proc. Kon. Ned. Akad. Wetenschap B 5 1 , 793 (1948). C. Itzykson and J.-B. Zuber: Quantum Field Theory Addison-Wesley (1980) section 3-2-4. [2] Itzykson-Zuber, ref. [1] section 4-3-4.

An Introduction to Zakharov Theory of Weak Turbulence

159

[3] For a detailed exposition of the theory, see V. Zakharov, Y. L'vov and G. Falkovich: Kolmogorov Spectra of Turbulence I: Wave Turbulence Springer (1992). However, the reader should be aware of the numerous misprints in this book. [4] See, e.g.: L. Landau and E. Lifechitz: Fluid Mechanics, Pergamon Press (1980), sections 12 and 62. D. Acheson: Elementary Fluid Dynamics, Oxford University Press (1990). T. Faber: Fluid Dynamics for Physicists, Cambridge University Press (1995), chapter 5. [5] R. Seliger and G. Whitham: Proc. Roy. Soc. A.305 1 (1968) [6] V. Zakharov: preprint [7] Landau-Lifschitz, ref. [4] section 25 [8] H. Yuen and B. Lake: Nonlinear Dynamics of Deep-Water Gravity Waves [8] V. Zakharov: preface to the Russian version of the book by Yuen and Lake. [9] V. Krasitskii: J. Fluid Mech. 272 1 (1994) [10] A. Dyachenko and V. Zakharov: Phys. Lett. A 190 144 (1994) [12] A. Dyachenko, Y. Lvov and V. Zakharov: Physica D 87 233 (1995) [13] See e.g.: E. Lifechitz and L. Pitaevskii, Physical Kinetics Pergamon Press (1981) section 29 [14] K. Hasselmann: J. Fluid Mech. 12 481 (1962) [15] A. Dyachenko and Y. Lvov, Am. Met. Soc. 3237 (1995) [16] See e.g.: R. Balian: Fom Microphysics to Macrophysics, Springer (1992), section 15.3 [17] Landau-Lifschitz, ref. [4], sections 33-34. U. Frisch Turbulence, Cam­ bridge University Press (1995)

P H E N O M E N A B E Y O N D ALL ORDERS A N D BIFURCATIONS OF REVERSIBLE HOMOCLINIC CONNECTIONS N E A R HIGHER RESONANCES ERIC LOMBARDI Institut Non Liniaire de Nice These lectures are devoted to phenomena beyond all orders in dynamical systems. We explain how these phenomena are connected with oscillatory integrals and how to obtain an exponentially small equivalent of an oscillatory integral when it involves solutions of nonlinear differential equations. The method proposed herein enables us to study the problem of existence of homoclinic connections to 0 for vector fields admitting a 02iw or a (iwo)2iwi resonance. These problems could not be solved by a direct application of the Melnikov method since in these cases the Melnikov function is given by an exponentially small oscillatory integral.

1 1.1

Introduction Phenomena

beyond all orders in dynamical

systems

In many physical problems one can construct "solutions" in term of power

* series of a small parameter e which reads Y(t,e) = £ enYn(t) + o(e*). In n=0

some cases, this power series is denned at any order but it diverges. This divergence may express that the system has a solution for which such an expansion misses some exponentially small term like e _ 1 / e Z(t). Such a term is said to lie beyond any algebraic orders. Such problems, in which these very small terms have great practical interest, are known in many branches of science including dendritic crystals growth, quantum tunneling, KAM theory, theory of water-waves (which was our original motivation for this work) and others. A collection of these apparently unrelated problems can be found in [20]. A first example is a model of crystal growth governed by the equation e26"'+6' = cos6.

(1)

For e = 0 this equation admits a front connecting the two fixed points ± f . The question is then to determine whether a front connecting ± f exists for the perturbed equation. This question was studied by Hammersley and Mazzarino in [9] and by Amick and McLeod in [2] who proved that for e ^ 0, such a front does not exist. This problem was also studied by Kruskal and Segur in [14], using Matching Asymptotic Expansions (M.A.E.). They give arguments 161

162

E. Lombardi

which suggest that for e ^ 0, there is no heteroclinic orbits connecting ± | . Their argument is formal, but it is very general and it can be used for a large class of problems including P.D.E. A second example is a perturbed KdV equation a&u

&u

B

du

du

n

for which one is interested in the existence of solitary waves, i.e. in the existence of traveling waves u(x, t) = I Y ((x - ct)y/c) such that Y(f) —► 0. £-+±oo

This amounts to look for a homoclinic connection to 0 of the fourth order equation (e = e'\/c) £2^L +

fL^Y

+ Y2 = 0

(2)

For e' = 0, the KdV equation admits a one parameter family of solitary waves explicitly given by u(x,t) = §ccosh~2 P * - ^ ^ ) , Hammersley and Mazzarino [8] and Amick and McLeod [3] proved that for e ^ 0 there is no homoclinic connection to 0. This equation was also studied by Pomeau et al. in [19] using M.A.E. combined with Borel summation. They give arguments who suggest that for e ^ 0 there is no homoclinic connection to 0, but that there exist homoclinic connections to exponentially small periodic orbits. A third example occurs in water wave theory when studying the existence of solitary waves for an inviscid, incompressible and irrotational fluid layer governed by Euler equation, under the influence of gravity and small surface tension (Bond number 6 < 1/3) for a Froude number F close to 1 (see [11] and [15]). The existence of generalized solitary waves with exponentially small ripples at infinity has been obtained in [21] and [15]. Moreover, the non existence of truly solitary waves for Bond number smaller and closed to 1/3 and for Froude number close to 1 was proved by Sun in [22]. A fourth example is a chain of nonlinear oscillators coupled with their nearest neighbors, governed by Xn + V'(Xn) = 1(Xn+1-2Xn

+ Xn^),

n€Z,

(3)

where Xn is a function of t e M; 7 is assumed to be positive and where V is smooth function such that V'(0) = 0 and V"(0) = 1. The problem is then to determine the existence of traveling waves, i.e. solutions of (3) of the form Xn(t) = x(t - TIT). G.Iooss and K.Kirchassgner proved in [12] that there exist in the parameter plane (r, 7) curves in the neighborhood of which there always

Phenomena Beyond All Orders and Bifurcations 163 +10)

o-ICO

n
U=0

u>0

, i +1(0,

,,

11 - ( C O ,

c >

ii<0

Figure 1. The 02iu> and the (iuo)2iwi resonances

exist nanopteron , i.e homoclinic connection to exponentially small periodic orbits, and there is generically no homoclinic connection to 0. Now, if we want to determine the common denominators between all these examples, what makes the appearance of exponentially small oscillations and the disappearance of hetero or homoclinic connections: we first observe that all these problems can be reformulated as dynamical systems

in finite dimensions (O.D.E. case, X € R n ) or infinite dimensions (P.D.E. case X € H, H Hilbert or Banach space), studied near a fixed point placed at the origin. Moreover, the parameter can be chosen such that the phenomena of appearance of exponentially small oscillations and disappearance of hetero or homoclinic connections occurs for /x = 0. Finally the time £ of the dynamical system is not necessarily the physical time. For instance for the traveling waves "£ = x - ct". A second observation is that all these systems are reversible, i.e. that the vector field V anticommutes with some symmetry S. The last observation is that for n = 0 when the phenomena of appearance or disappearance occurs, the spectrum of the differential DxV(0,Q) presents very specific configuration. A first example is the 02iw resonance which occurs for the perturbed KdV equation and for the true water water wave problem. In this case a first part of the spectrum of DxV(0, n) is bounded away from the imaginary axis and a second part admits the bifurcation described in Figure 1. A second example is the (kjo)2iwi resonance, which occurs for the chain of coupled nonlinear oscillators. The bifurcation of spectrum corresponding to this resonance is described in Figure 1. We can observe that for these two resonances, even after bifurcation, for n > 0 it remains on the imaginary axis

164 E. Lombardi

pair of purely imaginary eigenvalues which coexists with a set of hyperbolic eigenvalues with small real parts. Hence, a "rap! id o scUlatory parts and a slow hyperbolic part" coexist in such vector fields admitting a 02iu> or a (iu>0)iui resonance. Other examples of problems involving relevant exponentially small terms can be found in the Hamiltonian literature when studying the splitting of separatrices. This study was initiated by Poincare in [18]. Later on, a regular method for studying the separatrices splitting was proposed by Melnikov [17] and Arnold gave to this method an elegant form [4]. Starting from the work of Holmes, Marsden and Scheurle [10] the rapidly forced pendulum governed by the equation x = sin x + /iep sin -

(4)

became one of the most popular models involving this difficulty. Holmes, Marsden and Scheurle gave exponentially small estimates of the splitting for p > 8. This result was improved by several authors up to p=0 (see [6] for the case p = 0 and for a good historic of this subject). More recently Gelfreich gave a proof of the asymptotic formula of the splitting up to p = - 2 [7]. Here again, o "rapid oscillatory parts and a slow hyperbolic part" coexist in the system. 1.2 A little toy model : from phenomena beyond any algebraic order to oscillatory integrals To illustrate how phenomena beyond all orders are induced by the coexistence in a system of a "rapid oscillatory parts and a slow hyperbolic part'', we begin with a toy model in R3 which reads ( dA

Si = -"*> ^=u)A

+ p(e*-a2),

(5)

da , 2 — = tl - a1. ax where (A, B, a) 6 R3 and u is a positive fixed number. This system has two parameters, e > 0 which is small, and p which can be any real number. This system develops typical "phenomena beyond any algebraic order" which can be seen without any difficulty because all the bounded solutions can be computed explicitly. Hence, this system enables to understand what kind of mathematical difficulties are hidden behind these phenomena, and

Phenomena Beyond AH Orders and Bifurcations

165

what kind of mathematical tools have to be developed for dealing with any system where such phenomena are involved. To study this system it is more convenient to set Z = A + iB,

Q = £/3,

x = t/e,

and to rewrite (5) with these new coordinates. We obtain

' § = *!z + v*(i-/n (6)

M

This system is reversible, which means that if (Z(t),0(t)) is a solution, then S(Z(—<),/?(-<)) is another solution where S is the reflection given by S(Z,0) = (Z, -0). We call a reversible solution of (6) a solution which s&tisGesS{Z(-t),0(-t)) = (Z(t),0(t)). For p = 0, the system is uncoupled and the phase portrait is fully sym­ metric (See Fig. 2) : the truncated system admits two families of periodic orbits of arbitrary size given by

P£v(«) = (*e IM/e)+,v ,±l), (Jfc = 0 corresponds to the two fixed points) and a family of heteroclinic orbits connecting Pjf which read Hk^tM

= ( f c e ^ / ^ . t a n h ^ + fo)).

Among these solutions there is a one parameter family of reversible ones given by Hk,o,o with k e R (H-k,o,o(t) = #*,*,<)(*)) and there is a unique (up to a phase shift) front connecting (0, ±1) which reads h(t) = ifo,o,o(*) = (0, tanh t). At last, observe that the periodic orbits are exchanged by symmetry, i.e.

spk+jt)

=

Pk^(-t).

The question is then to determine how this phase portrait is deformed by the higher order terms (p ^ 0). In other words, do the previously found orbits persist for the perturbed system? For periodic solutions the answer is yes. Moreover, because of the partic­ ular form of the perturbation, these orbits persist without any deformations. Indeed P * is still a solution of the perturbed system. For more general per­ turbation, the periodic orbits are deformed, and the proof of their persistence can be given by the Lyapunov Schmidt method. For the heteroclinic connections the situation is not so clear. A first question is the persistence of the reversible connections.

166 E. Lombardi

A

Figure 2. Phase portrait for p = 0

A very first way to make up one's mind is to look for a reversible front connecting the two fixed points (0, -1) and (0,1) using a classical asymptotic approach: one can compute a formal reversible solution of (6) connecting (0, -1) to (0,1) under the form Y

(*) = (2> B 2n(0,tanht in>0

where Z0 = Zx = 0

and

Zn(t) = -

d"for n > 2. (iw)"-1 dtn~2 Vcosh2(*)J

So this first approach predicts the persistence of the reversible front (provided that the power series converges.) A second approach for studying the persistence of the reversible connec­ tion is to formulate the problem in terms of stable manifolds and to illustrate it geometrically. We denote by W,"^* the stable manifold of the periodic solution P£ of the truncated system. The manifold Wg+^ is a two dimensional cylin­ der in R3 (one dimension for time and one dimension for the phase shift) .The

Phenomena Beyond All Orders and Bifurcations

Jfc>0

167

Jfc = 0

Figure 3. Stable manifold of the periodic solution p£ of the truncated system

radius of this cylinder is the size of the periodic orbit, i.e. k. A fundamental remark is the following : Since P£ and P^ are exchanged by symmetry, there exists a reversible heteroclinic orbit connecting P^ and Pf if and only if the stable manifold of P£ intersects the symmetry space E+ = {Y/SY = Y}. In this case, E+ is a line in R 3 . For k > 0, the intersection consists in two points which leads to the two reversible heteroclinic orbits connecting the periodic orbits Pfc- and Pf. For k = 0, there is a unique point of intersection, which leads to the unique reversible front of the truncated system. Figure 3 represents the intersection between W*j.' and £+ in two case : k > 0 and k = 0 for a fixed value of e. Let us denote by W ^ the stable manifold of the periodic solution P£ of the perturbed system. The stable manifold W ^ is obtained by perturbation of the stable manifold Ws+^* of the periodic solution of the truncated system. On figure 3 we can observe that for k = 0 the situation is not robust: unless there is a miracle, there should not exist any reversible front for the perturbed system, whereas for k large enough, the two points of intersection between W ^ * and S+ should persist, i.e. there should exist two reversible heteroclinic

168 E. Lombardi

orbits connecting P£ and P£. The natural question is then to determine, for a fixed value of the bifurcation parameter e, the smallest size kc(e) of the periodic solutions P£ and P£ which admit a reversible heteroclinic orbit connecting themselves : is it 0 or not? And if it is not 0 we would like to compute A;c with respect to e. When kc > 0, a subsidiary question is then to determine the behavior when t goes to - c o of the one dimensional stable manifold Wg+0 of the fixed point (0,1). This second approach predicts the non persistence of the reversible front. So, we have two heuristic arguments (an asymptotic one and a geometrical one) which lead to two opposite predictions. However, here we can overcome this difficulty since all the bounded so­ lutions of the perturbed system can be computed explicitly. For a more gen­ eral perturbation, explicit computations cannot be done, and this exponential smallness of the Melnikov function leads to serious difficulties. One aim of this work is to give mathematical tools to study such general systems. Now, let us perform explicit computations to check the validity of our heuristic arguments. Lemma 1.1 The bounded solutions of the perturbed system (6) are and Yto,zo(t) = (zoe'"t/e \

+ ipe f ^*-)/'—-l ds, tanh(t +1 0 )) Jo cosh (s +1 0 ) /

where k,
The proof of this lemma is made in two steps. We first observe that the bounded solutions of (6b) are given by /3(r) = tanh(r -I-10). Then it remains to solve (6a) which is a linear oscillator of high frequency w/e forced by an explicitly known, analytic, exponentially decaying second term. Using the explicit formula giving Yto<Xo we easily compute the reversible solutions. Lemma 1.2 1. The perturbed system, admits a one parameter family of reversible hete­ roclinic orbits H\ explicitly given by Hx = (xeiut/c+ipe \

/V
Phenomena Beyond All Orders and Bifurcations 169 with A G R.

2. The solution H\ connects Pj^x*),-
* ia,(\.\

f+°° sin(wa/e) . Jo cosh (s)

,

wflere

,
.

f+co cosius/e) . Jo cosh (s)

5. When A uartes /rom — oo to +oo, &(A,e) tafces ttwce all the values in }kc(e),+oo[ and once the value kc(e) which reads — , ds = n . , . r ln , ~ irpcje u"/2e 2 Jo cosh (s) 2sinh(wir/2e)«-»o *^ This lemma confirms our former geometric arguments: for the perturbed system, and for each e > 0 fixed, there exists a smallest size kc(e) (expo­ nentially small) such that for every k, 0 < k < kc(e) there is no reversible heteroclinic orbit which connects P^^ and Pf , whereas for the truncated system, there exists heteroclinic orbits which connect two periodic orbits of the same arbitrary small size. The "surprise" is that this smallest size is ex­ ponentially small. It lies beyond any algebraic order of e. This phenomenon cannot be detected using a classical expansion of the solutions in powers of e. The next question is to determine the behavior of the one-dimensional stable manifold Wg+0 of the fixed point (0,1), when t goes to —oo. Here again, using the explicit formulas giving the solutions we obtain kc(e) = ps

L e m m a 1.3 A parameterization of the stable manifold W* 0 of the fixed point (0,1) is given by r+oo

Ys(t) = ( -ipe f ° V « - * > / £ — 1 — d s . t a n h w ) •

\

Jt

coslr(s)

/

with r+oo

K{e) = pe f " V 1 * " / ' \r— ds = . J " * . ~ J-oo cosh2(s) smh(wjr/2e) e-»o

2Trpue-"n/2t. ^

So the front does not persist. The stable manifold of (0,1) does not connect (0,-1) to (0,1), but it connects an exponentially small periodic orbit PRU) „/2 t o (0>1)- The stable manifold of (0,1) for the perturbed system develops exponentially small oscillations at - o o which cannot be detected with a classical asymptotic expansion of the solution in powers of e.

170 E. Lombardi

A third phenomenon beyond any algebraic order can be found when com­ puting the distance d(e) between the stable manifold Wg+0 of (0,1) and the symmetry line £+. Once again, this distance is given by an oscillatory integral and it is exponentially small. In all cases, the exponential smallness of kc(e), K(e) and d(e) is due to the fact that theses quantities are given by oscillatory integrals of the form

f+3°^gf{X(t))dt

n(e)=

where / is an analytic function and X(t) is a particular solution of the system. The toy model has been designed, so that all the solutions can be explicitly computed. Hence, in this particular case the integrand of the oscillatory integral is explicitly known whereas in general it is not true. So, in general we have to face the following problem: Problem 1.4 Assume we study a nonlinear differential equation in finite (O.D.E. case) or infinite (P.D.E case) dimensions of the form ^

= F(Y,t,e)

(7)

and that we want to compute 1(e) = /

r+0° i*t e—g(Y0(t,e))dt

./-OO

where u is positive; g is a given function, and YQ is a particular solution of (7) characterized by its initial value or more frequently by its behavior at infinity (for instance, it goes to a fixed point, a periodic orbit...). The problem is to determine what kind of information on YQ we need to know for being able to compute or at least to bound the oscillatory integral 1(e). In this lecture we present rigorous mathematical methods partially in­ spired by M.A.E. approach, which enable to compute the size of oscillatory integrals involving solutions of nonlinear differential equations. These tools enable us to prove typical results such as existence of homoclinic connection to exponentially small periodic orbit and non persistence of homoclinic con­ nection to 0 for reversible vector fields studied near a resonance. Section 4 deal with the 0 2+ iu; resonance and section 5 with the (iu>o)2 iu>i resonance. The crux point of the analysis is the description of the analytic continuation of the solutions which enable to catch exponentially small terms which are "hidden beyond all orders on the real axis".

Phenomena Beyond All Orders and Bifurcations

171

2 Exponential tools for evaluating mono frequency oscillatory integrals This section is devoted to the study of problem 1.4, i.e. to the computation of upper bounds and equivalents of oscillatory integrals

/ +00

J ao f (t, E)et'wt/E dt. involving solutions of nonlinear analytic differential equations. 2.1 Rough exponential upper bounds We begin with a very simple lemma which gives exponential upper bounds for mono-frequency oscillatory integrals. Lemma 2 . 1 (First Mono -Frequency Exponential Lemma) Let w, t, A be three real positive numbers. Let He be the set of functions f : l3jx]0,1] -+ C such that f (•, e) is holomorphic in B1 = {l; E C/IZm (t;) I < t} and

IIfII H-t

:= EE]s 1^ (If(6,E)IeAe(^)I) < +00. up Er^^

r +00 Then for every f E He and E E]0,1], I' (f, e) = f (t, E) e:':"'t/e dt satis-

J

fies

II71(f,E)I <

2 IIf IIH^ e -We /E . A

Proof. We only do the proof for It For I-, perform the change of time t' = -t in the integral and observe that f N f (-f, e) also belongs to Hl

and that IIf II H, = IIf IIHA . Let f E Hp , e, f' < 1 be fixed . Since f is holomorphic in B1 , the integral of fe"111 along the path 17 1 given in Fig. 4 , is equal to zero. Pushing R to +oo, we get / +0 I +(f, e) =

J ao f (it' + t, e) e'w01

+t)/E dt.

The estimates then follows, where the exponential comes from the oscillating term computed on the line Im (t) = $'. ❑

172

E. Lombardi

Lemma 2.1 gives a very efficient way to obtain exponential upper bounds because the membership to H£ is stable by addition, multiplication, "compo­ sition" , which can be summed up as follows Lemma 2.2 Hf is an algebra and If f € H£ and g is holomorphic in a domain containing the range of f and satisfies g(0) = 0, then g o f £ Hf. Remark 2.3 Lemma 2.1 is called mono-frequency because functions be­ longing to Hf cannot have a second "high frequency" like /o(f,e) = cos(w 0 f/e) cosh - 1 £. Indeed, for every £ > 0, ||/ 0 |L X = +oo holds because of the oscillation term. Oscillatory integral involving such functions are stud­ ied in Subsection 5.1. Remark 2.4 How to use Lemma 2.1 for solving problem 1.4 Our aim is to compute the size or at least an upper bound of the size of an integral of the form +00

/

9(Y0(t),t,e)e^cdt,

■oo

where g is an analytic function and Y0 is a particular solution of a differential equation of the form

f = F(Y,t,e) with F analytic with respect to {Y, t). To apply Lemma 2.1 we have to proceed in three steps:

Figure 4. Path Ti.

Phenomena Beyond All Orders and Bifurcations

173

• We must find the singularity which has the smallest imaginary part of e H F(•, £, •). This determines the width of the complex strip 131: t must be chosen such that F has no singularity in 131.

• Studying the holomorphic equation d = F'(1', E 131i we must extend the solutions of our first real equation into 131. This step is often performed using the Contraction Mapping Theorem in an appropriate space of functions of the type H11. • Then, we can apply Lemma 2.1, to obtain an exponential upper bound of I(e). This strategy works either in finite or infinite dimensions.

2.2 Sharp exponential upper bounds In order to determine the subtlety of Lemma 2 . 1 one can apply it to the basic f+00

example J(e) =

J using the residues, J(s) =

e"16 cosh-2 (t) dt which can be explicitly computed

7r w e sinh (-7rw/2e)

2 7rw eio

e-arw/2e

E

The function t; * cosh-2 ^ has a double pole at i 2 . This pole limits the size of the complex strip to f < i . Applying Lemma 2.1, one then find IJ(e)I

:5

1 II cosh2(l;)

e--/-Het

1 e`06 de > 0, Vt < 2 = co12 t

To obtain the right coefficient in the exponential we can set t = 2 - be. We get 6w bw e-war /2e ti e e-war/2e e-io 5 2 g2 sine Se

IJ(e)I e

With this "trick" we obtain the right coefficient in the exponential , but the polynomial degeneracy is not correct . This is due to the fact that II'IIHA is uniform in the strip 13, and does not describe what happens near the singularities. So we give in this section a refined version of the Exponential Lemma 2.1

174

E. Lombardx

which gives the right coefficient in the exponential and the right polynomial degeneracy. L e m m a 2.5 (Second M o n o - I r e q u e n c y Exponential L e m m a ) Let u>, 7, a, A be four positive numbers. Let S be in [1, +oo[. Let H^'g be the Banach set of functions such that f : (J B„-g€ x {e} —► C; f{-,'e) is Q<e
holomorphic in B„-ie

and | | / | | „ , H

x

< +oo where

*'.t

ll/ll w 7 .*:=

^p

[ ( | / ( { , e ) | ( k 2 + ^ r x 1 ( 0 + (l-X,(0)e A | 7 l ««>l)l

u»f/» x, ( 0 = 1 for \TZe (£) | < 1 and x, ( 0 = 0 otherwise. Then for every 7 > 1, there exits Af(7, A,o*,w) sucA tfia* /or every / € r+00

tfj;*

and e e]0,
/(*, e) e * ^ ' dt satisfies

J—oo

\I±(f,e)\<M-^e-»"\\f\\irt>.
Remark 2.6 Observe that \a2 + £ 2 T = |(f - ia)(f + icr)|^. Hence, roughly speaking, H^'g is the space of the holomorphic functions in Ba~ge which decays exponentially at infinity and which explodes at most as |f ± icr|—>" for £ near ±io\ Proof. The proof is done as for the First Mono Frequency Exponential Lemma 2.1 using the rectangular path drawn in Figure 4. The details are left to the reader. Remark 2.7 We check on our basic example J(e) that this lemma gives the right polynomial degeneracy, \J(e)\ < M

1 cosh'te)

r/2.2

2 I e —Wa« = M' sup ( --r-5— ^ - )1 -e-™/ *. -e [o.fi \sm*xj e £

R e m a r k 2.8 The membership to H^'g does not mean that functions have a pole in ±\o. Functions like (£,e) 1-4 e~~*2 which is entire; (f,e) H- cosh - 2 (f/2), (£,£) ^ cosh _1 (£ + ie2) cosh _1 (f - k 2 ) which respectively have poles at ±i7r and at ± i § ± ie 2 ; (£,e) .->• e i l t a n h ( 0 C osh~ 2 (0 which has an essential singu­ larity at if all belong to Hl'\. This comes from the fact that we work in

Phenomena Beyond All Orders and Bifurcations

II ! IIHo.n if (01

175

If(^)I <_ IIfIIH.ae aiRelfll

Q7I +iQl7

Figure 5. Description of f E Ho'a

B 2 _E and not in B 2 . This lemma simply describes how the function grows near i(a - 8e).

Ha'6

Remark 2.9 Unlike Ht \, is not an algebra. However the following lemma enables us to use it in nonlinear differential equations. Lemma 2 .10 For every f, g E HQ'6 , E^" f g lies in

Ha'6.

So, Lemma 2.5 can be used like the Exponential Lemma 2.5 to solve Problem 1.4 (see Remark 2.4) with holomorphic continuations of solutions in Ho'6 instead of HP . Here again, these continuations can be obtained by the Contraction Mapping Theorem. 2.3 Exponential equivalent : general theory In this section , our aim is to determine what kind of informations on a function f we must know how to compute an equivalent of an oscillatory integral It (f, E). Moreover , we must be able to obtain this kind of informations when f is not explicitly known and involves solutions of O.D .E or P.D.E. The membership to Ha' ' only gives upper bounds of I(f, e). A very first idea is to believe that the equivalent of the oscillatory integral 1(f, e), is given by the leading part of f on R. The following example ensures that this is not always true.

176

E. Lombard*

Example 2.11 Let / be given by f(t,e) = fi(t,e) 1 ~ —TTTTT cosh^t)

h&e)

+ fi(t,e)

with

e" hit,e) = cosh4(t)

&nd

The leading part of / on R iiis f\ (/ ~ f \ ) . The oscillatory integrals can be computed using the residues h :=/(/i,e) =

e

smh(wu/2e)' iru ' ',2 /a :=/(/*,£) = e sinh(7ru;/2£;) ( & ♦ * ) « - ■ Hence, /(/,*) ~ e-t-o

h

~ ^e-«-/ae e-fo e

/(/,,) ~Q IX + /

a

^(i

fori/>2, +

£)e-~/*

for,

= 2,

3

J(/,e) ~

J2

~ ^-re-"w/2£

for0
We define below a set of holomorphic functions / , for which we are able to to compute an equivalent of the oscillatory integral induced by / . Lemma 2.15 below, ensure that the relevant part of the function to compute the oscillatory integral induced by f is not its leading part on the real axis but its leading near the singularity. Definition 2.12 Assume 7, A > 0, a > 0, 6 > 1 , roo,mi > 1, 0 < r < 1 and v>Q. Denote by ££'j (m 0 ,mi,r,i/) the set of functions f such that i- f e H-%, 2. for every e e]0,£ 0 ] and every z £ Ei>r>e :=] - £ r - 1 , e r - 1 [ x ] - er-\ reads f(±(io + ez),e) = ±ft(z) where ft € H?° and ft € H^

-S[ f

+ ^ft(z,e)

(8)

tvith

H?° = { / : &6 -> C, / holomorphic in fij := Rx] - 00, -6[, _

:= sup (\f(z)\\z\m°)

< +00}

Phenomena Beyond All Orders and Bifurcations

177

and Kr

= {1 /

:

U (S«.r* x {e}\ -» C, /(-,£) holomorphic in Zs>r<€, v

e e]o,e 0 ]

'

|/||

:= sup (|/(*)||*r) <+00}. H; z€E«,r,«

1 -

*#

1/(01 <

e"-

,-A|KefOI

1 y_ / = ^7/0 +

1/(01 <

e"^ / l

F

Figure 6. Description of / 6 ££•£

Remark 2.13 The four functions $ and ff are uniquely defined since they are given by fjf(z) := limeV(±(i^+e«),e) and/^C*) = lime-',(e'1f(±(ia+ e-¥0

-•*)) ~ foiz))-

Let

us

e-fO

denote by P r the operators P r : / »-4 / *

Remark 2.14 The membership to # J ' ^ simply ensures that /(•,£) decays exponentially for \Re (f) | -> 00 and explodes as most as |£ T i^l 7 for £ close to ±\{a - Se). The membership to E^'f specifies the behavior of the function /(•,e) near ±i(cr - 6e). In the domain ) -er,er[x]a -er,
178

B. Lombardi

and for £ close to i(a-5e),

f(£,e) explodes like ^ft(i6)

= ^Pf{f)(i6).

So,

+

P r (/) is the "leading part of / near \{a - Se)". We call it the principal part of / . Similarly, P r ~(/) is the principal part of / near -i(cr - Se) Moreover, the membership to £j; 4 X (m 0 , mi, r, v) does not mean that func­ tions have poles at ±ia. All the following functions belong to the space E \,1 (2> 2> h 2)' b u t o n l v t h e t w 0 first ones have poles in i i f . This comes from the fact that we work in B* _ c and not in B». • The function / : (£,e) (->■ cosh~2(f) has poles of order 2 at ± f and • The function / : (f,e) ^ e 2 cosh _ 4 (f) has poles of order 4 at ±'f

and

P r ± (/) = ^ 4 ; • The function / : (f,e) 1-4 e - ^ is entire and P^{f) = 0; • The function / : (f,e) >-4 coslT 2 (f/2) has poles at ±iw and Pf(f) -1

2

_1

= 0;

2

• The function / : (£,e) 1-4 cosh (£ + ie )cosh (£ - ie ) has poles at ± i f ± ie2 and P r ± (/) = - z ~ 2 ; • The function / : (£,e) >-> e u t a n h ( 0 cosh _2 (f) has an essential singularity at if and Pf(f) = -z-*<Mfr. The membership to E^'s gives enough informations on a function to com­ pute an equivalent of the oscillatory integral involving it. Lemma 2.15 (Third Mono-Frequency Exponential Lemma) For even/7 > 1, 6 > 1, a,\ > 0, mo,mi > 1, 0 < r < 1, i/,w > 0 and every function f € ^ ( m o , m i , r , « / ) the oscillatory integrals I±{f,e) = / 0 + 0 °e ± i w i / t f(t,e)dt satisfy

^.')-7FT-(A*+A(«")) where A* := f

P?(f)(z)

eluzdz

forn>5

•/-iij+R

(K* does not depend on n) and v* := min(i/, (1 - r)(mo - 1), (1 - r)(y - 1)).

Phenomena Beyond All Orders and Bifurcations

179

The proof of this Lemma is given in [16]. This lemma ensures that the rel­ evant parts of / € E^'g, for computing an equivalent of the oscillatory inte­ grals J ± (/,£), are P r ± (/). Moreover, it gives a very efficient way to compute equivalents of oscillatory integrals involving solutions of non linear differential equations since the principal part P* of a sum (resp. of a product) is the sum (resp. the product) of the principal parts: Lemma 2.16 P* : E2'g(m0,mi,r,i/) every f,g£ 2-4

£ # , eVglies

H-» H™° are linear operators and for

in ETa$ and P f V / f f ) = Pr* (/)?*(
Exponential equivalent: strategy for nonlinear differential equation

To use the Third Mono-Frequency Exponential Lemma 2.15 for computing equivalents of oscillatory integrals 7 ± (/, e) when / involves solutions of real nonlinear differential equations, we must be able to prove that a given solution y of a real nonlinear analytic differential equation ~

= F(Y,t,e)

(9)

admits a holomorphic continuation which belongs to E%'f- We propose here a strategy to achieve this in three steps. This strategy is described for simplicity in the one dimensional case. It can be readily extended to n dimensional space

of type ft £%**. Step 1. Continuation far away from singularities. To prove the exis­ tence of a holomorphic continuation of y in T>a>r,e = #* \ {£ £ C/\TZe (£) | < \eT, \lm (f ± icr) | < | e r } , one can use the Contraction Mapping Theorem in the space of holomorphic functions defined in Vai, Pr- We first introduce two systems of coordinates in space and time: a first one (Y, £), which is the initial one on K, is called the outer system of coordinates; a second one (Y, z) := (e^Y, (f — \o-)/e) which is useful to describe what happens near icr is called the inner system of coordinates. For an orbit F(£) in the outer system of coordinates, let us denote by Y(z) = eyY(ia + ez) the corresponding orbit in the inner system of coordinates and vice versa. We use the same notations for complex domains: for a domain E of C, in the outer system of coordinates, let us denote by

180 E. Lombardi

S = (E - \a)je the corresponding one in the inner system of coordinates and vice versa. Step 1 gives a holomorphic continuation of 3>(f) in 2?ff,r,£- Then we must prove the existence of a holomorphic continuation of y(£) in Ea,r,e := ] - er, e r [ x ]a - er, a - 5e[ (see Fig. 8) which satisfies (8) + , i.e. the existence of a holomorphic continuation of y{z) in £,5,r,£ which satisfies 5>(z, e) = y0(z) + e"yi (z, e)

with

% € H?° and 5>i € 8%. (10)

Such a continuation in %,r,e is a holomorphic solution of the full system (9) rewritten in the system of coordinates dY - ~ - = F(Y,z,e):=e^F \J;Y(z),ia + ez,e\,

(11)

which satisfies (10) and which coincides on Ej, r , E n Va
Fo(Y,z):=F(Y,z,0)

(12)

in H™Q. This system is called the "inner system" and it represents "the relevant part of the system near ia". 1 I

.i i o L-ii1-56-

-io

Figure 7. Step 1 : continuation in domain TV,,-,!

Phenomena Beyond All Orders and Bifurcations

181

1a

Figure 8 . Step 2 : continuation in Ed,r,e

A

2 1/

0---^-'•

1/E1-r ll2e^`r' ^

Du.r.E

Figure 9. Domains Do,r,e and E6,,.,E

Step 2 . 2. Matching of holomorphic continuations. To achieve our procedure we must prove the existence of a solution Y of (9) which satisfies (10) where yo has been obtained at Step 2.1. Moreover Y must coincide with Y on E6,r,E fl DQ,r,e where y is the holomorphic continuation of y obtained at step 1. For that purpose, we can see the full equation as a perturbation of the inner equation dY = F(Y, z, e) = Fo (Y, z) + Fi (Y, z, e) dt

182 E. Lombardi

where the perturbation term is given by F\ (Y, z, e) := F(Y, z, e) - F(Y, z, 0). Then, we can solve the equation satisfied by 3^i using once more the contrac­ tion mapping theorem in H™*. Step 3. Continuation near — \a : symmetrization Step 1 and 2 give a holomorphic continuation of y in V* i e U X^ i e . We still denote by ^ this continuation. Moreover, ^ is real valued on the real axis. Hence, the isolated zero theorem ensures that y admits a holomorphic continuation in P f ,., £ U E i i i < e U S^TJ such that X = y@). Finally, since ^ ^ ^ is an isometry, Step 1, 2, 3 ensure that y belongs to E%j(mQ,mi,r,v) and that

Figure 10. Step 3 : continuation in Ea,r>e

Remark 2.17 This strategy of continuation step by step on different subdomains is a rigorous version for this problem of the formal theory of Matching Asymptotic Expansions (the words outer and inner systems of coordinates, and inner system comes from this theory). A clear use of this theory for this kind of problem, was done by Kruskal and Segur in [14]. 3 3.1

Resonances of reversible vector fields Definitions

Definition 3.1 Let V : U x A -> Rn : (u, A) •-► V(u, A) be a p-parameters family of vector fields in Rn defined in an open set UofW1 with parameter A

Phenomena Beyond All Orders and Bifurcations

183

lying in an open set A of W. The family is said to be reversible when there exists a symmetry S € GLn(R), S 2 = Id, which satisfies SV{u, X) = - V ( S u , X)

for every u € U and X 6 A.

In other words the symmetry is the same for all A € A. Definition 3.2 A fixed point u0 of a family of vector fields V(u, A) with u G U, A G A is a point u0 such that V(u0, X) = 0 holds for every A € A. When studying the dynamics of ^=V(u,A) u€Rn, at for a reversible family V{u, X) of vector fields near a reversible fixed point UQ where the spectrum of the differential L(X) = DuV(u0, X) at the fixed point has purely imaginary eigenvalues (non hyperbolic fixed point) for some critical value of the parameter Ac, a first question is the behavior of the eigenvalues of L(A) for A close to Ac- For a real reversible matrix L, if v G ker((L - aid)'), then Sv G ker((L + aid)9) and v G ker((L - aid)9). So, the eigenvalues of L(A) are given by pairs (a, -a) for a G (K U iK) \ {0} and by quadruplets (a,-a, a,-a) for a G C such that Ue{a)lm(a) ^ 0. Moreover, (a, -a) (resp. (a, -a, a, -a)) have generalized eigenspaces with the same Jordan de­ composition. So, if a(X) is a simple eigenvalues for A = Ac it cannot leave the imaginary axis for A close to Ac- So bifurcation of the spectrum can only occur for multiple purely imaginary eigenvalues. So, we define Definition 3.3 A reversible fixed point of a reversible family of vector fields V{u, A) is said to be resonant for X = Ac w^n "»e spectrum of DuV(u0, Ac) contains multiple purely imaginary eigenvalues. 3.2

Linear classification of reversible fixed points

One aim of this work is to study the dynamics of

for vector fields V{u) reversible with respect to some symmetry 5 near a reversible fixed point u0 when the spectrum of the differential L = DUV(u0) at the fixed point has purely imaginary eigenvalues (non hyperbolic fixed point). For simplicity we place the fixed point at the origin (u 0 = 0). We are specially .nterested in the existence of periodic orbits and homoclinic connections near

184 E. Lombardi

the origin. So we are only interested in the dynamics of V up to a linear change of coordinates, and more generally up to a change of coordinates of the form u = $(ui) with $(0) = 0 where $ is a diffeomorphism near the origin. So a very first step is to perform a linear change of coordinates, u = P(ui) with P e GL(RN) to obtain an equivalent equation ~

= V1(u1)

ueRN,

where Si = P^SP and Lx = P~XLP - I>UlVi(0) are simultaneously as simple as possible. Observe that Vi(izi) = P~1V{Pu\) and L\ are reversible with respect to the symmetry S\. Theorem 3.4 below gives the classification of the reversible pairs of N xN matrices, i.e. {(L,5) 6 (M(RN)f / S 2 = Id, LS = -SL} with respect to the equivalence relation ~ defined by the simultaneous conjugacy, i.e. (L,S) ~ (Li,S,) 4^3Pe

GL{RN),

Lx = P~lLP

and Sx =

P^SP.

Moreover it gives for each equivalence class a representative (L, S) where L is a Jordan normal matrix and S is the direct sum of blocks of the form Ip, -Iq, and where 5 is a diagonal matrix . In other words, this theorem says that it is always possible to perform a linear change of coordinates to obtain an equivalent equation governed by a reversible vector field Vi such that (Li,Si) is one of the previous represen­ tatives. Then we introduce a notation inspired by the notation of Jordan matrices used by V. Arnold in [5] to name each classes. To state the theorem we introduce the following convenient notations to describe matrices which are blockdiagonal and antiblockdiagonal: • for a k x k matrix M and a p x p matrix N we denote by M f\) N (resp. by M 0 N) the blockdiagonal matrix (resp. the antiblockdiagonal matrix)

• for kj x kj matrices Mj we denote by [\] Mj (resp. by 0 Mj) the block-

Phenomena Beyond All Orders and Bifurcations 185

diagonal matrix (resp. the antiblockdiagonal matrix) 'Mi 0

£K-

0

;'=I

0 \

•• 0

0 0 M,J

\/]Mj=

0 0 Mr, I 0 .•• 0

>=»

VMi 0

0

For a real matrix L, denote by fir. its minimal polynomial. It has t h e form

n(x) •.=n(* - *;)m(Aj) n ux -a^x - ^))

m(
where A; € R and Oj 6 C \ R. Denote by Jor(L) its real Jordan normal form which reads p m(\j)nk(Xj)

R

m(g,-)ni.(g,-)

Jor(L)=S S S M*i) Sj =Si fc=i S <=iS M°i) j=i *=i <=i where n^(^) is the number of Jordan blocks of size k corresponding to the eigenvalue £ and where J*(A) (resp. Jk{x + iy)) is the A; x k matrix (resp. the 2k x 2k matrix)

/A100\

o'-.'-.o oo'-.i

JkW =

( (*-

0

0

0

"••

o

\

Jk(x + iy) -

(13) 0

\000A/

•

•■■

«

° • ° £1/ Theorem 3.4 (Classification of real reversible matrices) 1. Let (L, S) be two N x N matrices such that S is a symmetry S2 = Id and SL = —LS holds. Then the minimal polynomial of L reads P

Q

tiL(X)=Xml[(X2 - \2)m^l[(X2 j=i

j=\

R

+ u2)m^J[[((X2 -a}){X2 i=i

-a)))m^

186

E. Lombardi

where \J,WJ > 0 and aj 6 C with He {a,) ,1m {a-,-) > 0. The numbers m, P, Q, R may be equal to 0. So the normal Jordan form of L reads

Jor(L) S

S Jt(0) S S S S

Q mjiwj) nk(uij)

R m(
SS N S W S S S j=l

«;=1

(M^)^M-^)) .

^

>.

N (.fc(»j)H-fc(-*i))

i=l *=1

<=1

/

<=1

2. (L,5) ~ (Jor(L),Sor(L,5)) to/iere Sor(L,S) is t/ie matrix m

fp>.(L,S)

fe=i \

9k(t,S)

\

<=i

/

<=i

iu/iere S* (resp. Sk,Sk) given by

S f c :=diag(l,-l,l,-l,---),

p

m(A,)n|.(A,)

i = i *=i

is the k x k (resp. 2k x 2k) diagonal matrix

S* = S(-1) J 'S2, J ^ S t j=o

with pk(L,S)

'=1

1

)^-

(14)

j=o

and qk(L,S) are given by inverse induction

pN{L,S) = dim(ker(L) n \m(LN-1) n ker(S - Id)), qN(L, S) = dim(ker(L) n i m ^ " - 1 ) n ker(5 + Id)), where im(L) is the image (or the range) of L and N Pk{L,

S) = dim(ker(L) n imCL*-1) n ker(S - Id)) -

£

Pi{L, S),

t=fc+i N

qk(L, S) = dim(ker(L) n imtL*- 1 ) n ker(5 + Id)) -

£ ft(L, 5). t=fc+i

5. 5o, (L,S) ~ (Li.Si) t/and on/y t/Jor(L) = Jor(Li) and (Pfc(S,L),
forl
Phenomena Beyond All Orders and Bifurcations

187

Remark 3.5 This theorem ensures that for reversible matrices L, L\ 6 GL(RN) (i.e. invertible), (L,S) ~ (LuSi) holds if and only if Jor(L) = Jor(Lj). In other words, for the reversible matrices which have the same invertible Jordan normal form there is only one possible symmetry up to si­ multaneous conjugacy. In this case, the theorem gives for each Jordan block the corresponding symmetry. However, for the reversible matrices which have the same non invertible Jordan normal form, there are several possible symmetries which leads to different equivalence classes for the simultaneous conjugacy. Each class is characterized by a non invertible Jordan normal form and by a sequence of positive integers (pk,Qk) with 1 < k < N which satisfy p* + g* = n*(0) where nfc(O) is the number of 0-Jordan block of size k in the Jordan normal form: for a class characterized by (p*,?*), the symmetry corresponding to "1.(0)

the direct sum of nt(0) 0-Jordan blocks of size k, i.e. ^

•/*(()), is given by

<=i

(fis*)s(£M*))3.3

Nomenclature

For denoting each class, we must introduce a name which describes the Jordan normal matrix and the sequence (pk,Qk), i-e. the structure of the symmetry corresponding to the 0-Jordan blocks. For denoting Jordan normal matrices we can use the nomenclature introduced by V. Arnold in [5]: A Jordan block Jk (A) corresponding to a real eigenvalue is denoted by A* and a Jordan block Jk (cr) corresponding to a pair of complex conjugate eigenvalues (a, a) is de­ noted byCT*CT*.Then a general Jordan normal matrix is denoted by the formal product of the name of its Jordan blocks. Hence, AoAoAf a2a2 represents the Jordan normal matrix Ji(Ao) S ^ i ( ^ o ) S " ^ 3 ^ 1 ) S " ^ 2 ^ ) For reversible Jordan normal matrices, we know that if A 6 R U iR \ {0} is an eigenvalue, then -A is also an eigenvalue with the same Jordan blocks. So we can omit the product of the name of the blocks corresponding to -A. Similarly, we simply write ak instead of akcrk(-ak)(-ak). So in what follows, AA2cr3 represents the reversible Jordan normal matrix

•MA) S Ji(~X) S J2(A) S J2(-A) S Jz{a) S 2 2

J3(-a).

To obtain very short name, instead of writing AAAA A , we write 3.A(2.A2) Finally for denoting a class up to simultaneous conjugacy we must add to the name of the Jordan normal form something which represents the sequence (Pfc> Ik) 1 < k < N (as already explained, this sequence describe the symmetry

188

E. Lombardi

corresponding to the 0 Jordan blocks). For that purpose we incorporate the sequence in the name of the 0 Jordan block : we denote by (p.0k+)(q.0k~) the class of the pair

_|j*(o),£|st s£l(-sfc)j ^/=1

1=1

1=1

)

Moreover, when p (resp. q) is equal to 0 we do not write the term (p.0fc+) (resp. (q-.O*-)). Hence, for example, we denote by 2.0 + 0 2_ (ia;) the class of the reversible pairs which are simultaneously conjugated to the pair (L,S) with

/00000 00000 00010 L= 00000 00 0 0

0 \ 0 0 0 0-u

VOOOOw 0 j

/10 0 00 01 0 00 00 -100 s = 00 0 10 00 0 01 \00 0 00

0 \ 0 0 0 0

-v

We name resonant fixed point using the nomenclature introduced for re­ versible pairs. So for instance, we say that u 0 is a 0 2 + iu resonant fixed point of V(u,X) for A = A,, or equivalently that V(u,\) admits a 02+k<; resonance at UQ for A = AC "" • V(uo» A) = 0 for A close to A^, • the spectrum of DuV(uo,hc) is {0,±iw} where 0 is double non semi simple eigenvalue and ±iw are simple eigenvalues. • S(tpo) = +ipo where
T h e 02+iu> resonance

This section is devoted to the 02+iu; resonance : let us study one parameter family of analytic vector fields in E 4 , du — =V(u,A), dt

uelR4, Ae[-A0,A0], Ao>0

(15)

Phenomena Beyond All Orders and Bifurcations

189

near 0, assuming that 0 is an equilibrium, i.e. V(0, A) = 0 for A € [—Ao, Ao]. In addition, the vector field is supposed to be reversible, i.e. there exists a reflection 5 € GL 4 (E) such that for every u and A, V(5u,A) = -SV(u,A) holds. We are interested in the neighborhood of the degenerate case when the spectrum of the differential at the origin D„V(0,0) is {±iu>,0} where 0 is a double non semi-simple eigenvalue, and we denote by {<po,ip\, ip+,o = 0, DuV(0,0)v?i = ¥>o, D u V(0,0)y± = ±ik>¥>±, and by (^SiVi.V+.Vl) t n e corresponding dual basis. There exist only two types of such vector fields, since necessarily Sipo = ±ipo holds. The vector fields corresponding to S<po = ¥>o are said to admit a 02+iu> resonance, the other ones are said to admit a 0 2_ io; resonance. In this section, we only study vector fields admitting a 02+'IUJ resonance due to their physical interest. Two last generic hypotheses on the linear and the quadratic parts of the vector field are made. The first one concerns the linear part of the vector field, c = (y>J,Dl; uV(0,0)tpoj / 0. It ensures that the bifurcation really occurs: we do not study the hyper-degenerated case when the double eigenvalue 0 stays at 0 for A ^ 0. The last hypothesis, a := | (y>i, D2J„V(0,0)[po, 0 with |Pfc*(*)| < Mk. It admits also a unique reversible homoclinic connections h to 0 and two families of reversible connections h{, k > 0, j - 1,2 homoclinic to P£(t) satisfying h{ = h and Using the Lyapunov-Schmidt method we check that the periodic orbits persists for the full system, i.e. for p — 1, the full system (16) admits a one parameter family of reversible periodic orbits Pk(t), k > 0 with |Pt(i)| < Mk. The question of the persistence of the homoclinic connections is far more

190

E. Lombardi

intricate. Indeed, roughly speaking, looking for a reversible orbit Y homoclinic to Pk (Y(t) —> Pk(t±(p)) under the form Y = h + Pk + v where v i s a t—>IOO

perturbation term, one has to solve an equation of the type +oo

/

elu^eg(h(t)

+ v{t)) dt.

(17)

-oo

This equation has solutions provided that k > |J(e)|. So, as for our toy model it appears a critical size kc(e) for the reversible homoclinic connec­ tions given by kc{e) = \I{e)\. The critical size kc(e) is the smallest size of a periodic orbit which admits a reversible connection homoclinic to it­ self. So we would like to know whether /(e) vanishes or not. For v = 0, 1(e) = M e (e) is the classical Melnikov function and it is exponentially small: Me(e) = J^&V'gMt)) dt^e-™'2'. The standard perturbation theory simply ensures that 7(e) = Me(e) + 0(e2) = e~™'2e + 0(e2). Such an estimates of 1(e) does not enables us to determine whether 1(e) vanishes or not. To answer this question, we first use the First Mono-frequency Exponential Lemma 2.1 to obtain an exponentially small upper bound of \I(e). This way we obtain T h e o r e m 4.1 For any t €]0,7r[, any p € [0,1] and for every e small enough, the full system (16) admits two families of reversible connections homoclinic to the periodic orbits Pk provided that k which is essentially the size of the periodic orbit, satisfies k > K(l)e2v~utlc. This theorems ensures that every vector field admitting a 0 2+ iw resonance ad­ mits reversible connections homoclinic to exponentially small periodic orbits. Then, with the Third Mono-frequency Exponential Lemma 2.15, we prove Theorem 4.2 There exists a real analytic function defined on [0,1], A(p) = J2 P n A n , such that for any p e [0,1] with A(p) ^ 0, then for e small enough, n>l

the full system does not admit any homoclinic connection to 0, which princi­ pal part is h. The first coefficient Ai is explicitly known in terms of Bessel functions and in terms of the coefficients of the power expansion of the re­ mainder R. Hence, if Ai / 0 (which is generically satisfied), there is at most a finite number of values of p for which the full system (16) admits, for e small enough, a homoclinic connection to 0 which principal part is h.

Phenomena Beyond All Orders and Bifurcations

191

So generically, vector fields admitting at 0 a 0 2+ iu; resonance at the origin do not admit any homoclinic connection to 0, they always admit homoclinic connections to exponentially small periodic orbits. The detailed proofs of Theorems 4.1 and 4.2 are given in [16]. The persistence results given by these two theorems are summed up in Figure 11. A sketch of proof for Theorem 4.2 is given in the following subsection. k k*\

* = K(fy? e"f k « A(/j)e e (expected critical size )

0 nonpersisteno Truncated system (p = 0)

Full system (p ^ 0, 0 < £ < ir)

Figure 11. Domain of existence of reversible homoclinic connections to a periodic orbit of size k for the truncated system and their domains of persistence and non-persistence obtained in this chapter

Sketch of proof for Theorem 4-2 We make a proof by contradiction. Assume that Y is a reversible homoclinic connection to 0 of (16) and denote by v = Y — h the perturbation term where h is the reversible homoclinic connection of the truncated system, i.e. of the system (16) with p = 0. Thus, the perturbation term v satisfies the equation dv — -DN(h).v

=

g(v,t,P,e)

(18)

with g(v,t,e)

= Q(v)+pR(h+v,e)

where Q(v,e) =

N(h+v,e)-N(h,e)-DN(h,e).v.

Then, the crux point of the analysis, is the following : for an antireversible function / which goes to 0 exponentially at infinity, there exists a reversible function u which goes to 0 exponentially at infinity and which satisfies ^-DN(h,e).u

=f

(19)

192

E. Lombard*

if and only if the function / satisfies the solvability condition

f

r+oo

(r.(t),f(t))dt =0 /o where the vector r_ is the solution of the homogeneous equation (19) with / = 0 explicitly given by Jo

r _ ( t ) : = (0,0,-sintf e (0,cosik(0)

with

rpe{t) = (» + ae) *+26etanh(|i).

Thus, if the full system (16) admits a reversible homoclinic connection Y to 0, then v = Y - h necessarily satisfies r+oo

I(e,p):=

/ Jo

(r-(t),Q(v,v)+PR(h{t)

+ v{t),u))dt

= 0.

Our aim is then to prove that T(e, p) is exponentially small but does not vanish which leads to a contradiction. The classical Melnikov approach is based on the idea that the leading part of I{e,p) comes from the leading part of Y on R. So one introduce the Melnikov function /•+oo

Me(p,e)=p

(r-(t),R{h(t),e))dt Jo .

r+oo = ij^

e l(( 7 +~)f+26eunh(ie))

(RB(h(t),e)+iRA(h(t),e))dt

where R = (Ra,Rp,RA,RB), expecting that it is the leading part of I(e,p). However, here Me{p,e) is given by an oscillatory integral which can be com­ puted explicitly using the residues and which is exponentially small U)7T

Me(p,e) = pe- e (A1+

O (e)). £->0

where Ai is explicitly known in terms of Bessel functions and in terms of the coefficients of the power expansion of the remainder R. Using classical perturbation theory, on can only obtain that I{e,p) = Me(p,e) + 0(e2). Such an estimates does not enables us to determine whether I(e, p) vanishes or not. So to obtain an equivalent of the oscillatory integral I{e,p) with Lemma 2.15, we follow the strategy proposed in subsection 2.4 to prove that Y = h + v belongs to i£j(2,3,I,i) x ^ j ( 3 , 4 , i , I ) x££j(2,3,l,I) x£$(2,3,*,!). for A e]0,1[ and some S > 1. This way we obtain that I{e,p) = {pAi + ] T p n A n ) e-Tn>2

Phenomena Beyond All Orders and Bifurcations

193

Observe that the leading part h on R of Y and the perturbation term v contribute to the size of the integral at the same order of e. For the numerical constant A(p) = £ n > 1 p n A n , the first term pAi comes from h and the other terms come from v. Thus generically A(p) ^ 0, so I(e, p) is exponentially small but does not vanishes and thus there is no reversible homoclinic connection to 0. 5

The (iu;o)2iu/i resonance

This third section is devoted to the (io>o)2iwi resonance : let us study a one parameter family of analytic vector fields in R 6 , ^ = V(u, A), « e R 6 , A 6 [-A0) A0], A0 > 0 (20) at near 0, assuming that 0 is an equilibrium, i.e. V(0, A) = 0 for A 6 [-Ao, Ao]. In addition, the vector field is supposed to be reversible, i.e. there exists a reflec­ tion S G GL 6 (R) such that for every u and A, V(5u, A) = -SV(u, A) holds. We are interested in the neighborhood of the degenerate case when the spectrum of the differential at the origin D u V(0,0) is {±iwo,±uJi} where ±iwo is a double non semi-simple eigenvalue, and we denote by (o,ip*,4>f) a basis of eigen­ vectors and generalized eigenvectors D„V(O,O)0* = ±ILJQ4>Q, DUV(0,0)ipp = ±iu0ip* + f. such that ^ = <$ and ip^ = ipQ- De­ note by ((<$'*,ip£'*,<j>f'*) the corresponding dual basis. Moreover, we make non resonance assumptions, 2 ^ ^p/q for p + q< 5; ^ 4. - ; ^i. £ N <=$> w* = min |wi - pu0\ > 0. Two last generic hypothesis on the linear and the cubic parts of the vector field are made. The first one concerns the linear part of the vector field, q° = (tpo'*,DluV{0>0)(j)^\ ^ 0. It ensures that the bifurcation really oc­ curs: we do not study the hyper-degenerated case when the double eigen­ values ±\uo stay on the imaginary axis for A ^ 0. The last hypothesis, 9° : = ( ^ ' ' . D L u ^ C 0 ' 0 ) ^ ' ^ ' ^ ] ) < ° ensures that the normal form sys­ tems admit homoclinic connections to 0. This last condition is the same as the one obtained in [13] for the (iw)2 resonance (also called 1:1 resonance ). One example of such vector fields in infinite dimensions occurs when studying traveling waves in a chain of nonlinear coupled oscillators (see [12]).

194

E.

Lombardi

To study the dynamics of (20) near the origin, one can use the normal form theory (see [1]) to obtain an equivalent system (u = $(Y,A) where * is a polynomial diffeomorphism of degree 3 close to identity) ^

= N(Y,e) + pR(Y,e)

(22)

where e = y/qfX; N is the normal form of order 3; R represents the higher or­ der terms and p is an additional parameter : p = 0 corresponds to the normal form system and for p = 1, (20) is equivalent to (22). For p = 0, the normal form system admits a one parameter family of reversible periodic orbits Pk* (t), fc > 0 with \Pk*{t)\ < Mk. It admits also two reversible homoclinic connections to 0 h and h and four families of reversible connections h{, t){, k > 0, j = 1,2 homoclinic to P£(t) and satisfying /ij = h, hj = h, h[(t) —> Pfc*(t ± tpj), fjfc W . ~Z* Fk (f ± 0 with \Pk(t)\ < Mk. The question of the persistence of the homoclinic connections is far more intricate. Indeed, roughly speaking, looking for a reversible orbit Y homoclinic to Pk (Y(t) t^x> Pk(t ± y)) under the form Y = h + Pk + v where v is a perturbation term, one has to solve an equation of the type +0O

/

e iu " t/e fl(/i(t) + v(t)) dt.

(23)

■oo

This equation has solutions provided that k > \I{e)\. For v = 0,1(e) ~ M e (e) is the classical Melnikov function and it is exponentially small. Indeed, it has the form

where u;* is defined in (21). Such an integral can not be estimated using the Third Mono-frequency Exponential Lemma 2.15 which gives exponen­ tial equivalent of oscillatory integral f*™eiUlt/c f{t,e)dt provided / admits a holomorphic continuation in a suitable space of bounded functions. Here, for every small t > 0, supje cosh - 1 (tf) cos(i&>0/e)l = +oo because of the sec£>0

ond rapid oscillations induced by w0. To solve this difficulty, we make here only a partial complexification of time : in subsection 5.1 we give a first BiFrequencies Exponential Lemma 5.3 which gives an exponential upper bound

Phenomena Beyond All Orders and Bifurcations

195

for a bi-oscillatory integral +0C

/

f(t,^,e)ei*»t'sdt

(25)

-oo

provided that / : (£, s,e) •-► /(£, s,e) belongs to a space bH£ of functions holomorphic with respect to f € C and 27r-periodic with respect to s G R. Such a lemma enables us to deal with integral of the type of (24). In subsection 5.2, we briefly explain how to prove that a solution y of a nonlinear differential equation can be written as y(t) := y(t,uot/e,e). Using this strategy we can finally prove Theorem 5.1 For any £ e]0, f [, any p G [0,1] and for every e small enough, the full system (22) admits four families of reversible connections homoclinic to the periodic orbits P* provided that k which is essentially the size of the periodic orbit, satisfies k > M(£)e~"'* // ' £ . This theorems ensures that every vector field admitting a (iu;o)2iu;i resonance with q° < 0 admits reversible connections homoclinic to exponentially small periodic orbits. To study the persistence of the reversible homoclinic connection to 0 (which corresponds to k=0) we must determine whether 1(e) defined in (23) vanishes or not. For that purpose, we give in section 2 a second Bi-Frequencies Exponential Lemma 5.6 which gives an exponential equivalent of bi-oscillatory integrals. Using this Lemma, we can prove Theorem 5.2 There exists a real analytic function defined on [0,1], \(p) = £ p n A n , such that for any p € [0,1] with \(p) ^ 0, then for e small enough, n>\

the full system does not admit any reversible homoclinic connection to 0, which principal part is h. The first coefficient Ai is explicitly known in terms of Bessel functions and in terms of the coefficients of the power expansion of the remainder R. Hence, if k\ ^ 0 (which is generically satisfied), there is at most a finite number of values of p for which the full system admits, for e small enough, a reversible homoclinic connection to 0 which principal part is h. So generically, vector fields admitting at 0 a (iwo)2iwi resonance with g° < 0 at the origin do not admit any reversible homoclinic connection to 0, whereas they always admit homoclinic connections to exponentially small periodic orbits. The detailed proofs of Theorems and Lemmas given in this section are available in [16].

196

5.1

E. Lombardi

Exponential asymptotics of bi-oscillatory integrals

Lemma 5.3 (First Bi-Frequencies Exponential Lemma) Let I, A be two positive numbers. Let bH$ be the set of functions f satisfying 1. f : Btx Rx]0,l] -> C; f ►-* /((,«,e) is holomorphic in Bt := {£ e C / | I m ( 0 | < 1}; a i-4 f(t,s,e) is 2ir-periodic and of class C 1 in R,(£, s) »-> /(£, s,e) is continuous in Bt x R; 2- ||/IL„ A :=

sup

( ( | / t t , « , e ) | + \%-(t,s,e)\)

e^^A

< +oo.

Let wo, <*>i, 6e two real positive numbers. Define w* by w* = min |wi — pu>o|. T/ien /or every f €bHf and e e]0,l],the bi-oscillatory integral I{f,e) defined by (25) satisfies

The Lemma 5.3 gives a very efficient way to obtain exponential upper bounds because the membership to bH( is stable by addition, multiplication, "compo­ sition": the space bH£ is an algebra and for every / ebH( and g holomorphic in a domain containing the range of / and satisfying g(0) = 0, go f ebH^ holds. For computing an equivalent of the bi-oscillatory integral, we need more information on / . Definition 5.4 Assume 7, A > 0, a > 0, 6 > 1 , mo.m! > 1, 0 < r < 1 and u>0. Denote by bEl'g (m 0 ,mi,r,iv) the set of functions f such that 1. f :

U

Ba-s£ x R x {e} -> C; { >->• {£,s,e) is holomorphic in Bf,

0
s >-»■ (£,s,e) is 2n-periodic and of class C 1 in R; (f,s) »-> (£,s,e) is continuous in B( x R and ||/|| ou , < +00 where

ll/IU := sup

0<E<<7

(|/«,*.e)| + |§j«,*,e)|)x

(C,«)6B<,-i,xR

(itt - w)(€ + i
Xl (fl)e*l*««>l)

«»** Xi ( 0 = 1 /or \Ke {$ | < 1 and *, ( 0 = 0 ot/iemwe. 5. /or every e e ]0,e0] and every z € E«,r,e := j - e 1 - 1 , ^ - ^ x ] - e r _ 1 , - o " [ , / reads /(±(Ur + « ) , « , e ) = — # ( * , « ) + — /,*(«,«, e)

Phenomena Beyond All Orders and Bifurcations

197

where ff £bH?° and f± GbH^ with

*&,"-= {/: ns

: -¥ C,

where Ug := 1 x ]-oo, -6[, z i-> f{z,s) holomorphic in Clg, s t-» /(z, s) 2n-periodic in R, 2, s) •-»• / ( z , s) contt'nuorw in tig x R := sup C|/(2r,s)||^|mo1) < +00} v y J * r ° (*..)€SixR

and b

ffTr= \J1

U (s«, P , e x R x {e}) -> C,

ee]0,eo]V

2 i-> f(z,s,e) is holomorphic in Ej.r.e, s •-> /(2,a,£) 2-rr-periodic in R, 2,3) »-» /(2,s,e) continuous in T^s
8tr

SUP (|/(2)||2|mi) < +00} X J
Remark 5.5 The two functions f0 and f\ are uniquely defined. Indeed, they are given by f*(z,s) := lime 7 /(±(io- + ez),s,e) and / * ( * ) = lim£ _ , / (£ 7 / ± (±(io- + ez),s,e)

- /,*(«)). Denote by Pf* the operators P,-* :

e—yO

Lemma 5.6 (Second Bi-Frequencies Exponential Lemma) Assume 7 > 1, 5 > 1,CT,A > 0, mo, mi > 1, 0 < r < 1, v,u> > 0. Let WQ, U>I 6e iu/o positive numbers. Define w* 6y w* = min|u;i -pu>o|- For ^ £ N + i , tAis minimum is reached for a unique value ofp denoted hyp*. When jjj1 € N+ 5, it is reached for 2 values of p and we denote by p* the smallest one. Then, for every function f EbE2's (m.Q,m\,r,v) the bi-oscillatory integral I(f,e) satisfies

' = -Fn-( A + .2D('")) where u* := min(t', (1 - r)(mo - 1), (1 - r)(7 - 1)) and A is given by

l.for%?N+l A := / dz e'u-2~ f f(z, s)eip-'ds J-in+K InJ-r,

with n > 6

198 E. Lombardi

where f = f£ when u>i - p*<^o > 0 and f = f0 when W\ - p*u>0 < 0; 2. for wi = (p* + \)u0, A := |


im't/j n > 6. This lemma ensures that the relevant parts of / ebEFa't for computing an equivalent of the oscillatory integrals /*(/,£) are P^C/). Moreover, it gives a very efficient way to compute equivalents of oscillatory integrals in­ volving solutions of non linear differential equations since the principal part P* of a sum (resp. of a product) is the sum (resp. the product) of the princi­ pal parts: P* :bE^ (mo,mi,r,v) t-tbH™° are linear operators and for every f,g £bE^, e-rfg lies in » £ $ and P?&fg) = P?U)P?(g). The exponential lemma 5.6, can be proved using the exponential lemma given in section 4 for mono-frequency oscillatory integrals. Observe that here, the interaction be­ tween the two frequencies u>o and wj changes the decay rate in the exponential: here we have w* instead of u)\ for the mono frequency case. 5.2 Strategy for non linear differential equations For using the Bi-Frequencies Exponential Lemma 5.3 to compute equivalents of bi-oscillatory integrals / ( / , e) when / involves solutions of real nonlinear differential equations, we must be able to prove that a given solution y of a real nonlinear analytic differential equation ^=F(Y,e)

(26)

can be written y(t) = y(t, ^,e) where y belongs to hH$. For that purpose we look for solutions of the first order P.D.E.

since every solution y of (27) gives a solution y of (26) by setting y{t) »(*. ^ . e ) - Observe that if yi((,s) = y2($,s) + $(( - e/uos) with 0(0) = 0 then j/i and j/ 2 lead to the same function y(t). This indetermination induced by the introduction of two times variables f and s lead to some technical difficulties. However, we can use the method of characteristics to transform (27) into an integral equation of the form y = T(F(y, e)) where T is a suitable integral operator. Then, we solve this last equation using the contraction

Phenomena Beyond All Orders and Bifurcations

199

mapping theorem in spaces of type bH(. For obtaining solution of (26) which reads 3^(0 = y(t, ^,e) where y belongs to bE%'g we must combine the above strategy with the one proposed in subsection 2.4 for obtaining solution of (26) in E^'t which is the suitable set of functions for computing equivalent of mono-frequency oscillatory integrals. References 1. Iooss G., Adelmeyer M., Topics in bifurcation theory and applications. Advanced Series in non Linear Dynamics, 3, World Scientific, 1992. 2. Amick C.J., McLeod J.B. Arch. Rational Mech. Anal. 109, 1990, 139171. 3. Amick C.J., McLeod J.B. A singular perturbation problem in water waves Stab. Appl. Anal, of Cont. Media, 1992, 127-148 4. Arnold V.I. Instability of dynamical systems with several degrees of free­ dom. Dokaldy Akad Nauk SSSR 156, 1964, 581-585 5. Arnold V.I. Geometrical methods in the theory of ordinary differential equations. Springer, 1983. 6. Delshams A., Seara T.M. An asymptotic expression for the splitting of separatrices of rapidly forced pendulum. Com. Math. Phys. 150, 1992, 433-463. 7. Gelfreich V.G. Reference system for splitting of separatrix. Nonlinearity 10, 1997, 175-193 Gelfreich V.G. Separatrix splitting for a high-frequency perturbation of the pendulum (1996, unpublished). 8. Hammersley J.M., Mazzarino G. Computational aspects of some au­ tonomous differential equations. Proc. Roy. Soc. London Ser. A 424 (1989) 9. Hammersley J.M., Mazzarino G. IMA J. Appl. Math. 42, 1989, 43-75 10. Holmes P., Marsden J., Scheurle J. Exponentially small splittings of sep­ aratrices with applications to KAM theory and degenerate bifurcations. Contemporary Mathematics Vol. 81, 1988, 213-244. 11. Iooss G., Kirchgassner K., Water waves for small surface tension: An approach via normal form, Proceedings of the Royal Society of Edinburgh 122A, 1992, 267-299. 12. Iooss G., Kirchgassner K., Traveling waves in a chain of coupled nonlinear oscillators. Comm. Math. Phys. to appear. 13. Iooss G., P&roueme M.C., Perturbed Homoclinic Solutions in 1:1 Reso­ nance Vector Fields, Journal of Differential Equations 102 1993, 62-88. 14. Kruskal M.D., Segur H., Asymptotics beyond all orders in a model of

200

E. Lombardi

crystal growth , Studies in Applied Mathematics 85 (1991) 129-181. 15. Lombardi E., Homoclinic orbits to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rational Mech. Anal. 137 (1997) 227-304. 16. Lombardi E., Oscillatory integrals and phenomena beyond any alge­ braic order; with applications to homoclinic orbits in reversible systems. Springer Verlag Lecture Notes in Mathematics To appear. 17. Melnikov V.K. On the stability of the center for time periodic perturba­ tions. Trans. Moscow Math. soc. 12, 1963, 1-57. 18. PoincarSH. Les methodes nouvelles de la mecanique celeste (vol. 2) Paris: Gauthier-Villars (1893). 19. Pomeau Y., Ramani A., Grammaticos B. Structural stability of the Korteweg-De Vries solitons under a singular perturbation. Physica D 31,1988, 127-134 20. Segur H., Tanveer S., Levine H., Asymptotics beyond all orders. NATO ASI Series B Vol. 284 (1991). 21. Sun S.M., Shen M.C. Exponentially small estimate for the amplitude of capillary ripples of generalized solitary wave J. Math. Anal. Appl. 172, 1993, 533-566. 22. Sun S.M. Non-existence of Truly solitary waves in water with small sur­ face tension. Proc. Roy. London, 1999, in press.

MATHEMATICAL MODELLING IN THE LIFE SCIENCES: APPLICATIONS IN PATTERN FORMATION A N D W O U N D HEALING PHILIP K. MAINI Centre for Mathematical Biology, Oxford

1

Introduction

Spatial and spatio-temporal patterns occur widely in the life sciences as well as in chemistry. Perhaps the best-known example of spatio-temporal pattern formation is the spontaneous generation of propagating fronts, target pat­ terns, spiral waves and toroidal scrolls in the Belousov-Zhabotinsky reaction, in which bromate ions oxidise malonic acid in a reaction catalysed by cerium, which has the states Ce 3 + and Ce 4 + . Sustained periodic oscillations are ob­ served in the cerium ions. If, instead, one uses the catalyst Fe 2 + and Fe 3 + and phenanthroline, the periodic oscillations are visualised as colour changes between reddish-orange and blue (see, for example, Murray, 1993, Johnson and Scott, 1996 for review). Similar types of patterning arise in physiology and one of the most widely-studied and important areas of wave propagation concerns the electrical activity in the heart (Panfilov and Holden, 1997) which stimulates muscle contraction resulting in the heart beating. Understanding the mechanisms underlying spatio-temporal pattern for­ mation is a central goal in embryology. Although genes control pattern for­ mation, genetics does not give us an understanding of the actual mechanisms involved in patterning. Many models of how different processes can conspire to produce pattern have been proposed and analysed. They range from gradienttype models involving a simple source-sink mechanism (Wolpert, 1969); to cellular automata models in which the tissue is discretised and rules are intro­ duced as to how different elements interact with each other (see, for example, Bard, 1981); to more complicated models which incorporate more sophisti­ cated chemistry, physics and biology, and which propose that patterns are set up due to self-organisation rather than as a consequence of externally imposed :ues. All of these models focus on the key question of how cells respond to and interact with signalling cues. This is a crucial question in a number of areas oi the biomedical sciences. For example, in tumour formation, one wishes to inderstand how the signalling cues involved in the regulation of cellular projesses no longer function properly. Many of the physical processes that occur 201

202 P. K. Main.

during the formation and spread of tumour cells are also of vital importance in wound healing, where their function benefits the organism rather than be­ ing destructive to it. Recent advances in molecular and cellular biology have led to the rapid development of experimental research into the biochemical mechanisms underlying the processes of wound healing. Wound healing is an enormously complex dynamic spatio-temporal process and new insights are being gained by focussing on the interaction of specific processes involved for a particular aspect of healing. In all the above areas, a number of complex mechanical and biochemical processes are interacting in a highly nonlinear way. Such systems are amenable to mathematical modelling and the role of the modeller is to suggest explana­ tions, based on biologically plausible mechanisms, of observed behaviour and to make experimentally testable predictions. In this article, I will review, in Section 2, some of the commonly used models for pattern formation in early development. In Section 3 I will describe some models that have been used to address particular phenomena in wound healing. At the end of each section, a discussion on the particular applications is presented. Section 4 contains a general discussion and points to future directions. 2

Models for pattern formation and morphogenesis

Cell fate and position within the embryo can be strongly influenced by en­ vironmental factors. Therefore, to answer questions on pattern formation, one must really address the issue of how the embryo organises the complex spatio-temporal sequence of signalling cues necessary to develop structure in a controlled and coordinated manner. Structure can form through tissue movement and rearrangement, cell-cell interaction, or in response to chemical signals. We first consider the latter type of model. 2.1

Chemical pre-pattern models

The simplest chemical pre-pattern model is the gradient model proposed by Wolpert (1969) in which a source-sink mechanism, coupled with diffusion and degradation, leads to a spatial gradient in a single morphogen. He proposed that this provided positional information for cells, which differentiated ac­ cording to a series of threshold values. More complicated spatial patterns can be generated due to the reaction and diffusion of a number of chemicals. This phenomenon is known as diffusion-driven instability and was first proposed by Turing in a seminal paper (Turing, 1952). The reaction kinetics he consid-

Mathematical Modelling in the Life Sciences

203

ered were stabilizing and diffusion is, of course, a homogenizing process. Yet combined in the appropriate way, these two stabilizing influences can conspire to produce an instability which can result in spatially heterogeneous chemical profiles - a spatial pattern. This is an example of an emergent property and is termed diffusion-driven instability, that is, a spatially uniform steady state, linearly stable in the absence of diffusion, becomes linearly unstable in the presence of diffusion.

To derive the partial differential equation form of Turing's model, let us first consider a single chemical, with concentration c(x, t) at position x E R3 and time t E (0, oo). Consider an arbitrary volume V C R3. Then rate of change of chemical in V = - a flux + net production i.e. dcdv = FAS + f (c)dv v v v where F is the flux of chemical per unit area and f (c) is net chemical production per unit volume. On using the divergence theorem and the fact that the volume V is arbitrary, this equation becomes

J

f

ac

at

J

= -V.F + f (c). (2)

The function F is determined by Fick's Law, which states that chemical flux is proportional to the concentration gradient, i.e. F = -DVc (3) where D is the diffusion coefficient. Therefore the reaction-diffusion equation governing the spatio-temporal evolution of c takes the form: ac

at

= V.(DVc) + .f (c)•

(4)

To complete the model formulation we need to specify initial conditions, c(x, 0) = co(x) and boundary conditions. The latter may typically be periodic, zero flux or fixed, depending on the phenomenon being modelled. The above derivation can be generalised easily to a system of interacting chemicals leading (in the case of constant diffusion coefficients) to au =

at

DV2u +

f(u), (5)

where u is a vector of chemical concentrations, u = (ul, u2, ..., u„ )T ; f = (fl (u), f2 (u), ..., fn(u))T and models chemical interaction; and D is an n x n diffusion matrix. In the simplest examples, D is a diagonal matrix. More generally, D can have off-diagonal terms to model cross-diffusion.

204 P. K. Maini

The classical reaction-diffusion model is a system of two chemicals, u and v, reacting and diffusing as follows: — = I>iV 2 u + /(u,t;)

(6)

— = D-2V2v + g{u,v).

(7)

We will assume zero flux boundary conditions. The functions / and g are rational functions of u and v (see examples later). Using standard linear analysis (see, for example, Murray, 1993) it can be shown that a spatially uni­ form steady state of the above system can undergo diffusion-driven instability if the following conditions hold: ( C . l ) /„ + gv < 0 (C.2) fugv

- fv9u > 0

(C.3) Dl9v

+ D2fu > 2^D1D2(fu9v

-

fv9u)

where all the partial derivatives are evaluated at the spatially uniform steady state. One possible scenario for pattern formation is that in which fu and gu are positive, the latter implying that u activates v, while gv and /„ are negative, so that v inhibits u (see, for example, Dillon et al, 1994). Condition (C.l) =* l/ul < lflt>|, s o fr°m ( c -3) Di < D2. That is, the activator diffuses more slowly than the inhibitor. This is an example of the classic property of many self-organising systems, namely short-range activation, long-range inhibition. In Turing's original model, / and g were linear so that, if the uniform steady state became unstable, then the chemical concentrations would grow exponentially. This, of course, is biologically unrealistic. Since Turing's paper, a number of models have been proposed wherein / and g are nonlinear so that when the uniform steady state becomes unstable, it may or may not evolve to a bounded, stationary, spatially non-uniform, steady state (a spatial pattern) depending on the nonlinear terms. These models may be classified into four types: (i) Phenomenological Models: The functions / and g are chosen so that one of the chemicals is an activator, the other an inhibitor. An example is the Gierer-Meinhardt model (1972), for which 2

/(u,u) = a - / ? u + — ,

g(u,v) = Su2 - nv

(8)

Mathematical Modelling in the Life Sciences

where a,0,y,S u.

205

and 77 are positive constants, u activates v and v inhibits

(ii) Hypothetical Models: Derived from a hypothetically proposed series of chemical reactions. For example, Schnakenberg (1979) proposed a series of trimolecular autocatalytic reactions involving two chemicals as follows *i

XT±A,

B-^Y,

2I + K-^3I.

Using the Law of Mass Action, which states that the rate of reaction is directly proportional to the product of the active concentrations of the reactants, and denoting the concentrations of X, Y, A and B by u, v, a and b, respectively, we have f(u, v) — faa - kxu + ktu2v,

g(u, v) = £36 - k^2v

(9)

where ki,...,k4 are (positive) rate constants. Assuming that there is an abundance of A and B, a and b can be considered to be approximately constant. (iii) Empirical Models: The kinetics are fitted to experimental data. For example, the Thomas (1975) immobilized-enzyme substrate-inhibition mechanism involves the reaction of uric acid (concentration u) with oxy­ gen (concentration v). Both reactants diffuse from a reservoir maintained at constant concentration uo and vo, respectively, onto a membrane con­ taining the immobilized enzyme uricase. They react in the presence of the enzyme with empirical rate ^ ^ ' J M , so that /(u,i>) = a{UQ -u)

g(u,v) = (3(v0 -v)-

-

«•+•+"/*.•

(10)

2/

Km +u + u'/K, where a, 0, Vm, Km and K, are positive constants. (iv) Actual Chemical Reactions: Although Turing predicted, in 1952, the spatial patterning potential of chemical reactions, this phenomenon has only recently been realised in actual chemical reactions. Therefore, it is now possible in certain cases to write down detailed reaction schemes and derive, using the Law of Mass Action, the kinetic terms.

206 P. K. Maim

The first Turing patterns were observed in the chlorite-iodide-malonic acid starch reaction (CIMA reaction) (Castets et al., 1990, De Kepper et al., 1991). The model proposed for this reaction by Lengyel and Epstein (1991) stresses three processes: the reaction between malonic acid (MA) and iodine to create iodide, and the reactions between chlorite and iodide and chloride and iodide. These reactions take the form MA + h -> IMA + r + H+ C102 + r -► CIO2 + \h CIO2 + 4 / - + 4H+ -> Cl~ + 2/ 2 + 2H20. The rates of these reactions can be determined experimentally. By making the experimentally realistic assumption that the concentration of malonic acid, chlorine dioxide and iodine are constant, Lengyel and Epstein derived the following model: — = ki - u - — — 2 + V2u at 1+u 90

-k

k3 u

{ -TT*)

+ cV2u

where u,v are the concentrations of iodide and chlorite, respectively and &ii &2, ^3 and c are positive constants. Murray (1982) calculates and compares the parameter spaces determined by (C.1)-(C3) for linear instability in the Gierer-Meinhardt, Schnakenberg and Thomas models. In many cases, particularly in one dimension, the results of the linear analysis carry over to the weakly nonlinear case but the fully nonlinear system can only be analysed by numerical simulation. There is now a great deal of literature on this subject and general results on the patterning properties of reactiondiffusion equations can be found in the books by Britton (1986), EdelsteinKeshet (1988), Fife (1979), Grindrod (1996), Murray (1993) and Segel (1984). The gradient models and the Turing-type models differ in two crucial aspects: In the gradient model, the chemical pre-pattern is set up by a sim­ ple process which can only produce a simple gradient. To use this gradient to generate complicated pattern, it is hypothesized that a complex series of thresholds exist and cells have the machinery to interpret multiple thresholds. In Turing's model, complex spatial patterns arise due to a complex chemical interaction, but the interpretation of the pre-pattern is via a single threshold (Nagorcka, 1989) and is therefore simpler than that in the gradient model.

Mathematical Modelling in the Life Sciences

207

2.2 Cell movement models Cell movement models are based on the assumption that a spatial pattern arises in cell density, and cells then differentiate in a density-dependent manner. Cell aggregation occurs when the cell dispersing factors (for example, diffusion) are overcome by aggregating factors such as chemotaxis (movement up chemical gradients), or factors generated by the mechanical interaction of cells with the extracellular matrix (ECM) on which they move. These include haptotaxis (movement up adhesive gradients) or passive convection arising as the result of deformation of the ECM due to cell traction. Chemotactic models have been analysed by a number of authors and shown to lead to spatial pattern formation (see, for example, Keller and Segel, 1971; Maini et al., 1991). These models involve reaction and diffusion, but spatial patterning arises in this case due to the advective term introduced by chemotaxis. The typical model takes the form:

On =

at

DnV2n - V.(X(u)nVu) + f (n, u) 8u = D,,V 2 u+g(n,u), 8t

(11)

(12)

where n(x, t), u(x, t) denote cell density and chemoattractant concentration, respectively, at position x and time t; D, Du are diffusion coefficients, f, g are terms incorporating production and degradation and X(u) is the chemotactic sensitivity. The latter varies depending on the mode of cell-chemoattractant interaction (Othmer and Stevens, 1997). The first partial differential equation model incorporating the role of mechanical cues in the formation of cell aggregation was proposed by Oster et al. (1983) and since then such models have been extensively studied (Murray, 1993). They consist of conservation equations for cells and extracellular matrix, which take the general form of the equations above, but the main difference is the force-balance equation, which is that for a viscoelastic material. The mechanical model proposed by Oster et al. (1983) has three dependent variables: n(x, t), p(x, t) and u(x, t) which represent, respectively, cell density, matrix density and matrix displacement at position x and time t. The cell equation is an

at

= - V.J + rn(N - n)

(13)

where net cell production is assumed to be of logistic form, r and N are positive constants, and J, the cell flux, is given by J = -DOn+anVp+n et .

208 P. K. Maini

The first term on the right-hand side models random motion and the third term accounts for passive advection. Cells can also move by attaching cell processes to adhesive sites within the matrix and crawling along. As cells exert forces on the extracellular matrix they generate adhesive gradients which serve as guidance cues to motion. The movement up such gradients is termed haptotaxis. The assumption that the number of adhesive sites is proportional to ECM density leads to the second term on the right-hand side, where a is the haptotactic coefficient, assumed to be a non-negative constant. Hence the equation for cell motion is -£ = DV2n - aV.(nVp) -V.(n~)+

rn(N - n).

(14)

The equation for ECM density is much simpler as the only contribution to matrix flux is advection, and matrix secretion is assumed negligible. Hence, p satisfies dp _ ( du\ v 15

it = - - {'*) ■

< >

To derive the equation for the matrix displacement, u(x, t), we first note that for cellular and embryonic processes, inertial terms are negligible in compar­ ison to viscous and elastic forces, that is, motion ceases instantly when the applied forces are turned off. Hence the traction forces generated by the cells are balanced by the viscoelastic forces within the ECM. Therefore the equilibrium equations are V.a + pF = 0 (16) where a is the composite stress tensor of the cell-ECM milieu and pF accounts for body forces. Oster et al. (1983) model the cell-matrix composite as a linear viscoelastic material with stress tensor <7 = <7 p +cr n .

(17)

Here ap is the usual viscoelastic stress tensor (see, for example, Landau and Lifshitz, 1970), de viscous

89,

E

(

v

\

elastic

where: 6 = V.u is the dilatation, e = |[Vu + Vu T ] is the stress tensor, I is the unit tensor, p,i,pL2 are the shear and bulk viscosities, respectively, and E, v are the Young's modulus and the Poisson ratio, respectively.

Mathematical Modelling in the Life Science!

209

The stress due to cell traction is modelled by

where r and A are positive constants. This satisfies the conditions that there is no traction without matrix and that traction per cell decreases with increasing :ell density (contact inhibition). If the cell-matrix composite is attached to an external substratum, for example a subdermal basement layer, then the simplest way to model the aody force is to assume F = su

(20)

where s is the modulus of elasticity of the substrate to which the composite is attached. With appropriate boundary conditions (for example zero flux on n and o, with u fixed) the above equations for cell density, matrix density and dis­ placement define a simple version of the more complicated mechanical model presented by Oster et al. (1983). Linear and nonlinear analyses, plus numerical simulation, show that modsis within this general mechanical framework can exhibit steady-state spa­ tial patterns (Perelson et al., 1986) and spatio-temporal patterns (Ngwa and Maini, 1995). Other movement models hypothesize that cells move to minimize energy (Cocho et al, 1987a,b; Steinberg, 1970; Sulsky, 1984). Such models can be set up mathematically and solved to show cell sorting and patterning behaviour consistent with a number of experimentally observed phenomena. 2.3

Cell rearrangement models

Theoretical studies in this area include the early purse-string model of Odell et al. (1981) for tissue folding in which, in response to a large deformation, cells were proposed to actively contract and in so doing cause a large de­ formation in neighbouring cells which, in turn, also contract, setting up a propagating contraction wave which leads to tissue folding. This model was applied to a variety of developmental problems, and provided the precursor to the "mechanochemical theory" of developmental pattern formation reviewed above. This approach emphasises the link between tissue mechanics and chem­ ical regulation, and has been applied widely in both developmental biology and medicine. Subsequently, Weliky and Oster developed a discrete-cell modslling approach in which morphogenesis occurs via mechanical rearrangement

210 P. K. Maini

of neighbours in an epithelial sheet. They assume that the boundary of the ep­ ithelial sheet is pulled over the surface of the egg and show that the resultant model can produce many experimentally observed aspects of both Pundulus epiboly and notochord morphogenesis in Xenopus laevis (Weliky and Oster, 1990; Weliky et al., 1991). More recently, Davidson et al. (1995) used a com­ putational finite element model to test various explanations for sea urchin invagination. In all these models individual cell movements within the tissue are deter­ mined by the balance of mechanical forces acting on the cell. Such models can exhibit tissue folding, thickening, invagination, exogastrulation and intercala­ tion, and have been shown to capture many of the key aspects of processes such as gastrulation, neural tube formation, and ventral furrow formation in DrosophUa. Models for tissue motion are not amenable to a mathematical analysis and tend to be highly computation-based. 2.4 Applications Turing considered the chemicals in his model to be growth hormones, so that the spatial pattern in chemical concentrations would result in spatially nonuniform growth and hence pattern. He applied his theory to account for budding in plant stems and to growth-induced shape changes in the early embryo which he proposed could account for gastrulation. Since his seminal paper, reaction-diffusion models have been proposed to account for a vast number of patterning processes in nature, too great to be completely reviewed here, so we consider only a few examples to give a flavour of the applications. Gierer and Meinhardt (1972) used their model to account for pattern for­ mation in Hydra and showed that this model could account for the remarkable regenerative properties exhibited by this organism. Reaction-diffusion models have been proposed to account for compartmentalisation in insect develop­ ment and to provide an explanation for the occurrence of various mutants (see Meinhardt, 1982, for review). However, for DrosophUa it now appears that patterning is due to a cascade of protein interactions that are consistent with the gradient-type models and are not of Turing-type. Meinhardt (1995) has shown that reaction-diffusion type models solved on a growing domain can produce the spectacular variety of patterns seen on shells (Figure 1), while Nijhout (1990) has shown that such models, together with a small number of sources and sinks, can exhibit the vast array of pig­ mentation patterns observed on butterfly wings. Shell patterns can also be produced by an integro-partial differential equation model aimed at capturing neural activity along the front of the growing shell (Ermentrout et al, 1986).

Mathematical Modelling in the Life Sciences

211

Figure 1 . (a) Spatial pattern exhibited on the shell Oliva porphyria and a simulation of a reaction-diffusion model on a large array of cells. (Reproduced with permission from Meinhardt , 1995).

Although this model is based on very different biology to the reaction-diffusion model and has a different mathematical form, it still falls under the general category of short-range activation, long-range inhibition models. Reaction-diffusion and cell movement models have been applied to animal coat markings (Bard, 1981; Cocho et al., 1987a,b; Murray, 1981; Murray and Myerscough, 1991) and to skeletal patterning in the limb, for which gradient models have also been proposed (see Maini and Solursh, 1991, for a critical review). The above models propose different scenarios for pattern formation and, to date, it is a highly controversial issue as to which model is the appropriate one. Chemical morphogens, the name given to the chemicals proposed to form pre-patterns, have yet to be unequivocally identified, so it is difficult

212 P. K. Maini

to fix parameter values. For the mechanical type model mentioned above, parameters are also difficult to determine. Moreover, the original model is based on the Kelvin model of linear viscoelasticity and there is no justification for taking this particular form. The Maxwell model will also give rise to pattern formation, but for different parameter values (Byrne and Chaplain, 1996). Thus, without constraints on the parameter values, one can produce similar patterns with each model and therefore cannot distinguish between them on that basis. It is also difficult to distinguish, from a biological viewpoint, between models. For example, in some cases one observes both a chemical pattern and a cell aggregation pattern but it is difficult biologically to determine which came first. The chemical pre-pattern model would say that the chemical pattern arose first and cells responded to this by differentiating, resulting in cell condensations. The cell-chemotaxis model scenario would be that both patterns arise simultaneously, while under the assumptions in the mechanical model, the patterns would be explained by cell aggregations first forming and then secreting a chemical pattern as a result of differentiation. Although these models are based on very different biological assump­ tions, many of them share common properties. For example, the patterning in reaction-diffusion and in many cell movement models arises from the inter­ action of the processes of short-range activation, long-range inhibition. On the one hand, this has the disadvantage of making it very difficult to use models to distinguish between mechanisms, on the other hand, it does mean that one can make general conclusions and predictions that are mechanism-independent. This leads to the idea of developmental constraints which proposes that only certain patterns are possible, regardless of the mechanism (Oster and Murray, 1989). Figure 2 illustrates one such developmental constraint. A key property of many development processes is their robustness in the face of naturally occurring random fluctuations. This has been a major prob­ lem for reaction-diffusion theory, as it is well-known that the patterns it pro­ duces are not robust (Bard and Lauder, 1974). In other words, Turing-type models can exhibit multiple stable solutions in large regions of parameter space. Recently it has been shown that boundary conditions can play a cru­ cial role in stabilizing patterns. For example, if one chooses fixed boundary conditions for one chemical and zero flux boundary conditions for the other, then this reduces the number of admissible solutions and thus diminishes the regions in parameter space in which one obtains multiple stable solutions. In effect, the boundary conditions serve to select only certain patterns (Dillon et al, 1994). In higher dimensions, this problem becomes more acute as one now has

Mathematical Modelling in the Life Sciences

213

(o)

Figure 2. Simulation of a cell-chemotaxis model of the form (11) - (12) showing the effect of domain size on cell density concentration (arrow denotes increasing cell density). As the domain narrows, the diamond pattern changes to a simpler, wavy stripe pattern. This is an example of a developmental constraint. (b) Examples of diamond patterns on snakes (i) Crotalus adamanteus; (ii) Coluber hippocrepis (note the effect of the tapering domain). Reproduced with permission from Murray and Myerscough, 1991.

214 P. K. Maini

the added problem of degeneracy. For example, for certain parameters, there may be two or more admissible solutions with the same linear growth rate and it is then not clear which solution is selected. Using nonlinear bifur­ cation analysis, Ermentrout (1991) showed that the nonlinear terms play a key role in pattern selection, with quadratic terms favouring spots, while cu­ bic terms favour stripes. More recently, Benson et al. (1998) have shown how a spatially-varying parameter can unravel such degeneracies and select one pattern over another. The role of spatially-varying parameters has re­ ceived little attention (although it should be noted that the Gierer-Meinhardt model was initially designed to explain how localized structures could arise in places determined by a pre-existing gradient) but they can play a crucial role in the patterning process. For example, Wolpert and Hornbruch (1990) showed experimentally that double-anterior chick limb buds gave rise to two humeri, even though the size of the bud was the same as that of a normal limb bud, which only produces one humerus. This contradicts the standard Turing model, which predicts that patterning complexity is intimately linked to domain size. Maini et al. 1992, showed that a Turing model with spatiallyvarying diffusion coefficients could give rise to results that are consistent with Wolpert and Hornbruch's experiments. Results of dye-spreading experiments suggest that the hypothesis of spatially-varying diffusion is very plausible (Brummer et al., 1991). These studies have all been carried out on the reaction-diffusion model system because it is the simplest, mathematically speaking, of the models reviewed in this section. It is still an open question as to whether the results on robustness, pattern selection and spatially-varying parameters carry through to the other model types. In all the above applications, patterns occur simultaneously throughout the whole domain. However, in some cases, patterns arise as the result of propagation. For example, in the alligator embryo, the pigmentation stripes occur as a propagating pattern moving down the body from head to tail. Murray et al. (1990), have shown that a cell-chemotactic model of the form discussed above can give rise to such patterns. They were able to make ex­ perimentally confirmed predictions on how the number of stripes varies with the length of the embryo and present biological evidence that supports the view that this is a cell movement process. More recently, Painter et al. (1999a) have studied the formation of the primitive streak. This is a novel patterning process wherein, during the blas­ toderm stage of early chick development, a column of cells, known as the primitive streak, advances to about 3/5 the way across the disk-shaped blas­ toderm, before regressing. They have shown how this could arise from a

Mathematical Modelling in the Life Sciences 215

Figure 3. Coupling two reaction-diffusion models can lead to a complicated pattern of stripes interspersed with spots, as observed in the thirteen-lined ground squirrel and Pomacanthus maculatus. (Reproduced with permission from Arag6n et al., 1998).

cell-chemotaxis type model and have made a number of testable predictions. 2.5

Coupling pattern generators

In many cases, pattern formation arises as the result of the interaction of more than one pattern generator. For example, epidermal-dermal interactions play a crucial role in the development of skin organs, such as hair, teeth, feathers and scales. Nagorcka et al. (1987) considered a tissue-tissue interaction model that coupled a mechanochemical cell movement model in the dermis with a

216 P. K. Maini

reaction-diffusion model in the epidermis. They showed that such a model can give rise to patterns on two different length scales, namely, large spots inter­ spersed with small spots, and these are similar to the scale patterns observed in certain lizards and to the feather germ patterns observed on certain birds. More recently, Arag6n et al. (1998) have shown how the coupling between two systems of reaction-diffusion models can produce complex patterns (see Figure 3). The life cycle of the cellular slime mould Dictyostelium discoideum serves as an excellent paradigm for morphogenesis as it exhibits the processes of signal transduction, signal relay, cell movement and aggregation, all of which play important roles in early embryonic development. Hence, for many years, it has attracted the interest of developmental biologists and theoreticians alike. Starvation conditions trigger a developmental programme which is initiated by cell-cell signalling via the extracellular messenger cyclic 3'5'adenosine monophosphate (cAMP). The chemotactic response to this signal leads, through the phenomenon of cell streaming, to the formation of a multicellular organism composed typically of 104 - 105 cells. This organism passes through an intermediate motile (slug) phase during which cells differentiate into pre-spore and pre-stalk types, before developing a fruiting body, aiding the dispersal of spores from which, under favourable conditions, new amoe­ bae develop. The comparative simplicity of morphogenesis in Dictyostelium has made it an attractive model system for the study of self-organisation, and many of the molecular and cellular mechanisms which are involved in cell aggregation, collective movement and differentiation have now been identified. Several models have been proposed to account for many of the afore­ mentioned phenomena. Here, we shall focus on a model for cell aggregation proposed by Hofer et al. (1995a,b) - we refer the reader to this paper for a fuller description of the biology and parameter values, as well as references to other models. The model takes the form: dn — = V • (/xVn - X{v)nVu) (21) Yt = mn)h(u,v)

- ((n) + S)f2(u)} + V 2 u

dv -QI = -9\{u)v + g2{u)(l - v),

(22) (23)

where n, u and v denote cell density, extracellular cAMP concentration and fraction of active cAMP receptors, respectively. The second and third equations are a simplified model of the cAMP-cell receptor dynamics (see Martiel and Goldbeter, 1987, for full details) modi­ fied by cell density effects; the rate of cAMP synthesis per cell is / i (u, v) =

Mathematical Modelling in the Life Sciences

217

(bv + v2)(a + u 2 ) / ( l + u2), where a and b are positive constants. This models autocatalytic production with saturation, mediated by receptor binding. The functions /2 and gi are assumed to be linear in u, to model linear degradation of cAMP and binding of active receptors to cAMP, respectively, while 52 is assumed to be a positive constant, accounting for the resensitization of the desensitized fraction of receptors at constant rate. This subsystem exhibits ex­ citable dynamics leading to the formation of spiral waves of cAMP concentra­ tion. Cell density effects are modelled by the factor 0(n) = n/(l—pn/(K+n)), where p and K are positive constants. This excitable system is coupled with a chemotaxis equation for cell density, with constant diffusion coefficient p, and chemotactic sensitivity x{v) = Xovm/{Am +vm), m > 1, which accounts for adaptation, where Xo and A are positive constants. Hence an appreciable chemotactic response requires a minimal fraction of active receptors, yet, for a large fraction of active receptors, the response saturates. Note that many models of chemotaxis assume x to be a constant. Under that assumption, cells would respond to a pulse of chemoattractant by moving towards the wave in the wavefront, then moving with the wave in the waveback, resulting in a net movement away from the source of attractant, rather than towards it. This is the so-called "chemotactic wave paradox" (Soil et al., 1993). The form of x{v) chosen above resolves this paradox (Hofer et al, 1994). Using experimentally determined parameter values, the above model cap­ tures the key features of the aggregation process (see Figure 4). The model is consistent with the observation that as streaming proceeds, the wavespeed and wavelength of the spiral patterns decrease (Gross et al., 1976). Previously, this has been explained by assuming that biochemical changes must be occur­ ring in the cell-cAMP system and, indeed, it has now been established that changes of this sort can occur. However, in the above model, this behaviour arises naturally, because as the cells form streams, they alter the conditions through which the cAMP waves are propagating. This is initially equivalent to increasing the excitability of the medium which, in turn, leads to an in­ crease in the rotation frequency of the spiral core. As a result (Tyson and Keener, 1988) the wavespeed and wavelength of the spiral patterns decrease. 2.6

Domain Growth

None of the above models account for domain growth. However, a number of recent studies have shown that domain growth can play a vital part in the pat­ tern formation process. For example, Kondo and Asai (1995) showed that the pigmentation pattern in the juvenile angelfish Pomacanthus semicirculatis, exhibits changes in the number of stripes as it grows. Briefly, when the wave-

218 P. K. Maini

v\V-

\ j

W/// :

/

'

(a)

(b)

Figure 4. Spatio-temporal evolution of (a) cell density, and (b) cAMP concentration in a numerical simulation of (21)-(23). (Reproduced with permission from Hofer et ai., 1995b).

Mathematical Modelling in the Life Sciences

219

length of the stripes increases twofold due to domain growth, further stripes are inserted to restore the wavelength to its original value. They showed that this is consistent with a reaction-diffusion model. More recently, Painter et al. (1999b) showed that a cell-chemotaxis model coupled with a reaction-diffusion model on a growing domain could account also for the fact that the inserted stripes are thinner than the original stripes. This phenomenon is not easily explained by a reaction-diffusion model. It is believed that domain growth also plays a role in the determination of the sites of tooth primordia in alligators. The first seven tooth primordia of the lower alligator jaw occur (taking one half of the jaw) in the spatio-temporal sequence 4-1-5-2-6-3-7, that is, the fourth primordia to form is the posteriormost while the seventh primordia to form is the anterior-most. Kulesa et al. (1996) have shown how a reaction-diffusion model coupled with an inhibitor model on a growing domain could explain this sequence. More recently, Crampin et al. (1999) have shown that domain growth can lead to the robust pattern selection of certain types of patterns in reactiondiffusion models, while Holloway and Harrison (1999) have shown that inter­ specific variation in branching patterns in certain plants can be accounted for by a system of reaction-diffusion equations on a growing two-dimensional domain in which growth is controlled by the concentration of one of the chem­ icals. 2.7

Discussion

Development of spatial pattern and form is unquestionably one of the central areas in biology. It is a very complex process that involves the interaction of a large number of components acting at different levels, yet the models presented in this section focus on only a small number of components. An important question to ask therefore, is that although it is clear that even these simple models offer theoreticians an enormous range of challenging problems from modelling, numerical computation and mathematical viewpoints, what does it actually say about the biology? The models presented here are really of two types. The first are conceptual, for example, it is remarkable that the simple mechanism of short-range activation, long-range inhibition can give rise to such an enormous range of patterns. One can think of this as being a primary mechanism for generating spatial pattern. Such models provide a framework in which one can test theories of patterning, as well as making experimentally testable predictions. The second type of model is where more is known about the biochemistry and parameter values. Pattern formation in Dictyostelium discoideum pro-

220 P. K. Maini

vides the most well-studied case of this type of model. Although the model described in this section is based on only three equations, the excitable sub­ system of the model arises from a much larger model which can be simplified by using the fact that there are a number of different timescales involved. The recent work by Weijer and coworkers has gone a long way in explaining a number of patterning phenomena that occur in Dictyostelium discoideum by including ideas from chemotaxis, excitable media and computational fluid dy­ namics (see Vasiev and Weijer, 2000, and references therein). Such single-cell systems, which are simpler to manipulate than embryos, may be important in providing crucial insights to some of the workings of more complicated multicellular organisms, and are therefore attracting more and more attention. For example, there is now a great deal of literature on the modelling of bacterial patterns (see Ben-Jacob et al., 2000, and references therein). All the above models are based on considering pattern formation at a macroscopic level. There is now an enormous amount known at the molecular level about pattern formation. One of the main future challenges is to develop models that can integrate the molecular and cellular levels. 3

Models for wound healing

Wound healing is an enormously complicated phenomenon involving different processes interacting on different spatio-temporal timescales (see, for example, the books by Clark and Henson, 1988, Asmussen and Sollner, 1993) and the method of healing varies depending on whether the wound is a surface wound (epidermal) or a deep wound (dermal), and on whether it is in the adult or the embryo. In this section we focus on a model for epithelial wound healing and one for dermal wound healing, and show how the ideas of modelling used in the previous section can be applied. 3.1

Corneal Wound Healing

Cell migration and proliferation are central to the healing of wounds in the corneal epithelium and biological evidence suggests that both processes are regulated by a protein called epidermal growth factor (EGF). The source of EGF is an area of controversy and the model of Dale et al. (1994) sets out to investigate the possibility that the exposed underlying tissue within the wound acts as an additional source to the overlying tear film layer. Here, we summarise the model of Dale et al. (1994) and refer the reader to the original and references therein for full biological and modelling details. The model is a pair of reaction-diffusion equations for the cell density n(x, t)

Mathematical Modelling in the Life Sciences

221

and EGF concentration c(x, t) at position x and time t and takes the form (suitably nondimensionalised) Cell Migration

^

Mitotic Generation

Diffusion

/M Production by Cells

Lo8s

"n"N

(24)

- j=^-fc

(25)

= V.(D^(c)Vn)'+ i ( c ) n ( 2 - » )

|=a^j+

N a t ur al

-

/

s

v ' Decay of Active EGF

where £>n(c) = ac+0, and DC, (x,6,a , 0 and c are all positive constants. The model assumes that the cell diffusion coefficient is increased by EGF and s(c) is an increasing function of c to account for the EGF-enhanced cell mitotis rate. The function f(n) is taken to be of the form f(n) = A + B(n), where

{

a

if n < 0.2

CJ(2 - 5n) if 0.2 < n < 0.4 and A and a are positive constants. The A accounts for the constant source 0 the function if n > B(n) 0.4 is chosen to model EGF of EGF due to the tear film, while

and A and a are positive constants. The A accounts for the constant source of EGF due to the tear film, while the function B(n) is chosen to model EGF production due to wounding. The authors show that the detailed form of this function is not important to the behaviour of the model. Using parameter values derived from the biological literature, this model system is solved on a one-dimensional domain (a realistic approximation for the case of surface slash wounds, where cell fronts move in from either side of the slash to close the wound) to investigate the behaviour of travelling wave solutions which move from a region where the cell density and EGF concentration are at their unwounded levels, n = c = l ( a s x - » - c o ) , into a region of no cell density, n — 0, with the EGF concentration at its wounded level, c = f(0)/6 (as x -> +oo). An important conclusion from this work is that a realistic speed of healing is only attained if the function B(n) is included (Figure 5). Hence the conclusion is that for the wound healing scenario envisaged by this model, biologically realistic healing times can only be achieved by assuming that the tear source of EGF is supplemented by EGF production in the wounded tissue. For biologically realistic parameter values an analytical approximation to the minimum wavespeed can derived as J8/3s(££z-). An important biological implication of this result is that the rate of healing of corneal epithelial wounds can be increased by increasing the cell diffusion coefficient or the secretion rate of EGF. However, increasing the chemical diffusion coefficient does not have a

222 P. K. Maini

Figure 5. Numerical solution for the corneal wound healing model showing cell density and EGF concentration profiles as functions of space at equal time intervals, (a) The tear film is the only source of EGF, that is, /(n) = A. (b) An additional source of EGF is included, that is, /(n) = A + B(n). Note that in (b) healing occurs much more rapidly. (Reproduced with permission from Dale et at., 1994).

Mathematical Modelling in the Life Sciences 223

significant effect. The model can also be used to make experimentally testable predictions on how the speed of healing will change as the result of topical application of EGF. 3.2

Dermal healing

A crucial aspect of wound healing concerns the mechanical interaction of cells with their external environment. Cells deform and remodel the extracellular matrix on which they move and ECM materials, in turn, affect cellular prop­ erties and cell orientation. Using the mechanochemical model framework (see Murray, 1993, for review, and Murray et o/., 1988, Murray and Tranquillo, 1992) Olsen et al. (1995) developed a mechanochemical model for dermal wound healing. We refer the reader to the original paper for full details, in­ cluding experimental justification for each term within the model. The model consists of two cell types - fibroblasts and myofibroblasts, densities, n and m, respectively; a generic growth factor, concentration c, and ECM, density p. These quantities obey the general conservation equation f

= V'JQ + /C

(26)

where Q is the quantity in question, the first term on the right-hand side models motion with flux J Q , and the second term models production and degradation. To complete the model, a force balance equation is needed to account for the mechanical interaction of the cells with the ECM. For simplicity, we present here the one-dimensional version of the model, where x is space and t is time. The fibroblast cell equation takes the form: „ d2n d , , .dc du. _. , ., =Dn [x{c n) +n ]+R{c)n{1 )

dn

m

^-^

' ^ m

n, k\cn +k2m dnn

-K -c^Tc

-

,

,

-

(27) Implicit in these equations is the assumption that there are three main factors contributing to cell flux: random diffusion, with constant diffusion coefficient, Dn; chemotaxis with chemotactic sensitivity x(c,n), and advection in response to the displacement, u(x, t), of the ECM. The four remaining terms on the right-hand side model cell kinetics and include logistic cell growth with linear rate enhanced by growth factor, fibroblast conversion to myofibroblast phenotype mediated by growth factor, conversion from myofibroblast back to fibroblast cell type, and cell death. The myofibroblast equation takes the form dm d , du, _ . , .„ m. kicn . , .„„.

-at = d-x[-m-di]

+£

'* ( c ) m ( 1 " K] + cTTc - hm ~ ^ m -

(28)

224 P. K. Maini

Here, it is assumed that the dominant contribution to myofibroblast flux is advection, and that mitosis takes the same form as that for fibroblasts, mod­ ulated by a constant scale factor e r . The growth factor satisfies the equation dc

at

_ d2c

= Dc

w

+

d ,

[ c

du,

]

ai ~ m

nt

+ S{n m c)

' '

x

~

J

dcC

-

(29)

Implicit in this equation is the assumption that the dominant contributions to growth factor flux are random diffusion, with constant diffusion coefficient Dc, and advection. The remaining terms on the right-hand side model biosynthesis and degradation. The ECM moves primarily by advection and satisfies the equation dp

d ,

du.

d-t = d-x[-pm]

„. + B{n m c p)

> -' '

<30)

where B(n,m,c,p) represents ECM biosynthesis and degradation. Finally, modelling the ECM as a linear, isotropic, viscoelastic material, the displacement u satisfies the force balance equation

where the first two terms on the left-hand side model viscous and elastic forces, respectively, and the third term models cell traction forces. These forces are balanced by the body forces F(p,u). Note that this is very much a simplifi­ cation as a more realistic model should include anisotropy and plasticity. The above five equations, with appropriate initial and boundary conditions (see below) constitute the mechanochemical model frame­ work. Solving these equations in one spatial dimension is an approx­ imation to "slash" wounds. Numerical simulations show that these equations admit solutions in which a travelling front of cell density moves into the wound, causing it to heal. Using biologically realistic forms for the functions x(c)n),R(c),S{n,m,c),B(n,m.c,p),T(n,p),F{p,u), and estimates derived from experimental data for the parameters Dn,Dc,K,ki,k2,Ck,dn,dm,dc,€r,n and E, it can be shown that this model exhibits solutions for the decay of growth factor and rate of wound closure that closely agree with experimental results (see Olsen et cd., 1995, for full details). This model can be used to investigate abnormal wound healing. The full model is very complicated so simpler caricatures are considered. To investi­ gate the potential of the above model framework to exhibit spatially-varying contracted steady states, corresponding to fibrocontractive diseases, Olsen et

Mathematical Modelling in the Life Sciences 225

al. 1998, considered a simpler version of the model which focusses only on the mechanical aspects of the interaction. The non-dimensionalised version of this caricature model takes the form dn

_ d2n

Dn

m= w dp

d3u

^d2u

dT(n,p)

d

\

+

d .

du,

,,

d-x[-nm] + n{1-n)

J^y

.

,„„.

(32)

(VK\

.

. .

This caricature considers only the fibroblast cell type and assumes a simple form for logistic growth. It also assumes that there is negligible synthesis and degradation of ECM on the timescale of wound closure. This is a reasonable assumption to make in the stages prior to tissue remodelling during the process of wound healing. By defining the initial wound space as - 1 < x < 1 and using symmetry at x = 0 (the wound centre), we may restrict attention to the semi-infinite domain 0 < x < oo. The boundary conditions are thus 5H(0,t) = 1^(0,0 = u ( 0 , t ) = 0 and n(oo,«) = p(oo,«) = 1, u(oo,i) = 0. ox ox The initial conditions are n(x,0) = H(x-l),

p(x,0) = Pi + (l-pt)H{x-l),

u(x,0) = 0,

where the initial ECM density pi inside the wound is due to the early, pro­ visional wound matrix which is low in collagen and satisfies 0 < Pi < 1, and H(•) is the Heaviside step function. Consider now the healed steady state, n = 1. Linearising the matrix equation about the initial profile, we have (fH(l-du/dx),0<x1

(35)

as suggested by the small-strain restriction (since the convective flux should be small). Substituting this into the steady state equation for u results in a second order ordinary differential equation for u which can be written in the (rescaled) form u' = v

(36)

226 P. K. Maini

,

SOiU(l -v) , n r V r /i —VT> 0 < X < 1

su(\ -v)

1-ra-i;)' where T(p) =

&"'p'

, X>1

, and u satisfies the boundary conditions u(0) =

u(oo) = 0. Here it is assumed that the body force, F(p,u), is due to external tethering to the basement membrane and have modelled it by a linear spring, that is, F(p, u) = sup, where a is a constant. Standard phase plane analysis of (36)-(37) shows that for x > 1, the origin is a saddle (centre) iff T(l) < 1 (> 1). Linear stability analysis of the caricature model (32)-(34) shows that a necessary (but not sufficient) condition for the healed steady state to be stable is T ( l ) < 1 (see Olsen et a/., 1998). As this must be the case for the model to be realistic biologically, we have that the origin of the ordinary differential equation system (36)-(37) is a saddle, and the boundary condition u(oo) = 0 implies that the solution must converge towards the origin along the stable manifold as x tends to oo. By tracing backwards in x from infinity along the stable manifold, the solution reaches a point in the (u, u')-phase plane corresponding to x = 1 where u = U\, say. This must match the solution for 0 < x < 1. Now, at the wound centre, «(0) = 0, but v(0) is unspecified and is de­ termined by matching to the "outer" solution at x = 1. For 0 < x < 1, it can be shown that the origin can either be a saddle or a centre, depending on the form of the function T(TI, p) and the values of the other parameters. If the origin is a saddle point, then the solution in 0 < x < 1 will be either monotonic increasing with increasing gradient or monotonic decreasing with decreasing gradient. If the origin is a centre, then the solution in 0 < x < 1 may be oscillatory. Figure 6 illustrates the qualitative construction of such a solution and Figure 7 illustrates various possible forms of steady state so­ lutions based on this construction. Modelling the traction term, r(n,p), by T(TI, p) = Tonp/(T2 + p2), where TO and T are constant parameters, to account for the fact that traction forces depend on adhesion between cell surface re­ ceptors and binding sites on collagen fibres, but the ability of a cell to extend and retract protrusions within a collagen substrate is inhibited at relatively high collagen densities, steady states for (32)-(34) of the form illustrated in Figure 7(a)-(e) can be found by numerical simulation. Hence, this caricature model enables us to more fully understand the properties of the full model and shows clearly that the model can exhibit spatially-varying contracted steady states, and is thus consistent with clinical observations on normal healing. For full details, see Olsen et al. 1998.

Mathematical Modelling in the Life Sciences 227 V

S

** X • V.

—

- _ — ***

•

S •

I^LA \•

/ u

V-

I t , '

"s ^^

/ / / / / /

- -o-

\ \ \ \ \v

/ //

\u

/ //

0<x
\W

x>l

Figure 6. Qualitative illustration of a possible solution trajectory to (36)-(37) for the case in which the origin is a centre for 0 < x < 1 and a saddle point for x > 1 with u(x) -»• 0 from below as i -► oo. See also Figure 7(b). Dashed curves denote phase trajectories, with the contracted solution curve highlighted by solid arrows. See text for details. (Reproduced with permission from Olsen et al., 1997).

We now consider the application of the model to fibroproliferative wound healing disorders. These disorders are characterised by the generation of ab­ normally large amounts of tissue during the healing process, leading to, for example, keloid scarring. Numerical simulations of the full model show that it can exhibit solutions in which an excess of cells is observed, corresponding to a pathological state. To understand this more fully, a caricature model of the full system is, again, investigated. In this case, however, we focus purely on the chemical aspects of the mechanochemical framework by considering the cell-chemical sub-model d2n dn ^r- = Dn 2 dx at 32c dc ^- = D, 2 dx dt

X(c,n.

dx Kcnc + 7 + c -dcc,

dc +
PC

n 1 +Q + c n ( l - - ) - d „ n (38) (39)

where x( c > n ) = a/(P + c)2> aii^ OC,0,P,Q,KC and 7 are positive constants (see Olsen et al., 1996 for full details). This caricature model has two uniform steady states, (n, c) = (0,0), (K, 0) corresponding, respectively, to the trivial, or non-healing, state, and the nor-

228 P. K. Maini

Figure 7. Possible qualitative forms of the solution u(i) of the boundary value problem (36)-(37), representing contracted tissue displacement profiles. The point (u,«') = (0,0) must be a saddle point for x > 1 in the («, u')-phase plane, with u increasing to zero and u' decreasing to zero monotonically along the stable manifold in the top-left quadrant as x -f oo. For 0 < x < 1, the origin may be either a saddle point, in which case the profiles for u and u' are monotonic decreasing as shown in (a), or a centre, in which case « and u' oscillate about the origin as shown in (b-f); within this region, any number of oscillations is possible—for example, (f) is equivalent to (b) modulo one period. Note that the above steady-state profiles but with reversed signs of « and u' are also admissible solutions of (36)-(37), representing expanded tissue displacement profiles since u(l) would be positive. Recall that x = 1 is the initial wound boundary. (Reproduced with permission from Olsen et ai., 1997).

mal dermal state. For appropriate parameter values, two other steady states exist which have both n and c non-zero, with n > K. These are the patholog­ ical, or diseased, steady states. Results from bifurcation analysis of (38)-(39), in the absence of diffusion, show that for a critical value, KJ, of KC (which can be found in terms of the other parameters) the dermal steady state remains locally stable but loses global stability as the pathological steady states ap­ pear. At KC = «c, the dermal steady state loses stability and the pathological state with higher cell density level becomes globally stable. A travelling wave analysis of the model shows that trajectories from the dermal state to the pathological state are possible and a minimum wavespeed

Mathematical Modelling in the Life Sciences

229

Figure 8. Numerical simulations of (38)-(39) showing progression to pathological steady state (a), and cessation and regression for the case where KC is reduced to zero after a certain time (b). (Reproduced with permission from Olsen et al., 1997).

can be determined. Numerical simulations of the system show that such travelling waves do exist, but that reducing KC can cause the waves to stop and to regress (see Figure 8). This suggests that the reduction of the rate at which cells secrete growth factor can cause the disease to regress back to the normal dermal state. More detailed analysis of this model determines analytically how the bifurcation values of nc depend on the other parameters in the model. In particular, the model exhibits hysteresis and therefore, counter-intuitively, KC must be reduced considerably in order to progress from the diseased state to a healed state. This provides a clinically-testable method to help reduce this type of fibroproliferative disorder. It should be noted that although the above studies were carried out for simplified versions of the full mechanochemical model, the results do indeed

230 P. K. Mami

hold for the full model. 3.3

Discussion

The above models have been presented to illustrate how continuum type mod­ els can be used to address problems in normal and abnormal wound healing. The model for corneal wound healing has recently been extended to be more biologically realistic by including more than one cell type. The resultant model is a system of coupled integro-partial differential equations which ex­ hibits mitotic profiles that are more biologically realistic than those observed in the original model (Gaffney et al., 1999). The mechanochemical modelling framework can be extended to include cell alignment and matrix orientation, the latter is thought to play a crucial role in determining the severity of scar tissue formation (Olsen et al., 1999, Dallon et al., 1999). This particular modelling framework assumes that the extracellular matrix is a linear viscoelastic material. However, it is clearly more complicated than that and a more realistic model, considering the ECM as a viscoelastic-plastic material has recently been proposed (Tracqui et al., 1995). None of the above models investigate angiogenesis, the process by which new blood vessels form. This is of great interest also in tumour formation, where after reaching a certain size, limited by the availability of nutrients via simple diffusion, a tumour can only grow further by establishing its own blood supply via the release of so-called tumour angiogenesis factor. This is what allows the tumour to grow and undergo metastasis, causing the growth of secondary tumours which are usually fatal. For the mathematical modelling of angiogenesis in wound healing, the reader is referred to the paper by Byrne and Chaplain (1995), while the papers by Pettet et al. (1996) and Olsen et al. (1997) address wound healing angiogenesis. 4

Conclusions

In this paper I have considered the problems of spatial pattern formation in biology and of wound healing by illustrating a few applications. These seem­ ingly unrelated processes share the common underlying theme of cell response to, and interaction with, signalling cues. The models are conceptually simple and closely related, yet exhibit a bewildering array of behaviours, from spiral waves, spots and stripes, with application in pattern formation, to travelling waves of invasion in wound healing. The material in Sections 2 and 3 has illustrated how mathematical models

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can be used to gain important insights to the underlying mechanisms respon­ sible for spatio-temporal pattern formation in biology and normal/abnormal wound healing and to make experimentally testable predictions. It is clear that even these so-called simple models pose challenging problems to the mathe­ matician. Many of the results presented here were obtained from numerical simulation. An important future aim will be to make some of these results mathematically realistic. Over the past decade there have been huge advances in molecular biology. A key problem for the next decade will be to combine this knowledge with research in cellular biology to gain a fuller understanding of the systems being studied. Important scientific advances in this area will only be made by a truly interdisciplinary approach and it is clear that mathematical modelling and numerical computation will have important roles to play in this endeavour. It is also clear that biology and medicine will continue to be a source of novel, difficult and challenging problems for mathematicians and numerical analysts. References 1. J.L. Arag6n, C. Varea, R.A. Barrio and P.K. Maini, Spatial patterning in modified Turing systems: Application to pigmentation patterns on marine fish, FORMA, 13, 213-221 (1998) 2. P.D. Asmussen and B. Sollner: Wound care: Principles of wound healing. Beierdorf Medical Bibliothek (1993) 3. J.B.L. Bard, A model for generating aspects of zebra and other mam­ malian coat patterns, J. theor. Biol., 93, 363-385 (1981) 4. J.B.L. Bard and I. Lauder, How well does Turing's theory of morphogen­ esis work? J. theor. Biol., 45, 501-531 (1974) 5. D.L. Benson, P.K. Maini and J.A. Sherratt, Unravelling the Turing bi­ furcation using spatially varying diffusion coefficients, J. Math. Biol., 37, 381-417 (1998) 6. E. Ben-Jacob, I. Cohen, I. Golding and Y. Kozlovsky, Modeling branching and chiral colonial patterning of lubricating bacteria, In: Proceedings of IMA Workshop on Pattern Formation, H.G. Othmer and P.K. Maini, (eds) to appear (2000). 7. N.F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, London (1986) 8. F. Brummer, G. Zempel, P. Buhle, J.-C. Stein and D.F. Hulser, Retinoic acid modulates gap junction permeability: A comparative study of dye spreading and ionic coupling in cultured cells, Exp. Cell. Res., 196, 158-163 (1991)

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9. H.M. Byrne and M.A.J. Chaplain, Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions, Bull. Math. Biol., 57, 416-486 (1995) 10. H.M. Byrne and M.A.J. Chaplain, On the importance of consti­ tutive equations in mechanochemical models of pattern formation, Appl. Math. Lett, 9, 85-90 (1996) 11. V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimen­ tal evidence of a sustained Turing-type equilibrium chemical pattern, Phys. Rev. Lett, 64(3), 2953-2956 (1990) 12. R.A.F. Clark and P.M. Henson (eds.), The Molecular and Cellular Biology of Wound Repair, Plenum Press, New York (1988) 13. G. Cocho, R. P6rez-Pascual and J.L. Rius, Discrete systems, cell-cell interactions and color pattern of animals. I. Conflicting dynamics and pattern formation, J. theor. Biol, 125, 419-435 (1987a) 14. G. Cocho, R. Perez-Pascual, J.L. Rius and F. Soto, Discrete systems, cell-cell interactions and color pattern of animals. I. Clonal theory and cellular automata, J. theor. Biol, 125, 437-447 (1987b) 15. E.J. Crampin, E.A. Gaffney and P.K. Maini, Reaction and Diffu­ sion on Growing Domains: Scenarios for Robust Pattern Formation, Bull. Math. Biol, 6 1 , 1093-1120 (1999) 16. P.D. Dale, P.K. Maini and J.A. Sherratt, Mathematical modelling of corneal epithelium wound healing, Math. Biosciences, 124, 127-147 (1994) 17. J. Dallon, J. Sherratt and P.K. Maini, Mathematical modelling of extra­ cellular matrix dynamics using discrete cells: Fiber orientation and tissue regeneration, J. theor. Biol, 199, 449-471 (1999) 18. A. Davidson, M.A.R. Koehl, R. Keller and G.F. Oster, How do sea urchins invaginate? Using biomechanics to distinguish between mechanisms of primary imagination, Dev., 121, 2005-2018 (1995). 19. P. De Kepper, V. Castets, E. Dulos and J. Boissonade, Turing-type chem­ ical patterns in the chlorite-iodide-malonic acid reaction, Physica D, 49, 161-169 (1991) 20. R. Dillon, P.K. Maini and H.G. Othmer, Pattern formation in generalised Turing systems: I. Steady-state patterns in systems with mixed boundary conditions, J. Math. Biol, 32, 345-393 (1994) 21. L. Edelstein-Keshet, Mathematical Models in Biology, New York, Ran­ dom House (1988) 22. B. Ermentrout, Stripes or spots? Nonlinear effects in bifurcation of reaction-diffusion equations on the square, Proc. Roy. Soc. Lond., A434, 413-417 (1991)

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23. B. Ermentrout, J. Campbell and G. Oster, A model for shell patterns based on neural activity, The Veliger, 28, 369-338 (1986) 24. P. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Led. Notes in Biomath., 28, Springer-Verlag, Berlin, Heidelberg, New York (1979) 25. E.A. Gaffney, P.K. Maini, J.A. Sherratt and S. Tuft, The mathe­ matical modelling of cell kinetics in corneal epithelial wound healing, J. theor. Biol., 197, 15-40 (1999) 26. A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12, 30-39 (1972) 27. P. Grindrod, The Theory of Applications of Reaction-Diffusion Equa­ tions: Pattern and Waves, Oxford University Press (1996) 28. J.D. Gross, M.J. Peacey and D.J. Trevan, Signal emission and signal propagation during early aggregation in Dictyostelium discoideum, J. Cell Sci., 22, 645-656 (1976) 29. T. Hofer, P.K. Maini, J.A. Sherratt, M.A.J. Chaplain, P. Chauvet, D. Metevier, P.C. Montes and J.D. Murray, A resolution of the chemotactic wave paradox, Appl. Math. Lett., 7, 1-5 (1994) 30. T. Hofer, J.A. Sherratt and P.K. Maini, Dictyostelium dis­ coideum: cellular self-organization in an excitable biological medium, Proc. Roy. Soc. Lond. B 259, 249-257 (1995a) 31. T. Hofer, J.A. Sherratt and P.K. Maini, Cellular pattern formation during Dictyostelium aggregation, Physica D, 85, 425-444 (1995b) 32. D.M. Holloway and L.G. Harrison, Algal morphogenesis: modelling inter­ specific variation in Micrasterias with reaction-diffusion patterned catal­ ysis of cell surface growth, Phil. Trans. R. Soc. Lond., B354, 417-433 (1999) 33. B.R. Johnson and S.K. Scott, New approaches to chemical patterns, Chem. Soc. Rev., 265-273 (1996) 34. E.F. Keller and L.A. Segel, Travelling bands of bacteria: a theoretical analysis, J. theor. Biol., 30, 235-248 (1971) 35. S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376, 765-768 36. P.M. Kulesa, G.C. Cruywagen, S.R. Lubkin, P.K. Maini, J. Sneyd, M.W.J. Ferguson and J.D. Murray, On a model mechanism for the spa­ tial patterning of teeth primordia in the Alligator, J. theor. Biol., 180, 287-296 (1996) 37. L. Landau and E. Lifshitz, Theory of Elasticity, Pergamon Press, New York (1970) 38. I. Lengyel and I.R. Epstein, Modeling of Turing structures in the chlorite-

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iodide-malonic acid-starch reaction system, Science, 251, 650-652 (1991) 39. P.K. Maini and M. Solursh, Cellular mechanisms of pattern formation in the developing limb, Int. Rev. Cytology, 129, 91-133 (1991) 40. P.K. Maini, D.L. Benson and J.A. Sherratt, Pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients, IMA J.Math.Appl.Med. & Bioi, 9, 197-213 (1992) 41. P.K. Maini, M.R. Myerscough, K.H. Winters and J.D.Murray, Bifurcat­ ing spatially heterogeneous solutions in a chemotaxis model for biological pattern formation, Bull. Math. Biol, 53, 701-719 (1991) 42. J.L. Martiel and A. Goldbeter, A model based on receptor desensitization for cyclic AMP signaling in Dictyostelium cells, Biophys. J., 52, 807-828 (1987) 43. H. Meinhardt, Models of Biological Pattern Formation, Academic Press (1982) 44. H. Meinhardt, The Algorithmic Beauty of Sea Shells, Springer-Verlag (1995) 45. J.D. Murray, A pre-pattern formation mechanism for animal coat mark­ ings, J. theor. Biol., 88, 161-199 (1981) 46. J.D. Murray, Parameter space for Turing instability in reaction-diffusion mechanisms: a comparison of models, J. theor. Biol., 98, 143-163 (1982) 47. J.D. Murray Mathematical Biology, Springer-Verlag (1993) 48. J.D. Murray, D.C. Deeming, D.C. and M.W.J. Ferguson, Size dependent pigmentation pattern formation in embryos of Alligator mississippiensis: time of initiation of pattern generation mechanism, Proc. Roy. Soc. Lond., B 239, 279-293 (1990) 49. J.D. Murray, P.K. Maini and R.T. Tranquillo, Mechanochemical mod­ els for generating biological pattern and form in development, Physics Reports, 171, 59-84 (1988) 50. J.D. Murray and M.R. Myerscough, Pigmentation pattern formation on snakes, J. theor. Biol, 149, 339-360 (1991) 51. J.D. Murray and R.T. Tranquillo, Continuum of fibroblast-driven wound contraction: inflammation-mediation, J. theor. Biol, 158, 135-172 (1992) 52. B.N. Nagorcka, Wavelike isomorphic prepatterns in development, J. theor. Biol, 137, 127-162 (1989) 53. B.N. Nagorcka, V.S. Manoranjan and J.D. Murray, Complex spatial pat­ terns from tissue interactions — an illustrative model, J. theor. Biol, 128, 359-374 (1987) 54. G.A. Ngwa and P.K. Maini, Spatio-temporal patterns in a mechani­ cal model for mesenchymal morphogenesis, J. Math. Biol, 33, 489-520

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(1995) 55. H.F. Nijhout, A comprehensive model for colour pattern formation in butterflies, Proc. R. Soc. Lond. B 239, 81-113 (1990) 56. G.M. Odell, G. Oster, P. Alberch and B. Burnside, The mechanical basis of morphogenesis. I. Epithelial folding and invagination, Devi. Biol., 85, 446-462 (1981) 57. L. Olsen, J.A. Sherratt and P.K. Maini, A mechanochemical model for adult dermal wound contraction and the permanence of the contracted tissue displacement profile, J. theor. Biol., 177, 113-128 (1995) 58. L. Olsen, J.A. Sherratt and P.K. Maini, A mechanochemical model for adult dermal wound contraction: On the permanence of the contracted tissue displacement profile, J. theor. Biol., 177, 113-128 (1995) 59. L. Olsen, J.A. Sherratt and P.K. Maini, A mathematical model for fibroproliferative wound healing disorders, Bull. Math. Biol., 58, 787-808 (1996) 60. L. Olsen, P.K. Maini and J.A. Sherratt, Spatially varying equilibria of mechanical models: application to dermal wound contraction, Math. Biosciences, 147,113-129 (1998) 61. L. Olsen, J.A. Sherratt, P.K. Maini and F. Arnold, A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis, IMA J.Math.Appl.Med. & Biol, 14(4), 261-281 (1997) 62. L. Olsen, P.K. Maini, J.A. Sherratt and J. Dallon, Mathematical mod­ elling of anisotropy in fibrous connective tissue, Math. Biosciences, 158, 145-170 (1999) 63. G.F. Oster and J.D. Murray, Pattern formation models and development, Zool., 251, 186-202 (1989) 64. G.F. Oster, J.D. Murray and A.K. Harris, Mechanical aspects of mesenchymal morphogenesis, J. Embryol.exp. Morph., 78, 83-125 (1983) 65. H.G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57, 1044-1081 (1997) 66. A.V. Panfilov and A.V. Holden (eds), Computational Biology of the Heart, Chichester, John Wiley & Sons (1997) 67. K.J. Painter, P.K. Maini and H.G. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bull. Math. Biol. (to appear) (1999a) 68. K.J. Painter, P.K. Maini and H.G. Othmer, Stripe formation in juvenile Pomacanthus explained by a generalised Turing mechanism with chemotaxis, PNAS, 96, 5549-5554 (1999b)

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69. A.S. Perelson, P.K. Maini, J.D. Murray, J.M. Hyman and G.F. Oster, Nonlinear pattern selection in a mechanical model for morphogenesis, J. Math. Bioi, 24, 525-541 (1986) 70. G. Pettet, M.A.J. Chaplain, D.L.S. McElwain and H.M. Byrne, On the role of angiogenesis in wound healing, Proc. Roy. Soc. Lond., B263, 1487-1493 (1996) 71. J. Schnakenberg, Simple chemical reaction systems with limit cycle be­ haviour, J. theor. Bioi, 8 1 , 389-400 (1979) 72. L.A. Segel, Modelling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge, Cambridge University Press (1984) 73. D.R. Soil, D. Wessels and A. Sylwester, The motile behavior of amoebae in the aggregation wave in Dictyostelium discoideum, In: Experimental and Theoretical Advances in Biological Pattern Formation, H.G. Othmer, P.K. Maini and J.D. Murray, (eds.), Plenum Press, London, pp 325-328 (1993) 74. M.S. Steinberg, Does differential adhesion govern self-assembly processes in histogenesis? Equilibrium configurations and the emergence of a hier­ archy among populations of embryonic cells, J. exp. Zool., 173, 395-434 (1970) 75. D. Sulsky, S. Childress and J.K. Percus, A model of cell sorting, J. theor. Bioi, 106, 275-301 (1984) 76. D. Thomas, Artifical enzyme membranes, transport, memory and oscil­ latory phenomena, in Analysis and Control of Immobilized Enzyme Sys­ tems, ed. D. Thomas and J.-P. Kernevez, Springer, Berlin, Heidelberg, New York, 115-150 (1975) 77. P. Tracqui, D.E. Woodward, G.C. Cruywagen, J. Cook and J.D. Murray, A mechanical model for fibroblast-driven wound healing, J. Bioi Sys­ tems, 3, 1075-1085 (1995) 78. A.M. Turing, The chemical basis of morphogenesis, Phil Trans. Roy. Soc. Lond. B 327, 37-72 (1952) 79. J.J. Tyson and J.P. Keener, Singular perturbation theory of traveling waves in excitable media (a review), Physica D, 32, 327-361 (1988) 80. B. Vasiev and C.J. Weijer, Modelling Dictystelium Discoideum mor­ phogenesis, In: Proceedings of IMA Workshop on Pattern Formation, H.G. Othmer and P.K. Maini, (eds) to appear (2000). 81. M. Weliky, S. Minsuk, R. Keller and G.F. Oster, Notochord morpho­ genesis in Xenopus laevis: simulation of cell behaviour underlying tissue convergence and extension, Dev., 113, 1231-1244 (1991) 82. M. Weliky and G.F. Oster, The mechanical basis of cell rearrange­ ment. I. Epithelial morphogenesis during Fundulus epiboly, Dev., 109,

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373-386 (1990) 83. L. Wolpert, Positional information and the spatial pattern of cellular differentiation, J. theor. Biol, 25, 1-47 (1969) 84. L. Wolpert and A. Hornbruch, Double anterior chick limb buds and models for cartilage rudiment specification, Development, 109, 961-966 (1990)

RELATIVE EQUILIBRIA A N D CONSERVED QUANTITIES IN SYMMETRIC HAMILTONIAN SYSTEMS JAMES MONTALDI Institut Non Liniaire de Nice

1

Introduction

In this introduction, we first recall the basic phase space structures involved in Hamiltonian systems, the symplectic form, the Poisson brackets and the Hamiltonian function and vector fields, and the relationship between them. Afterwards we describe a few examples of Hamiltonian systems, both of the classical 'kinetic+potential' type as well as others using the symplectic/Poisson structure more explicitly. There are many applications of the ideas in these notes that have been investigated by different people, but which I shall not cover. The major ex­ ample, the one for which classical mechanics was invented, is the gravitational iV-body problem. But there are many others too, such as rigid bodies, cou­ pled rigid bodies, coupled rods, underwater sea vehicles, . . . not to mention the infinite dimensional systems such as water waves, fluid flow, plasmas and elasticity. The interested reader should consult the books in the list of refer­ ences at the end of these notes. Note that many of the items in the list of references are not in fact referred to in the text! 1.1

Hamilton's equations

The archetypal Hamiltonian system describes the motion of a particle in a potential well. If the particle has mass m, and V(x) is the potential energy at the point x (in whatever Euclidean space), then Newton's laws state that

mi =

-W(x).

In the 18th century, Lagrange introduced the phase space by defining y = x and passing to a first order differential equation, and Hamilton carried this further by introducing his now-famous equations q= p=

dH/dp, -dH/dq,

where q replaces x, and p = mi is the momentum, and H{q,p) = ±n\p\2 + V{q) 239

240 J. Montaldi

is the Hamiltonian, or total energy (kinetic + potential). This first order system is equivalent to Newton's law above, as is easily checked. The principal advantage of Lagrange/Hamilton's approach is that it is more readily generalized to systems where the configuration space is not a Euclidean space, but is a manifold. Such systems usually come about because of constraints imposed (eg in the rigid body the constraints are that the dis­ tances between any two particles is fixed, and the configuration space is then the set of rotations and translations in Euclidean 3-space). The other advantage of the Hamilton's approach is that it lends itself to generalizations to systems that are not of the kinetic + potential type, such as the model of a system of N point vortices in the plane or on a sphere, which we will see below, or Euler's equations modelling the "reduced" motion of a rigid body. These two generalizations lead to defining the dynamics in terms of a Hamiltonian on a phase space, where the phase space has the additional structure of being a Poisson or a symplectic manifold. The 'canonical' Poisson structure is given by it

Q1

_ v^ d/

d

9

dg df

where / and g are any two smooth functions on the phase space. The canonical symplectic structure is given by, n

w = 2_J dpj A dqj = da, where a = 5Z"=i Pj A Mi is the canonical Liouville l-form.a We shall see other examples of Poisson and symplectic structures in the course of these lectures. The Hamiltonian vector field XH is determined by the Hamiltonian H, a smooth function on the phase space, in either of the following ways: F = {H, F} w(v,XH)=dH(v),

W

where F = XH(F) is the time-rate of change of the function F along the trajectories of the dynamical system. Combining these two expressions gives "Not all authors agree on the choices of signs in the definition of the symplectic form or the Poisson brackets, so when using formulae involving either from a text or paper, it is necessary first to check the definitions. Our choice ensures that A"{/,9} = [Xf, Xg].

Relative Equilibria and Conserved Quantities in Symmetric Hamittonian Systems

241

the useful formula relating the symplectic and Poisson structures, u,(Xf,Xg) = {f,g}

(1.2)

for any smooth functions / , g. The first property of such Hamiltonian systems is their conservative na­ ture: the Hamiltonian function H is conserved under the dynamics and so too is the natural volume in phase space (Liouville's theorem). This has an important effect, not only on the type of dynamics encountered in such sys­ tems, but also on the types of generic bifurcations that can occur. Indeed, the first consequence of these conservation laws (energy and volume) is that one cannot have attractors in Hamiltonian systems, and in particular the notion of asymptotic stability is not available. A further feature of Hamiltonian systems is that symmetries lead to con­ served quantities. The two best-known examples of this are rotational sym­ metry leading to conservation of angular momentum, and translational sym­ metry to conservation of ordinary linear momentum. These further conserved quantities could in principle complicate the types of dynamics and bifurca­ tions one sees. However a well defined process called reduction (or symplectic reduction) can be used to replace the symmetry and conservation laws by a family of Hamiltonian systems, parametrized by these conserved quantities, on which there are general Hamiltonian systems whose bifurcations are those expected in generic systems. That said, there is a further complication which is that the phase space(s) for these reduced systems may be singular, or change or degenerate in a family, and we are only just beginning to understand the effects of these degenerations on the dynamics and bifurcations. 1.2

Examples

Spherical pendulum A spherical pendulum is a particle constrained to move on the surface of a sphere under the influence of gravity. As coordinates, one can take spherical polar coordinates 9, <j> (6 measuring the angle with the downward vertical, and <$> the angle with a fixed horizontal axis). Of course, this system has the defect of being singular at 0 = 0, n. The kinetic energy is T(q,q) = ! m £ 2 ( < J 2 + s i n 2 0 ^ ) , while the potential energy is V(q) = mg£(l - cos9). The momenta conjugate to the spherical polars are pe = dC/86 = me29\

p* = dC/d4> = ml2 sin2 6 4>,

242

J. Montaldi

where £ = T - V is the Lagrangian, and the Hamiltonian is then

The equations of motion are given as usual by Hamilton's equations. In particular, one can see from these equations that the angular momentum p^, about the vertical axis is conserved since H is independent of <j> (one has PO = dH/d

where Zj is a complex number representing the position of the j'-th vortex. This system is Hamiltonian, with the Hamiltonian given by a pairwise inter­ action depending on the mutual distances:

This is clearly not of the form 'kinetic+potential'. The Poisson structure is

{/> s} = E A 7 l (/** - »*/*) 3

and the symplectic form is w(u, v) = £V Xj (UJVJ - UjVj). Being of dimension 3, the Euclidean symmetry has 3 conserved quantities associated to it—see equation (6.2). A similar model can be obtained for point vortices on the sphere, providing a simple model for cyclones and hurricanes in planetary atmospheres. See Section 6.

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

1.3

243

Symmetry

A transformation of the phase space T : V -»• V is a symmetry of the Hamil­ tonian system, if (i) H{Tx) = H{x) for all

xeV,

(ii) T preserves the symplectic structure: T"u — u, or (ii') T preserves the Poisson structure: {/ o T, g o T}(x) — {/,

g}(T(x)).

There are three basic ways that, symmetries affect Hamiltonian systems: (a) The image by T of a solution is also a solution; (b) A solution with initial point fixed by T lies entirely within the set Fix(T,P), where Fix(T,V) = {x6 V \Tx = x). (c) If T is part of a continuous group, then the group gives rise to conserved quantities (Noether's theorem). The first of these is clear, and in fact is also true of more generalized symmetries for which H o T — H is constant, and T*w = a j ( c a constant). This occurs for example for homothcties of the plane in the planar point vortex model described above. The second (b) is less obvious, but very well-known; it follows from a very simple calculation as follows. If Tx = x then at(x) = <7t(Tx)

=Tat(x),

where at is the time t flow associated to the Hamiltonian system, and so <Jt(x) is fixed by T. If T is part of a compact group, then not only is Fix(T, V) invari­ ant, but it is a symplectic submanifold, and the restrictions of the Hamiltonian and the symplectic form (or Poisson structure) to Fix(T, V) is a Hamiltonian system which coincides with the restriction of the given Hamiltonian system— an exercise for the reader. This technique of restricting to fixed point spaces is sometimes called discrete reduction. In these notes we will be concentrating on the effect of (c). The central force problem provides the basic motivating example of this. 1.4

Central force problem

Consider a particle of mass m moving in the plane under a conservative force, whose potential depends only on the distance to the origin (a sim­ ilar analysis is possible for the spherical pendulum). It is then natural

244 J. Montaldi

to use polar coordinates (r, cj>) which are adapted to the rotational sym­ metry of the problem, so that V = V(r). The velocity of the particle is x = (rcos + (r sin (£)<£, rsin - (rcos)4>), so that the kinetic energy is

The Lagrangian is given by £ = T - V, and the Hamiltonian is H = T + V with associated momentum variables given by pr = dC/dr = mf and p^ = d£/d(fi = mr24>. Substituting for the velocities r and 0 in terms of the momenta determines the Hamiltonian to be:

H(r,<j>,pr,P) = — (pl + _ p 2 j

+

y(r).

Then Hamilton's equations with respect to these variables are |

r=ipr,

p r = - - L y p $ + v'( f .) (1.4)

The last equation says that p^ is preserved under the dynamics. In fact, p^ is the angular momentum about the origin r = 0. Since p$ is preserved, let us consider a motion with initial condition for which p0 = /j. Then (r,pr) evolve as

liV = - ^

a +

V'(r)

(L5)

This is in fact a Hamiltonian system, with Hamilton f/^(r,p r ) obtained by substituting fj, for p^. So that ^ ( r , p r ) = ^ - p ^ + - ^ + V(r). This is a 1-degree of freedom problem, called the reduced system, with "effective" potential energy V

"(»-) = ^ 2 + V ( r ) ,

and for a given potential energy function V(r), one can study how the be­ haviour of the system depends on p. For example, with the gravitational potential V(r) = - 1 / r , one obtains an effective potential of the form in Figure 1 below, where the fist graph shows the potential V(r) as a function of r, while the second and third show V„(r) for increasing values of p,.

Relative Equilibria and Conserved Quantities in Symmetric

Hamiltonian

Systems

245

It is clear that for /j, — 0 there is no equilibrium for the reduced system, while for ft > 0 there is an equilibrium, at r^ satisfying V^(r^) = 0—here rM = 3n2/m. Indeed, in this example it is a stable equilibrium for the effective potential has a minimum at the rM.

Figure 1. Effective potential for increasing values of fi, for V(r)

-1/r

Since r = rM and pr = 0 is an equilibrium of the reduced system, it is natural to substitute these values into the original equation (1.4). The two remaining equations are then

This describes a simple periodic orbit in the original phase space: {r,4>,pr,p
246

J.

Montaldi

1.5

Lie group actions

These notes assume the reader has a basic knowledge of actions of Lie groups on manifolds. Here I recall a few basic formulae and properties that are used. A useful reference is the new book by Chossat and Lauterbach [5]. Let G be a Lie group acting smoothly on a manifold V, and let g be its Lie algebra. We denote this action by (g, x) t-» g ■ x. The orbit through x is G -x = {g-x | g G G}. To each element ( e g there is associated a vector field on V which we denote £p. It is defined as follows fr(z) = ^ | t

= 0 (exp(<0-z)-

The tangent space to the group orbit at x is then g • x = {&{%) I £ € g}. A simple calculation relates the vector field at x with its image at g ■ x: dgz£v(x) = (Adgt)-p(g-x),

(1.6)

where Ad9 £ is the adjoint action of g on f, which in the case of matrix groups is just Adg S = g$g-\ The adjoint representation of g on g is the infinitesimal version obtained by differentiating the adjoint action of G: ad

*V

=

Jt\t = o Ade "P('«) ^ = & ''l'

Dual to the adjoint action on g is the coadjoint action on g*: (Coad9 n, TJ) := {n, Ad 9 -i 77) ,

(1.7)

and similarly there is the infinitesimal version, (coad,c (i, rj) := (n, ad_ 4 7?) = (/*, [77,

fl).

(1.8)

Examples 2.5 describe the coadjoint actions for the groups SO(3),SE(2) and SL(2). Given x € V, the isotropy subgroup of x is Gx = {g€G\g-x

= x}.

The Lie algebra gx of Gx consists of those £ 6 g for which fp(z) = 0, and the fixed point set of K Fix{K,V) = {x£V\K-x

= x},

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consists of those points whose isotropy subgroup contains K. It is not hard to show that it's a submanifold of V. Moreover, those points with isotropy precisely K form an open (possibly empty) subset of Fix(K,V). Stratification by orbit type If G acts on a manifold V, then the orbit space V/G is smooth at points where Gp is trivial, and more generally where the orbit type in a neighbourhood of p is constant. More generally, for each subgroup H < G one defines the orbit type stra­ tum V(H) to D e the set of points p for which Gp is conjugate to H. This is a union of G-orbits, and its image in V/G is also called the orbit type stratum (now iii the orbit space). These orbit type strata are submanifolds of V and V/G, and they fit together to form a locally trivial stratification (i.e. locally it has a product structure). For dynamical systems, the importance of this partition in to orbit type strata, is that for an equivariant vector field, the strata are preserved by the dynamics. Slice to a group action A slice to a group action at x 6 V is a submanifold of V which is transverse to the orbit through x and of complementary dimension. If possible, the slice is chosen to be invariant under the isotropy subgroup Gx (this is always possible if Gx is compact). A basic result of the theory of Lie group actions is that under the orbit map V -> V/G the slice projects to a neighbourhood of the image of G ■ x in the orbit space. Principle of symmetric criticality This principle is the variational version of discrete reduction, and provides a useful method for finding critical points of invariant functions. It states that, if G acts on a manifold V, and if / : V -> R is a smooth invariant function, then x £ F\x(G,V) is a critical point of / if and only if it is a critical point of the restriction f\Fix(G,v) °f / to Fix(G,V). One proof is to use an invariant Riemannian metric to define an equivariant vector field V / , which being equivariant, is tangent to Fix(G, V). For a full proof, valid also in infinite dimensions, see [58].

2

Noether's Theorem and the Momentum Map

The purpose of this section is to bring together facts about symmetry and conserved quantities that are useful for studying bifurcations. They are all found in various places, more or less explicitly, but not together in a single source. Furthermore, there appear to be misconceptions about whether "nonequivariant" momentum maps cause extra problems. Essentially, anything true for the equivariant ones remains true for non-equivariant ones (which are

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in fact equivariant, but for a modified action, as we shall see below). Many of the details of this chapter can be found in the books [16,8,10,13]. Examples 2.1 We will be using 3 examples of symplectic group actions in this section to illustrate various points. These arise in the models of systems of point vortices. (A) G = SO(3) acts by rotations on the sphere V - S 2 , with symplectic form given by the usual area form with total area 4-K (for example in spherical polars, u = sin(#) d$Ad(p). The Lie algebra so(3) can be represented by skewsymmetric matrices, and the vector field corresponding to the skew-symmetric matrix f is simply x t-¥ £x. (B) G = SE(2) acts on the plane V = R 2 , with its usual symplectic form w = dx A dy. This group acts by translation and rotations; indeed, SE(2) ~ R 2 x SO(2) (semidirect product), where R 2 is the normal subgroup of translations of the plane, and SO(2) is the group of rotations about some point, e.g. the origin. The Lie algebra se(2) is represented by constant vector fields (corresponding to the translation subgroup) and by infinitesimal rotations. (C) G = SL(2) = SL(2, R) acts by isometries on the hyperbolic plane V = H. There are several ways to realize this action, of which perhaps the best-known is to use Mobius transformations on the upper-half plane. However, we will use one that is more in keeping with the others, which is to represent the hyperbolic plane as one sheet of a 2-sheeted hyperboloid in R 3 : U = {{x,y,z)

6 R 3 | z7 - x2 - y2 = 1, z > 0}.

(2.1)

(The hyperbolic metric on H is induced from the Minkowski metric (dx2 + dy2 - dz2) on R 3 .) The symplectic form on H is given by w x (u,v) = - — - f l ( x ) - u A v ,

(2.2)

where R(x,y,z) = (x,y, -z) and u , v 6 Tx7i. One way to realize the action of SL(2) on V. is to embed 7i into the set of 2 x 2-trace zero matrices s l ^ R ) :

x« ( ; ) - * = ( , ! ,

»_+/)■

(,s>

Then A ■ x = AkA~l, for A e SL(2). Note that the image of the embedding consists of those matrices X of trace zero, unit determinant and such that A'12 > vY2i. Under this identification, the symplectic form (2.2) becomes the Kostant-Kirillov-Souriau symplectic form on the coadjoint orbit (see Example 2.5(C)).

D

Relative Equilibria and Conserved Quantities

2.1

in Symmetric

Hamiltonian

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249

Noether's theorem

With such a set-up, the famous theorem of Emmy Noether states that any 1-parameter group of symmetries is associated to a conserved quantity for the dynamics. In fact one needs some hypothesis such as the phase space being simply connected, or the group being semisimple (see [8] for details). For example, the circle group acting on the torus does not produce a globally well-defined conserved quantity. How do these conserved quantities come about? We already have a pro­ cedure for passing from Hamiltonian function to Hamiltonian vector field, and here we apply the reverse procedure. For each £ G g let fa : V —> R be a function that satisfies Hamilton's equation

dfc =«(&>,-),

(2.4)

if such a function exists. Of course each of these fa is only defined up to a constant, since only dfa is determined. Such functions are known as momentum functions, and a symplectic ac­ tion for which such momentum functions exist is said to be a Hamiltonian action. See [8] for conditions under which symplectic actions are Hamilto­ nian. Theorem 2.2 (Noether) Consider a Hamiltonian action of the Lie group G on the symplectic (or Poisson) manifold V, and let H be an invariant Hamil­ tonian. Then the flow of the Hamiltonian vector field leaves the momentum functions fa invariant. PROOF:

A simple algebraic computation: XH{fa)

= {H,fa}

= -{fa,H}

= -ZV(H)

This last equation holds because H is G-invariant.

= 0. □

Momentum map We leave questions of dynamics now, and consider the structure of the set of momentum functions. The first observation is that the momentum functions 0£ can be chosen to depend linearly on £. For example, let { £ i , . . . , £d} be a basis for g, and let fax,..., fad be Hamiltonian functions for the associated vector fields. Then for £ = ai£i + • • • + a<j£d, one can put, fa = a,ifa^+•■■ +adfad. It is easy to check that such (j>^ satisfy the necessary equation (2.4). Thus, for each point p e P w e have a linear functional £ >->fa(p),which we call $(p), so that

$(p) =

(fai(p),---,faM)

250 J. Montaldi

is a map $ : V -¥ g*, where g* is the vector space dual of the Lie algebra 9. Such a map is called a momentum map. The defining equation for the momentum map is,
(2.5)

for all p € V all v e TPV and all £ 6 g. An immediate and important consequence of (2.5) is: ker(d$ p ) = ( g . P r im(d* p ) = gpx C g*

(2-6) (2.7)

Here, U" is the linear space that is orthogonal to U with respect to the symplectic form. In particular, the momentum map is a submersion in a neighbourhood of any point where the action is free (or locally free: g p = 0). The fact that $ is only determined by the differential condition (2.5) means that it is only defined up to a constant. It follows that if $ is a momentum map, then $1 is a momentum map if and only if there is some C E g* for which, for all p € V,

We return to the possibility of choosing a different $ below. E x a m p l e s 2.3 Here we see momentum maps for three symplectic actions that arise for the point-vortex models, firstly for point vortices on the sphere, secondly for those in the plane and thirdly on the hyperbolic plane. We also give the general formula for momentum maps for coadjoint actions. (A) Let G - SO(3) act diagonally on the product V = S2 x . . . x S2 (N copies). On the r factor we put the symplectic form <jj = AjWo, where u>o is the canonical area form on the unit sphere (fs2 u>o — 4TT). Then a momentum map is given by $(xi,...,xN)

= ^2\jXj,

(2.8)

3

where the Lie algebra so(3) (consisting of skew symmetric 3 x 3 matrices) is identified with R 3 in the "usual way": B € so(3) corresponds to b € R 3 satisfying Bu = b x u for all vectors u 6 R 3 . It is clear that this momentum map is equivariant with respect to the coadjoint action, which under this identification becomes the usual action of SO(3) on R 3 . (B) An analogous example is G = SE(2) acting on a product of TV planes, with symplectic form u — ©jAjWj, where LJJ is the standard symplectic form

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

251

on the j t h plane. If we write SE(2) as R 2 x SO(2) (semidirect product), then the natural momentum map is

*(*x,..., xN) = I £ A, JXj, i £,. A; \Xj |2 j . where J = I

(2.9)

_ J is the matrix for rotation through 7r/2.

(C) A further analogous example is G = SL(2) acting on a product of N copies of the hyperbolic plane, V = %N, with symplectic form u; = ®j\jWj, where Uj is the standard symplectic form on the j t h plane, see (2.2). The natural momentum map is given by, $(xi,...,xN)

= ^TXjXj. (2.10) i D Remark 2.4 Consider any group acting on a manifold X, the configuration space. Classical mechanics of the "kinetic + potential" type takes place on the cotangent bundle of a configuration space, and in this setting the given action on X induces a symplectic action on the cotangent bundle, by the formula 9-{x,p)

= (9-x,

(dgx)-Tp),

where A~T is the inverse transpose of the operator A. Such actions are called cotangent actions or cotangent lifts and they always preserve the canonical symplectic form u on the cotangent bundle. The momentum map for cotangent actions always exists, and is given by (*(x,p), 0 = (P, &>(*)>, where the pairing on the right is between T*X and TXX. For example, if V = T*R 3 is the phase space for a central force problem, which has SO(3) symmetry, then after identifying so(3)* with R 3 as above, the momentum map $ : V -> so(3)* is just the angular momentum. 2.2

Equivariance of the momentum map

A natural question arises: since a momentum map $ : V -> g* is defined on a space V with an action of the group G, is there an action of G on g* for which $ commutes with (intertwines) the two actions? The answer was given in the affirmative by Souriau [16]. Usually, but not always, this turns out to be the coadjoint action of G o n g " , and before stating Souriau's result we give some examples of coadjoint action.

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Examples 2.5 The coadjoint actions for the groups described in Examples 2.1 are as follows. (A) For G - SO(3), the Lie algebra g = so (3) consists of the 3 x 3 skewsymmetric matrices, and via the inner product (4, B) = ti(ABT) we can identify the dual g* with g. An easy computation shows that the coadjoint action is given by CoadA ft = Afj,AT. With the usual identification of so (3) with R 3 (see Example 2.3) the coadjoint action is just the usual action of SO(3) by rotations, so that the orbits for the coadjoint actions are spheres centered at the origin, and the origin itself. For each of the 2 types of orbit, the isotropy subgroup is either all of SO(3), or it is a circle subgroup SO(2) of SO(3). This variation of the symmetry type of the orbits has interesting repercussions for the dynamics, and in particular for the families of relative equilibria. (B) Consider now the 3-dimensional non-compact group G = SE(2) of Euclidean motions of the plane. As a group this is a semidirect product R 2 x SO(2), where R 2 acts by translations and SO(2) by rotations about some given point (the "origin"). Elements (u, R) e R 2 x SO(2) can be iden­ tified with elements R u 0 1 by introducing homogeneous coordinates. A calculation shows that the adjoint action is Ad ( u , f i ) (v,B) = ( i ? v - B u , B), since B and R commute. The coadjoint action is then Coad {Ui/0 (i/ > ^) = {Ru, ip + RvuT).

(2.11)

Note that since elements of so (2) are skew-symmetric, it follows that only the skew-symmetric part of tfj + RvnT is relevant. One can therefore replace rl> + RuuT by %l> + \{Rv\xT - uvTRT). A nice representation of this using complex numbers is given in Section 6. The coadjoint orbits are again of two types: first the cylinders with axis along the 1-dimensional subspace of g* which annihilates the translation sub­ group (or its subalgebra), and secondly the individual points on that axis; that is, the points of the form (0, V). In this case the two types of isotropy subgroup for the coadjoint action are firstly the translations in a given direc­ tion (orthogonal to v), so isomorphic to R, or in the case of a single point

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

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Figure 2. Coadjoint orbits for SO(3) and for SE(2)

on the axis, it is the whole group. In both cases the isotropy subgroup is non-compact, a fact to be contrasted with the modified coadjoint action to be defined below. (C) For G = SL(2). Let A 6 SL(2) and /x e sl(2)*, then Coad^ fi =

A~TfiAT.

Notice that the determinant of the matrix in sl(2)* is constant on each coad­ joint orbit. In fact the coadjoint orbits are of 4 types: (i) the 1-sheeted hyperboloids (where GM ~ R), (ii) one sheet of the 2-sheeted hyperboloids (where G^ ~ SO(2)), (iii) each sheet of the cone with the origin removed (where G^ ~ R), and (iv) the origin itself (where GM = SL(2)). □ It is easy to see that the momentum maps given in Examples 2.3(A,C) are equivariant with respect to the coadjoint actions described above, which is not surprising in the light of the theorem below. However this is not always true in the case of G = SE(2). To describe the action that makes $ equivariant we follow Souriau and define the cocycle 6

-

G

^ f (212) v g ^4 $(# -x) - Coad 9 $(x), ' It is of course necessary to show that this expression is independent of x, which it is provided V is connected. We leave the details to the reader: it suffices to differentiate with respect to x and use the invariance of the symplectic form. The map 6 defined above allows one to define a modified coadjoint action, by Coad£/i:=Coad f l /i + % ) .

(2.13)

A short calculation shows that this is indeed an action. Moreover, this action is by affine transformations whose underlying linear transformations are the coadjoint action.

254 J. Montaldi

Theorem 2.6 (Souriau) Let the Lie group G act on the connected symplectic manifold V in such a way that there is a momentum map $ : V -► g*. Let 6 : G -4 g* be defined by (2.12). Then $ is equivariant with respect to the modified action on g* : $(g-x)

= Coa.d"g${x).

Furthermore, if G is either semisimple or compact then the momentum map can be chosen so that 9 = 0. For proofs see [16] or [8]; the first proof of equivariance in the compact case appears to be in [48]. Examples 2.7 In Examples 2.3 we gave the momentum maps for the three point-vortex models, and pointed out that for SO(3) and SL(2) the mo­ mentum map is equivariant with respect to the usual coadjoint action (not surprisingly in view of the theorem above since SO(3) is compact and SL(2) is semisimple). However, this is not always true in the planar case: (B) Consider the momentum map given in Example 2.3(B) for the point vortex model in the plane:

To find the action that makes this momentum map equivariant, we compute

\jJXj, I Zj \j\xs|2 + Ej *jA*j.u) +

* ( M + u , . . . , AxN + u) = (A£ 3

'

2

+ A(Ju, I|u| ) = Coad ( u M ) #(ar l t . ..,xN) + A(Ju, i | u | 2 ) , (2.14) where A = £V -\? £ R, and Coad is given in (2.11). The cocycle associated to this momentum map is thus given by 9(u,A) = A(Ju, | | u | 2 ) . If A = 0 then $ is equivariant with respect to the usual coadjoint action, while if A ^ 0 it is equivariant with respect to a modified coadjoint action. Furthermore, one can show that in this latter case there is no constant vector C G g* for which $ + C is coadjoint-equivariant. Indeed, it is enough to see that the orbits for this modified coadjoint action are in fact paraboloids, with axis the annihilator in g* of R 2 c g and these are not translations of the coadjoint orbits, which are either cylinders or points. See Figure 3. Furthermore, a short calculation shows that the isotropy subgroups for this action are all compact: G^ ~ SO(2), for all n e se(2)*, which is quite different from the coadjoint action. □

Relative Equilibria and Conserved Quantities in Symmetric Hamiitonian Systems

255

Figure 3. Modified coadjoint orbits for SE(2)

2.3

Reduction

Since by Noether's theorem the dynamics preserve the level sets of the mo­ mentum map $, it makes sense to treat the dynamical problem one level set at a time. However, these level sets are not in general symplectic submanifolds, and the system on them is therefore not a Hamiitonian system. However, it turns out that if one passes to the orbit space of one of these level sets of $, then the resulting reduced space is symplectic, and the induced dynamics are Hamiitonian. Historically, this process of reduction was first used by Jacobi, in what is called "elimination of the nodes". However, its systematic treatment is much more recent, and is due to Meyer [47] and independently to Marsden and Weinstein [43]. As usual suppose the Lie group G acts in a Hamiitonian fashion on the symplectic manifold V, and let $ : V -t g* be a momentum map which is equivariant with respect to a (possibly modified) coadjoint action as discussed above. Consider a value \i 6 0* of the momentum map. Then since $ is equivariant, the isotropy subgroup Gu of this modified coadjoint action acts on the level set < J>~ 1 (/J). Define the reduced space to be ^:=*-1(/x)/G„.

(2.15)

That is, two points of $ _ 1 ( A 0 axe identified if and only if they lie in the same group orbit. This defines Vu as a set, but to do dynamics one needs to give it more structure. If the group G acts freely near p (i.e. Gp is trivial), then Vu is a smooth manifold near the image of p in Vu. This is for two reasons: firstly $ is a submersion near p by (2.7), and secondly the orbit space by Gu will have no singularities. This is the case of regular reduction. We discuss singular reduction briefly below. It is important to know whether the induced dynamics on the reduced

256 J. Montaldi

spaces are also Hamiltonian. The answer of course is "yes". To see this it is necessary to define the symplectic form CJ^ on V^. Let u,v e TpV^, be projections of u,v e TVV, and define u^(u,«)

:=UJ(U,V).

Of course, one must show this to be well defined, non-degenerate and closed: exercises left to the reader! The Hamiltonian is an invariant, function on V so its restriction to $ _ 1 (/i) is G^-invariant and so induces a well-defined function on V^, denoted H^. The dynamics induced on the reduced space is then determined by a vector field Xu which satisfies Hamilton's equation: dH^ = u^X^,—). The orbit momentum map It is clear that V^ = $ _ 1 (/x)/G can also be defined by V» = ^~1(Oli)/G, where O^ is the coadjoint orbit through /x. Since $ _ 1 (C M )/G C V/G it is natural to use the orbit momentum map: if : V/G -+ Q'/G defined by

4 4V/G - A g*/G

(2.16)

where the vertical arrows are the quotient maps. Then V^ = <^_1(CM). This is very useful for studying bifurcations as the momentum value varies. However, it may not be so useful if G„ is not compact, for there the orbit space g*/G is not in general a reasonable space (it is not even Hausdorff for example for the coadjoint action of SE(2) near the V-axis).

2.4

Singular reduction

If the action of G on V is not free, then the reduced space is no longer a manifold. However, it was shown by Sjamaar and Lerman [67] that if G is compact then the reduced space can be stratified—that is, decomposed into finitely many submanifolds which fit together in a nice way—and each stratum has a symplectic form which determines the dynamics on that stratum. In fact these strata are simply the sets of constant orbit type in $ _ 1 (/i), or rather their images in V^. For the case of a proper action of a non-compact group see [20]. The theorem follows from the local normal form of Marie and Guillemin-Sternberg [8,42].

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

2.5

257

Symplectic slice and the reduced space

Recall that a slice to a group action at a point p € V is a submanifold 5 through p satisfying TPS © g • p = TPV. If Gv is compact, it can be chosen to be invariant under Gp. The slice, or more precisely S/Gp, provides a local model for the orbit space V/G. In the symplectic world, one needs to take into account the symplectic structure, and one wants the "symplectic slice" to provide a local model for the reduced space P M . In the case of free (or locally free) actions this is fairly straightforward, but in the general case, this is more delicate because the momentum map is singular. For this reason, the symplectic slice is usually taken to be a subspace of TPV rather than a submanifold of V. Definition 2.8 Suppose Gp is compact. Define N C TPV to be a Gp invariant subspace satisfying TpP = TV © g • p. The symplectic slice is then Ni

:=NDker{d$p).

It follows from the implicit function theorem that local coordinates can be chosen that identify a transversal to G^-p within the possibly singular set $ - 1 ( p ) with a subset of the symplectic slice N\. 3

Relative Equilibria

An equilibrium point is a point in the phase space that is invariant under the dynamics: p G V for which XH(P) — 0, or equivalently dHp = 0, and one way to define a relative equilibrium is as a group orbit that is invariant under the dynamics. Although geometrically appealing, this is not the most physically transparent definition. Definition 3.1 A relative equilibrium is a trajectory y(t) in V such that for each t G R there is a symmetry transformation gt G G for which i(t) = ft-7(0). In other words, the trajectory is contained in a single group orbit. It is clear that if a group orbit is invariant under the dynamics, then all the trajectories in it are relative equilibria; and conversely, if *y(t) is the trajectory through p, then g ■ j(t) is the trajectory through g ■ p and consequently the entire group orbit is invariant as claimed above. For TV-body problems in space, relative equilibria are simply motions where the shape of the body does not change, and such motions are always rigid rotations about some axis. Proposition 3.2 Let $ be a momentum map for the G-action on V and let H be a G-invariant Hamiltonian on V. Let p G V and let fj, = $(p). Then

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the following are equivalent: 1 The trajectory -y(t) through p is a relative equilibrium, 2 The group orbit G • p is invariant under the dynamics, 3 3 f e g such that j(t) = exp(^) • p,

V« € R,

4 3£ € 0 such that p is a critical point of H^ = H — <j>^, 5 p is a critical point of the restriction of H to the level set $ - 1 ( / i ) . Remarks 3.3 (i) The vector £ appearing in (3) is the angular velocity of the relative equilibrium. It is the same as the vector £ appearing in (4). The angular velocity is only unique if the action is locally free at p; in general it is well-defined modulo QP. (ii) If $ - 1 (//) is singular then it has a natural stratification (see §2.4), and condition (4) of the proposition should be interpreted as being a stratified critical point; that is all derivatives of H along the stratum containing p vanish at p. (iii) Notice that (3) implies that relative equilibria cannot meandre around a group orbit, but must move in a rather rigid fashion. It follows from this equation that the trajectory is in fact a dense linear winding on a torus, at least if G is compact. The dimension of the torus is at most equal to the rank of the group G. For G — SO(3), the rank is 1, and this means that any RE that is not an equilibrium is in fact a periodic orbit. The equivalence of (1) and (2) is outlined above. Equivalence of (1) and (3): (3) => (1) is clear. For the converse, since y(t) lies in G • p so its derivative XH(p) = 7(0) lies in the tangent space gp = Tp(G-p). Let f € Q be such that XH(P) = £v(p)- Then by equivariance XH{9-P) = (Ad9 0v(9-p), see (1.6). Let gt = exp(tf)- Then since Ad 9l £ = £, PROOF:

jt{9t ■ p) = &{9t ■ p) = XH(gt ■ p). That is, 11-» gt ■ p is the unique solution through p. Equivalence of (3) and (4): (3) is equivalent to Xn(p) — (,vip)- Using the symplectic form, this is in turn equivalent to dH{p) = d(j>^{p). Equivalence of (4) and (5): If $ is submersive at p then $ - 1 (/f) is a submanifold of V and this is just Lagrange multipliers, since d^ = (d$(p), f). In the case that $ is singular, the result follows from the principle of symmetric criticality (see the end of the Introduction for a statement). If $ is singular at p, then by (2.7) gp ^ 0. Consider the restriction of H to Fix(G p ,'P). By the

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

259

theorem of Sjamaar and Lerman, the set $ _ 1 (/i) is stratified by the subsets of constant orbit type—see §2.4. Consider then the set VGP of points with isotropy precisely Gp (this an open subset of Fix(Gp,V) containing p), and restrict both $ and H to this submanifold. Now $ restricted to VG is of constant rank, so that $ _ 1 (/x) C\VGV is a submanifold. The result then follows as before, since by the principle of symmetric criticality H restricted to VGT has a critical point at p if and only if H has a critical point at p. O One of the earliest systematic investigations of relative equilibria was in a paper of Riemann where he classified all possible relative equilibria in a model of affine fluid flow that is now called the Riemann ellipsoid problem (or affine rigid body or pseudo-rigid body)—see [25] and [66] for details. The configuration space is the set of all 3 x 3 invertible matrices, and the symmetry group is SO(3) x SO(3). In that paper he identified the 6 conserved quantities and found geometric restrictions on the possible forms of relative equilibria using the fact that the momentum is conserved, and that the solutions are of the form given in (3) of the proposition above. In terms of general group actions, the geometric condition Riemann used is the following, which follows immediately from Proposition 3.2 above together with the conservation of momentum. Corollary 3.4 Let p G V be a point of a relative equilibrium, of angular velocity £, and let 9 be the cocycle associated to the momentum map, then coad* $(p) = 0. If G is compact, so the adjoint and coadjoint actions can be identified and 9 = 0 for a suitable choice of momentum map, this means that the angular velocity and momentum of a relative equilibrium commute. For example, for any system with symmetry SO(3) this means that at any relative equilibrium, the angular velocity and the value of the momentum are parallel. Definition 3.5 A relative equilibrium through p is said to be non-degenerate if the restriction of the Hessian d^H^ip) to the symplectic slice Ni is & nondegenerate quadratic form. Definition 3.6 A point n € jj* is a regular point of the (modified) coadjoint action if in a neighbourhood of /x all the isotropy subgroups are conjugate. Examples of regular points are: all points except the origin for the coad­ joint action of SO(3), all points except the special axis for the coadjoint action of SE(2), and all points for the modified coadjoint action of SE(2) described in Example 2.7(B). The following result, the first on the structure of the families of relative equilibria, was observed by V.I. Arnold in [2]. The proof is an application of

260 J. Montaldi

the implicit function theorem. Theorem 3.7 (Arnold) Suppose that p lies on a non-degenerate relative equilibrium, xoith Gv = 0 and n = $(p) a regular point of the (modified) coadjoint action. Then in a neighbourhood of p there exists a smooth family of relative equilibria parametrized by /i 6 g*. Lyapounov stability A compact invariant subset S of phase space is said to be Lyapounov stable if "any motion that starts nearby remains nearby", or more precisely, for every neighbourhood V of S there is another neighbourhood V C V such that every trajectory intersecting V is entirely contained in V. A compact subset is said to be Lyapounov stable relative to or modulo G if the V and V above are only required to be G-invariant subsets. The principal tool for showing an equilibrium to be Lyapounov stable is Dirichlet's criterion, which is that if the equilibrium point is a non-degenerate local minimum of the Hamiltonian, then it is Lyapounov stable. The proof consists of noting that in this case the level sets of the Hamiltonian are topologically spheres surrounding the equilibrium, and so by conservation of energy, if a trajectory lies within one of these spheres, it remains within it. It is reasonably clear firstly that it is sufficient if any conserved quantity has a local minimum at the equilibrium point, not necessarily the Hamiltonian itself, and secondly that the local minimum may in fact be degenerate. These observations lead to the notion of... Extremal relative equilibria A special role is played by extremal relative equilibria. This is partly due to Dirichlet's criterion for Lyapounov stability, and partly because of their robustness. A relative equilibrium is said to be extremal if the reduced Hamiltonian H^ on V^ has a local extremum (max or min) at that point. This is usually established by showing the restriction to the symplectic slice of the Hessian of H^ = H - fa to be positive (or negative) definite, for some f for which H^ has a critical point at p, see Proposition 3.2. It is of course conceivable that a relative equilibrium is extremal while the Hessian matrix is degenerate. The simplest case of this is for H{x, y) = x2 +j/ 4 in the plane. In fact this arises in the case of 7 identical point vortices in the plane. The configuration where they lie at the vertices of a regular heptagon is a relative equilibrium, and the Hessian of H on the symplectic slice is only positive semi-definite. However, a lengthy calculation shows that the relevant fourth order terms do not vanish, and the reduced Hamiltonian does indeed have a local minimum there. For the following statement, recall that any momentum map is equivariant with respect to an appropriate action of G on Q* (Section 2), and for y, e g*

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we write G^ for the isotropy subgroup of n for that action. Theorem 3.8 ([48,34]) Let G act properly on V with momentum map $ , and suppose p € $ - 1 ( / i ) is an extremal relative equilibrium, with G^ compact. Then, (i) The relative equilibrium is Lyapounov stable, relative to G; (ii) There is a G-invariant neighbourhood U ofp such that, for all / / € $(U) there is a relative equilibrium in U H $ - 1 ( / i ' ) . The proof is mostly point-set topology on the orbit space and using the orbit momentum map (2.16), though part (ii) uses the deeper property of (local) openness of the momentum map. In fact the proof of (i) also holds if /i is a regular point for the (modified) coadjoint action, even if G^ is not compact. By Proposition 2.4 of [35] this result can be refined to conclude that the relative equilibrium is stable relative to G^, as described by Patrick [59]. Note that the compactness of Gu ensures that g,, has a G,,-invariant complement in g, as required in [35]. A consequence of using point-set topology is that there is very little information on the structure of the family of relative equilibria; for such information see Section 5. The crucial remaining point is how to determine whether a given relative equilibrium is extremal. In the case of free actions this was done by the so-called energy-momentum and/or energy-Casimir methods of Arnold and Marsden and others. Recently this was extended to the general case of proper actions: Proposition 3.9 (Lerman, Singer [35]) Letp £ $ - 1 ( / i ) be a relative equi­ librium satisfying the same hypotheses as the theorem above, and let £ £ g be any angular velocity of the RE as in Proposition 3.2. If the restriction to the symplectic slice of the quadratic form a?H^ is definite, then the RE is extremal, and so Lyapounov stable relative to G^. 4

Bifurcations of (relative) equilibria

In this section we take an extremely brief look at the typical bifurcations of equilibria in families of Hamiltonian systems as a single parameter is varied. These results also apply to relative equilibria, provided the reduced space is smooth in a neighbourhood of the relative equilibrium, and failing that, it applies to the stratum containing the relative equilibrium. Bifurcations of relative equilibria near singular points of the reduced space have not been investigated systematically.

262 J. Montaldi

4-1

One degree of freedom

Saddle-centre bifurcation This is the generic bifurcation of equilibria in 1 d.o.f. See Fig. 4. For A < 0 there are two equilibrium points: at (x,y) = (±y/^X, 0), one local extremum and one saddle. As A -► 0 these coalesce and then disappear for A > 0. One notices also a homoclinic orbit for A < 0, connecting the saddle point with itself, which can also be seen as a limit of the family of periodic orbits surrounding the stable equilibrium. In terms of eigenvalues of the associated linear system, this bifurcation can be seen as a pair of simple imaginary eigenvalues (a centre) decreases along the imaginary axis, collide at 0 and "then" emerge along the real axis (a saddle). This description is slightly misleading as the centre and saddle coexist. At the point of bifurcation, the linear system is I

_ 1; that is, it has non-zero nilpotent part.

Note that this bifurcation is compatible with an antisymplectic (i.e. timereversing) symmetry (x,y) -* (x, -y).

Figure 4. Saddle-centre bifurcation

Symmetric pitchfork This is usually caused by symmetry, and is 2 saddle centre bifurcations occurring simultaneously. On one side of the critical pa­ rameter value, there are 3 coexisting equilibria, while on the other side there is only one. In fact there are two types of pitchfork: a supercritical pitchfork involves two centres collapsing into a central saddle, leaving a single centre, and a subcritical pitchfork involves two saddles collapsing into a central centre, leaving a single saddle. See Figures 5 and 6. These results and normal forms are derived from Singularity/Catastrophe Theory.

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Figure 5. Supercritical Pitchfork

m )M( )H A<0

A= 0 H(x,y) = \y2 - x* - Ax2 + h.o.t.

A>0

Figure 6. Subcritical Pitchfork

4-2 Higher degrees of freedom FVom the point of view of bifurcations of equilibria, or of critical points of the Hamiltonian, the results for 1 degree of freedom carry over to higher dimensions, by adding a sum of quadratic terms in the other variables. So for example the saddle-centre bifurcations has as "normal form", #A(X, y) = |j/f + I*? + \Xl + I £ ^ ( ± 3 $ ± vj) + h.o.t. Whether any of the equilibria are stable depends of course on the signs of the quadratic terms. On the other hand, it is a much more subtle question as to whether any of the associated dynamics, such as the heteroclinic connections, survive this

264 J. Montaldi

passage into higher dimensions. For the saddle-centre bifurcation, see [22] and for the time-reversible case, see the lectures of Eric Lombardi in this volume, and for more detail [39]. 5

Geometric Bifurcations

In this section we discuss bifurcations of the families of relative equilibria (RES) due to degenerations in the geometry of the momentum map. One understands fairly well now the geometry of the family of relative equilibria in the neighbourhood of a point where the reduced phase spaces change in dimension, which occurs at special values of the momentum map, provided however that the group action on the phase space is (locally) free, so that the momentum map is submersive. However, the general structure of relative equilibria near points with continuous isotropy is not so well understood, although some recent progress has been made. Notation Throughout the theorems below, we will suppose that p 6 V is a point on a non-degenerate RE, that the angular velocity of this RE is £ and the momentum value is p. Recall that an RE is said to be non-degenerate if the Hessian dPH^ restricted to the symplectic slice is a non-degenerate quadratic form. In an important paper [60], George Patrick investigates the structure of the set of relative equilibria as quoted in the following theorem. He also studied the nearby dynamics and introduced the notion of drift around relative equilibria in terms of the linearized vector field there, but we will not be describing that aspect here. Theorem 5.1 (Patrick [60]) Assume G is compact, Gp is finite and G$ !~l Gfj. is a maximal torus. Then in a neighbourhood of p the set of relative equilibria forms a smooth symplectic submanifold ofV of dimension dim(G) + rank(G). Note that, as matrices, since f and p commute they are simultaneously diagonalizable. Consequently, they are both contained in a common maximal torus, so that G$ nG M always contains a maximal torus. The condition of the theorem is therefore a generic condition. For example, for SO(3) the condition is satisfied if and only if £ and // are not both zero. This theorem has been refined by Patrick and Mark Roberts [62], where they show that assuming a generic transversality hypothesis and that as before Gp is finite, the set of relative equilibria near p is a stratified set, with the strata corresponding to the conjugacy class of the group G^CiC^.

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The following result is more of a bifurcation theorem, as it is aimed at counting the number of relative equilibria on each reduced phase space, near a given non-degenerate RE. Recall that, if fj. is a regular point for the coadjoint action and Gp is trivial, then by Arnold's theorem (Theorem 3.7) there is a unique RE on each nearby reduced space. Theorem 5.2 (Montaldi [48]) Assume G^ is compact and Gp trivial. Then for regular fi' near fj, there are at least w{Gll) REs on the reduced space V^, where w{Gil) is the order of the Weyl group ofG^. Furthermore, if £ is regular then there are precisely this number of relative equilibria on V^. (For non-regular / / see [48]-) Note that if GM is a torus, then w{Gli) = 1, and this result reduces to Arnold's in the case that G is compact. The lower bound of w(G) for general £ follows from the Morse inequalities on the coadjoint orbits, and so presupposes that the RES on V^ are all non-degenerate. Without that assumption one can use the Lyusternik-Schnirelman category giving a lower bound of i d i m ( 0 , , ) - l - l . The proof of this result relies on the local normal form of Marie [42] and Guillemin-Sternberg [28]. Here I will outline the idea of the proof in the case that fj, = 0. The idea is to use the reduced Hamiltonian rather than the augmented Hamiltonian H$. So, the relative equilibria in question are critical points of the Hamiltonian restricted to V^, and near p one has 7V ~7>o x C V CV0

xg'.

The reduced space Vo can be identified with the symplectic slice Ni (see §2.5), so that by hypothesis p £ Vo = Vo x {0} is a non-degenerate critical point of the restriction of HtoVo- Write coordinates (j/,f) € V x g*. Then for each v, the function H(•, u) has an isolated non-degenerate critical point y = y(v). Define h : g* -¥ R by h{v) =

H(y{u),u).

Then one can show that the restriction h^ of h to O^ has a critical point at v iff H\v , has a critical point at (y(v),v). In this manner, the problem is reduced to finding critical points of the restrictions of a smooth function h to coadjoint orbits O^, and then one can use Morse theory or LyusternikSchnirelman techniques. The above proof assumes that Gp is trivial. However, if Gp is finite, then the proof can be modified to show that the nearby relative equilibria correspond to critical points of a smooth Gp-invariant function h, constructed in the same manner as above, but on Vo x g* rather than on the full orbit space Vo XG, 9*- This time though, different critical points correspond to

266

J. Montaldi

the same relative equilibrium if they lie in the same G p -orbit on O^. This argument gives the following 'equivariant' version of Theorem 5.2. Theorem 5.3 (Montaldi, Roberts [50]) // Gp is finite, then it acts on nearby coadjoint orbits O^. If a subgroup S of Gp is such that Fix(£,0„<) consists of isolated points, then they are all relative equilibria. This result uses the fact that H is G p -invariant, and not that Gp acts symplectically. In [50] this bifurcation result is applied to finding relative equilibria of molecules, and in [37] it is applied to finding relative equilibria of systems of point vortices on the sphere, and in both we use antisymplectic symmetries of H as well as symplectic ones. An action of G where the elements act either symplectically or antisymplectically, i.e. g'uj = ±u, is said to be semisymplectic [51]. In [50], the stability of the bifurcating relative equilibria is also calculated using these methods. The reader should also see [65,36,62,57] for further developments. 6

Examples

Let us look at three examples of symmetric Hamiltonian systems. • Point vortices on the sphere. • Point vortices in the plane. • Molecules (as classical mechanical systems). The motivation for choosing these models is that the first is relatively simple: it has no points where the action fails to be free (unless there are only 2 point vortices), and the group is compact. The bifurcations that arise are therefore of the types described in Section 5. The second is similar, except that the group of symmetries is no longer compact. The study of the classical mechanics of molecules is also of interest in molecular spectroscopy, where it is common practice to label particular quan­ tum states in terms of the corresponding classical behaviour. We treat this example extremely briefly! 6.1

Point vortices on the sphere

The model is a finite set of point vortices on the unit sphere in 3-space. A point vortex is an infinitesimal region of vorticity in a 2-dimensional fluid flow, though we ignore the fluid, and just concentrate on the vortices. The equations of motion for this system were obtained by V.A. Bogomolov [21]. A study of the dynamics of 3 point vortices has been carried out by Kidambi

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

267

and Newton [30] and by Pekarsky and Marsden [63]. The case of N identical point vortices has been treated in [37]. If xi,..., xw are the distinct locations of these point-vortices (unit vectors in R 3 ) , then the differential equation describing their motion is

>-2-A*i-sfc-z/ Here A i , . . . , XN are the strengths of the vortices: each Xj is a non-zero real number. It turns out that this vector field is Hamiltonian, with H(x) = - — ^ n

XjXk log (1 - Xj ■ xk),

}«■■

and Poisson structure {/>ff}(x) = - 5 2 *Jxdjf i where djf = djf(x) 2

Xj € S , and x =

x djg ■ Xj,

€ R 3 is the differential of / with respect to the point (XI,...,XN).

The phase space is V = S2 x S2 x . . . x S2 \ A, where A is the 'big diagonal' where at least one pair of points coincides, which is removed to avoid collisions. The symplectic form on V is given by u> = Xiu>\ ® • • • © AJVU>/V,

where <jj is the standard area form on the j t h copy of S2. This system has full rotational symmetry G = SO(3), and hence a 3component conserved quantity. After identifying so (3) with R 3 as usual, this momentum map is the so-called centre of vorticity:

*( x ) =^2^JxiThere are a number of immediate general consequences for this system that can be drawn from the Hamiltonian structure. For example, if all the vorticities are of the same sign, then as x -► A in V, so H(x) -* +00 and it follows that H attains its minimum at some point; this point is necessar­ ily an equilibrium point, and the set on which this minimum is attained is Lyapounov stable (one would expect this set to be a finite union of SO(3)orbits). Moreover, for any /x € so(3)* cz R 3 the same argument can be applied on $ - 1 ( / i ) , and the minimum on that set is necessarily a relative equilibrium, by Proposition 3.2.

268 J. Montaldi

However, if the vorticities are of mixed signs, then there is no general statement about the existence of equilibria or relative equilibria, except for TV = 3 where all RES are known (see below). If there are some identical vortices, then there is an extra finite symmetry group—a subgroup of the permutation group SN- This extra finite symmetry group is used to considerable effect in [37], from which Figures 7 and 8 are taken. Moreover, the time-reversing symmetries obtained from the reflexions in 0(3) are also used together with the principle of symmetric criticality to prove the existence of many types of relative equilibria. For example, let C/» denote the group of order 2 generated by reflexion in the horizontal plane. Then x e Fix(C h ) if all the vortices are on the equator. An application of the arguments above shows that if all the vorticities are of the same sign then there is a point on FixtCh.'P) where the restriction of the Hamiltonian attains its minimum, and by the principle of symmetric critical­ ity (see §1.5) it follows that this is an equilibrium point for the full system (though no longer a local minimum in general). And the same extension to this argument as before provides relative equilibria in Fix(C h , $ - 1 (/*)). In general there will be several relative equilibria in each component of Fix ( Q , , $ - 1 (/x)). However, it is shown in [37] that if all the vorticities are of the same sign then in each component of Fix(CA, V) there is a unique equilibrium point. 2 vortices If there are only 2 point vortices, then every solution is a relative equilibrium. Indeed, if $(x) = fi ^ 0 then the two points rotate at the same angular velocity about the axis containing fi. The only possible case where \i — 0 is if Ai = A2 and xx = - z 2 which is an equilibrium point. 3 vortices There have been two recent studies of the system of 3 point vortices, by Kidambi and Newton [30] (who also treat the 2 vortex case in an appendix) and by Pekarsky and Marsden [63]. The former describes not only the relative equilibria, but also self-similar collapse, where triple collision occurs in finite time with the three vortices retaining their same shape up to similarity. The latter describes the REs and their stability using the techniques described in these lectures (many of which were in fact developed by Marsden and co-workers). One of the results of Kidambi and Newton is that there are two classes of RE: those lying on a great-circle and those lying at the vertices of an equilateral triangle, and any RE belongs to one of these classes. Moreover, every equilateral triangle on the sphere is an RE, as we shall prove below. The following results are mostly taken from [30] and [63], and some are discussed below. Denote A1A2 + A2A3 + A3A1 by <72(A). P r o p o s i t i o n 6.1 For the system of N = 3 point vortices on the sphere,

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

C„*R)

269

qjOR-p)

Figure 7. Relative equilibria for 3 identical vortices on the sphere

(i) There exist equilibria iff 02(A) > 0, and they always lie on a great circle. (ii) All equilateral configurations are RES, and they are Lyapounov stable modulo SO(2) if a2(\) > 0, and are unstable if a2{\) < 0. (Hi) Self-similar collapse occurs iff (72(A) = 0. (iv) The configurations where the vortices lie on a great circle, at the vertices of a right-angled isosceles triangle is always an RE. Furthermore, the triangle with x\ at the right-angle is Lyapounov stable provided

\l + \l>2a2(\). The phase space is of dimension 6 in this case, and so the orbit space 7 7 /SO(3) is of dimension 3, and points in the orbit space correspond to the shapes of the triangle formed by the 3 vortices. The obvious set of coordinates consisting of the three pairwise distances has a problem for great circle con­ figurations since nearby such a configuration these distances do not determine the configuration uniquely. Indeed, these three distances are 0(3) invariants, as they do not distinguish the orientation, and configurations on a great circle have non-trivial isotropy for the 0(3)-action so it is not surprising that these coordinates have a problem there. For a good set of SO(3)-invariants, one must use the oriented volume as well, which is what is done in the two works cited above. However, the three distances do form a good set of coordinates away from the great circle configurations, and we will restrict our attention to those. So, let n be the chord distance \\x2 - X3II etc. Then the Hamiltonian and the orbit momentum map (2.16) are given by H(rur2,r3)

= - ^ l o g ( r 3 ) - ^ l o g ( n ) - ^Mog(r2)

(n,r 2 ,r 3 ) = | $ | 2 = (Ai + A2 + A3)2 - AxA2r2 - A2A3r2 -

X^rj

(6.1)

270

J. Montaldi

The reduced spaces are then PM := ¥> -1 (A* 2 )- The relative equilibria are determined by the critical points of the restriction of H to the reduced spaces, and are therefore critical points of H - r}
0 A3Ax 0

0 \ 0 . AtAj/

Now, the tangent space to ¥>-1(/x2) is spanned by (Ai,-A 2 ,0), (Ai,0,-A 3 ), and a computation then shows that the Hessian of the reduced Hamiltonian is definite if and only if 0. The computations for the configurations of vortices lying on great circles are longer, and I will not go into them further here; details can be found in the original papers [30,63]. R e m a r k s 6.2 (i) The case CT2(A) = 0 remains to be understood. (ii) It is not known whether any other form of collapse (i.e. not self-similar) can occur. (iii) A further bifurcation occurs at equilateral RES which lie on a great circle which to my knowledge has not been investigated. 4 vortices Much less is known in general about the case of 4 vortices. It is shown in [63] that the configuration where the four vortices lie at the vertices of a regular tetrahedron is always a relative equilibrium. It is also easy to show (from the equations of motion) that a square lying in a great circle is always an RE, independently of the values of the vorticities. However, in contrast to the 3-vortex case, squares not lying in great circles are not REs unless all the vortices are identical. In the case that all the vorticities coincide, a classification of symmetric REs is given in [37], from which Figure 8 is taken. Moreover, using the implicit

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

CJR.R')

271

qj[2R)

Figure 8. Relative equilibria for 4 identical vortices on the sphere

function theorem one can show that many of the REs shown to exist in [37] persist under small perturbations of the Hamiltonian, and so in particular under small changes of the values of the vorticities. There is one interesting occurrence of a symmetric pitchfork bifurcation: consider the family of square configurations, on the co-latitude 0—type Civ{R) in the figure. When the ring is close to the North pole (say), the RE is stable. As 6 increases, one pair of eigenvalues approaches 0, and at 0 = arccos(l/v / 3) there is a pitchfork bifurcation (of type I in the terminology of §4.1). The bifurcating pair of relative equilibria are of type C2t,(/?, R')Stability of a ring of vortices It was shown by Dritschel and Polvani [64] that the stability of a single ring of identical vortices depends on the latitude. They show that if 9 is the angle subtended by any of the vortices with the axis of symmetry of the ring (the colatitude), then the configuration of a regular

272 J. Montaldi

ring of N identical vortices is linearly stable as follows N range of stability N range of stability 3 all 0 4 cos* 9 > 1/3 5 cos2 9 > 1/2 6 cos2 9 > 4/5 while for N > 6 the ring is never stable. See also [38], where it is shown that the rings are not only linearly stable but Lyapounov stable. Bifurcations The changes in stability that occur in the table above involve bifurcations of the relative equilibria, and indeed the supercritical pitch­ fork bifurcation described in Section 4, since they all involve a loss of C2 symmetry. Consider for example the case of N = 4 identical vortices. A single ring (square) near the pole is a Lyapounov stable relative equilib­ rium. As 6 -> cos _ 1 (l/\/3), so one of the eigenvalues tends to 0, and for 9 > c o s _ 1 ( l / \ / 3 ) , there appears a new family of relative equilibria consisting of vortices alternately above and below the vortices in the now-unstable square configuration. These bifurcating RES—denoted C2v(R, R') in [37]—are then Lyapounov stable. Other stability transitions have been observed in [37,38], but the corresponding bifurcations have not been studied. The other type of bifurcation that occurs in this problem is the geometric bifurcation due to the different geometry of the reduced spaces for p. = 0 and p. ■£ 0—see Section 5. Consider a relative equilibrium pe on p = 0, for example the ring of N identical vortices on the equator. This corresponds to a point with symmetry DNh in the phase space (the dihedral group in the equatorial plane together with inversion in that plane). Nearby reduced spaces are then locally of the form V^ ~ V0 x O^, where O^ is the coadjoint orbit through p., which here is a sphere, as described very briefly in Section 5. The relative equilibria on 7?M near p e are the critical points of some function h : O^ -* R, and moreover this function is invariant under some action of T>nh ~ D ^ x C2 on Op. An analysis of this action shows that there must be critical points with symmetry of types C^„ and C 2 „; see Figure 9. The corresponding relative equilibria have configurations of types CNV (R) (a regular ring) and if N = 2m then C2v{mR) and C2v((m - l)iJ,2p), while if JV = 2m +1 then C2v(mR,p). Here the configuration C2v (mR, £p) consists of m pairs and (. poles all lying on a common great circle, while the great circle is rotating rigidly about an axis containing the poles. See Figure 8, and see [37] for details. 6.2

Point vortices in the plane

This system has a much older history than the model of vortices on the sphere, going back to Helmoltz and KirchhofF, and has been studied by many people

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273

Figure 9. Level sets of a typical function on a sphere with symmetry D3/,, showing half of the 8 critical points

since; for reviews see [17,3,15]. Consider an ideal fluid in the plane whose vorticity is concentrated in N point vortices, of strengths A i , . . . , \ N - These points move according to the differential system z

i ~ o~ 2_,

A*

where Zj is a complex number representing the position of the j-th vortex, after identifying the plane with C, see (1.3). The Hamiltonian for this system is

H(z) = --7-^2 47T

X Xk lo

i

S \zi ~ 2*l

The symmetry here is the group of 2-D Euclidean motions SE(2)—which is not compact. The corresponding conserved quantity (momentum map) is

$ ( « i , . . . , * * ) = (i^XjZj,

|EAil2il2J-

(6-2)

From the geometric point of view, this is interesting because if the total vorticity A = £ , \j is non-zero, the coadjoint action on se(2)* must be mod­ ified in order that the momentum map be equivariant (see Section 2). If we identify SE(2) with C x U(l) and u(l) with R, then the modified coadjoint action (2.7) becomes Coadf UiS) (i/,^) = (e*v, rp + %(eievu)) + A (tu, £|u| 2 ) where 9(z) is the imaginary part of z.

(6.3)

274 J. Montaldi

If A = 0 then the orbits are points on the ^-axis, and cylinders around that axis, while if A ^ 0 the orbits are all paraboloids—see Examples 2.5 and 2.7 respectively and Figures 2 and 3. Indeed, one can show that the modified coadjoint orbits are given by the level sets of / : C x R -> R defined by f(v,tl>) = \v\2-2AiP,

(6.4)

at least if A ^ 0. If A = 0 then the non-zero level sets are the cylindrical orbits, but the zero level set is the whole V-axis. 2 vortices in the plane As in the case of 2 vortices on the sphere, here they are also always relative equilibria. It is simple to prove from the differential equation that if A = Xi + A2 ^ 0 then the two point vortices rotate about the fixed point (Ai^ + A 2 z 2 )/A. On the other hand, if A = 0 then they translate together towards infinity, in the direction orthogonal to the segment joining them. These two types of motion are both relative equilibria. From a geometrical point of view, the reduced spaces are all just single points, so the corresponding motions are indeed all relative equilibria, and furthermore the relative equilibria are trivially extremal, and so provided G„ is compact (ie. A ^ 0, so G^ ~ SO(2)) they are stable modulo GM. 3 vortices in the plane The classical work on three planar point vortices is a beautiful paper by J.L. Synge [69]. We approach this problem as we did for three point vortices on the sphere. That is, points in the quotient of the phase space (R 2 ) 3 ~ C 3 by SE(2) correspond to shapes of oriented triangles. Again we ignore the orientation, which only causes problems near collinear configurations. Then a point in the quotient space is determined by the three lengths (rj ,r2,r3), and on that space 47rH(r 1 ,r 2 ,r 3 ) = -XiX2\og(r3) - A 2 A 3 log(n) - X3XX log(r 2 ) V(ri,r 2 ,r 3 ) = - A ^ r 2 - A2A3r2 - XzXxrl,

lR

.. W

the second is just / o $. It is remarkable that apart from constant term in ip, these are identical to equations (6.1) for 3 point vortices on the sphere. The non-collinear relative equilibria are given by the critical points of H restricted to the level-sets of tp, and the computation has already been done. Thus the relative equilibria are again equilateral triangles, of side r say, with Lagrange multiplier n = l/{Airr2) again. Note however, that this time the relation between rj and the "angular velocity" f is not so simple: 0 = d(H -

W)(p)

= d(H - nf o *)(p) = dH(p) - n df(ti)d*(p),

so that f = n df(^). Thus if $(p) = (u, ij)) then i = 27,(1/, - A ) .

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One consequence of this expression for f is that if A = 0 then the "angu­ lar velocity" is in fact rectilinear motion, with constant velocity f = 2TJI> = (l/27rr 2 )i/, where v = i V XjZj, which in modulus is independent of the rep­ resentative triangle. Note that if the 3 vortices are not collinear, then u ^ 0 so that the function / separates the relevant coadjoint orbits even in the case A = 0. On the other hand, if A ^ 0, and n ^ 0, then the relative equilibrium is a periodic orbit. Furthermore, the Lyapounov stabilities modulo G of these equilateral relative equilibria are the same as for the spherical case. The symplectic reduction for a particular class of planar 4-vortex prob­ lems has recently been considered by Patrick [61]. In particular he treats the case A — 0 so that the momentum map is coadjoint-equivariant; it would be interesting to see how the results are affected by changing to A ^ 0. 6.3

Molecules

Consider a molecule consisting of N atoms. The Born-Oppenheimer approxi­ mation consists of ignoring the movement of the electrons (which is reasonable as they are so light). The system then has 3N degrees of freedom, the 3 di­ rections of motion for each nucleus, or 3A^ — 3 after fixing the centre of mass. The rotational symmetry of the system gives rise to the conservation of angular momentum J. If we put J = 0, then there are 3A^ — 6 degrees of freedom which describes the shape of the molecule, and correspond to the vibrational motions. In other words, the J = 0 reduced space is of dimension 6A^ - 12. The simplest motions beyond the equilibria are the periodic orbits, which near the stable equilibrium are given by Lyapounov's theorem and its generalizations. For J / 0, the motion has a rotational aspect, and the simplest type of motion is the relative equilibrium. Beyond that are motions that are a combination of rotations and vibrations, so-called rovibrationcd states. The reduced spaces for J ^ 0 are of dimension 6N - 10. Consider now the simplest interesting case of a triatomic molecule. The reduced spaces are of dimension 6 (for J = 0) or 8 (for J ^ 0). There is one complication that we will not discuss here, namely the reduced space for J = 0 is singular at points corresponding to collinear equilibria. Consider then an equilibrium of a triatomic molecule which is not collinear. The geometric bifurcation methods discussed in Section 5, and at the end of §6.1, show that there are relative equilibria which bifurcate from the equilibria, and in fact there at at least 6 such families of RE parametrized by ||J|| and corresponding to critical points of functions on a sphere. The stabilities of these bifurcating families are discussed in [50]. For a complete investigation into the relative

276

J. Montaidi

equilibria of a specific molecule with 3 identical atoms—namely Hj—see [33]. Another interesting example is the tetra-atomic molecule ammonia NH3. This has an equilibrium where the three hydrogen atoms form an equilateral triangle, and the nitrogen atom is slighly above (or below) the centre of the triangle—so that the molecule is nearly planar. The analysis is similar to the triatomic case, with 6 families of RES that bifurcate from each equilibrium. The stability analyses should also be similar to the triatomic case. However, the presence of two very close stable equilibria (with the nitrogen atom on one side and on the other of the hydrogen-plane) and an unstable planar equilibrium betewen them suggests that there will be further bifurcations. An interesting "semilocal" analysis could be obtained by adding a new parameter A so that the equilibria undergo a pitchfork bifurcation (of type I) at A = 0, with the genuine system corresponding to say A = - 1 and an artificial one for A > 0 with the symmetric planar equilibrium being stable. One needs to investigate how the pitchfork bifurcation within the reduced space Vo, obtained by varying A, couples with the geometric bifurcation, obtained by varying ||J||. References Books

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