Direction-Reversing Traveling Waves in the Even Fermi ...brink/dirrev.pdf · The relation between...

DOI: 10.1007/s00332-002-0497-xJ. Nonlinear Sci. Vol. 12: pp. 479–504 (2002)

© 2002 Springer-Verlag New York Inc.

Direction-Reversing Traveling Waves in the EvenFermi-Pasta-Ulam Lattice

B. RinkMathematics Institute, Utrecht University, P.O. Box 80.010, 3508 TA Utrecht,The Netherlandse-mail: [email protected]

Received October 9, 2001; accepted April 1, 2002Online publication July 18, 2002Communicated by P. J. Holmes

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with pe-riodic boundary conditions and quartic nonlinearities. Due to special resonances anddiscrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouvilleintegrable. The normal form equations can easily be solved if the number of particles inthe lattice is odd, but if the number of particles is even, several nontrivial phenomena oc-cur. In the latter case we observe that the phase space of the normal form is decomposedin invariant subspaces that describe the interaction between the Fourier modes with wavenumber j and the Fourier modes with wave number n

2 − j . We study how the level setsof the integrals of the normal form foliate these invariant subspaces. The integrable foli-ations turn out to be singular and the method of singular reduction shows that the normalform has invariant pinched tori and monodromy. Monodromy is an obstruction to theexistence of global action-angle variables. The pinched tori are interpreted as homoclinicand heteroclinic connections between traveling waves. Thus we discover a class of solu-tions of the normal form which can be described as direction-reversing traveling waves.The relation between the FPU lattice and its Birkhoff normal form can be understoodfrom KAM theory and approximation theory. This explains why we observe the impactof the direction-reversing traveling waves numerically as a relaxation oscillation in theoriginal FPU system.

Key words. Fermi-Pasta-Ulam lattice, Birkhoff normal form, singular reduction, Hamil-tonian Hopf bifurcation, traveling waves, approximation theory, KAM theory

AMS subject classification (2000). 34C14, 34C23, 34C26, 34C29, 34M35, 37J15,37J35, 37J40, 70H08, 70H09, 70H33

480 B. Rink

1. Introduction

The Fermi-Pasta-Ulam (FPU) lattice was introduced by E. Fermi, J. Pasta, and S. Ulam in[9]. It is a discrete model for a nonlinear string. This string is modeled by a finite numbern of equal point masses representing the material elements of the string. The point massesconstitute an anharmonic lattice. They interact via identical nonlinear forces betweenthe nearest neighbours. In this paper we shall consider the FPU lattice with periodicboundary conditions. This lattice models a circular string.

The periodic FPU lattice can mathematically be described as follows. The parti-cles in the lattice are labeled by an element in Z/nZ such that the 0-th and n-th par-ticles are identified. For j ∈ Z/nZ, let qj ∈ R be the position of the j-th particle.The space of positions q = (q1, . . . , qn) of the particles is R

n . The space of positionsand conjugate momenta is the cotangent bundle T ∗

Rn of R

n , the elements of whichare denoted (q, p) = (q1, . . . , qn, p1, . . . , pn). T ∗

Rn is a symplectic space, endowed

with the canonical symplectic form dq ∧ dp =∑nj=1 dqj ∧ dpj . Any smooth function

H : T ∗R

n → R induces the Hamiltonian vector field X H given by the defining rela-tion ιX H (dq ∧ dp) = d H . In other words, we have the system of ordinary differentialequations qj = ∂ H

∂pj, pj = − ∂ H

∂qj.

The Hamiltonian function of the periodic FPU lattice with quartic nonlinearities isthe sum of kinetic energy and potential energy. Because particles interact only with theirnearest neighbour, the Hamiltonian is

H =∑

j∈Z/nZ

1

2p2

j + W (qj+1 − qj ), (1.1)

in which W : R → R is a potential energy function of the form

W (x) = 1

2x2 + 1

4x4. (1.2)

We would like to view the solutions of the FPU system (1.1) as a superposition of waves.Therefore it is natural to make the following Fourier transformation:

qj =∑

1≤k< n2

√2

nωk

(cos

(2π jk

n

)qk + sin

(2π jk

n

)qn−k

)

+ (−1) j

√2n

q n2+ 1√

nqn. (1.3)

The term with the subscript n2 appears only if n is even. The real numbers ωj are defined

as

ωj := 2 sin

(jπ

n

), j = 1, . . . , n − 1. (1.4)

The mapping q �→ q is a linear isomorphism of the position space Rn . It induces a

symplectic transformation (q, p) �→ (q, p) on T ∗R

n . The Fourier coordinates (q, p)

Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice 481

are known as phonons or quasi-particles. In phonons, the Hamiltonian (1.1) reads

H = 1

2p2

n +n−1∑j=1

ωj

2( p2

j + q2j )+ H4(q1, . . . , qn−1). (1.5)

H4 denotes the quartic part of H . An exact expression for H4 in terms of the qj can becomputed by combining (1.1) and (1.3).

The pair of conjugate coordinates (qj , pj ) is called the j-th normal mode. For 1 ≤j ≤ n

2 , we say that the j-th and the (n− j)-th normal mode have wave number j , becauseboth are Fourier coefficients of waves with wave number j , that is, waves with wavelength 1/ j times the length of the lattice. The j-th and (n− j)-th normal mode also havethe same linear vibrational frequency: ωj = ωn− j . But the j-th and (n − j)-th normalmode are out of phase, because one describes a cosine wave in the lattice and the othera sine wave.

We observe that the Hamiltonian (1.5) is independent of qn . Therefore, the totalmomentum pn = 1√

n(p1+· · ·+pn) is a constant of motion and the pair (qn, pn) describes

the motion of a free particle. Moreover, the equations of motion for the remainingvariables (q1, . . . , qn−1, p1, . . . , pn−1) are completely independent of (qn, pn). The latterare hence usually neglected. We end up with the Hamiltonian system for the variables(q1, . . . , qn−1, p1, . . . , pn−1) on T ∗

Rn−1 given by Hamiltonian (1.5) where the term 1

2 p2n

is considered constant now. Thus we removed the total momentum from the equationsof motion. This construction is equivalent to the Marsden-Weinstein reduction of thesymmetry induced by the flow of X pn = ∂

∂qn, cf. [1] or [16].

Because ω1, . . . , ωn−1 > 0, we conclude using the Morse-lemma [1] that the originis a stable equilibrium of the Hamiltonian system on T ∗

Rn−1 induced by (1.5). It corre-

sponds to an equidistant configuration of the n particles in the lattice. In the sequel wewill study the flow of the vector field induced by (1.5) on T ∗

Rn−1 in a neighbourhood

of this equilibrium.The linearised equations of motion induced by the quadratic part of the Hamiltonian

(1.5) can easily be solved. Each solution of the linearised equations is a superpositionof harmonic oscillations with frequencies ωj . The normal mode energies ωj

2 ( p2j + q2

j )

are constants. The linear (or “harmonic”) FPU lattice is completely integrable and thesolutions lie on invariant tori of dimension n − 1. But the presence of the quartic termsin the Hamiltonian (1.5) destroys the integrability. In the anharmonic lattice, the normalmodes interact and they may exchange energy. Fermi, Pasta, and Ulam expected that amany particle system such as the FPU lattice would be ergodic, meaning that almost allorbits densely fill up an energy level set of the Hamiltonian. Ergodicity would have to leadto “thermalisation” or equipartition of energy between the various normal modes of thesystem. FPU’s nowadays famous numerical experiment [9] was intended to investigatehow thermalisation would take place. The result was astonishing: It turned out that therewas no sign of ergodicity of thermalisation at all. Instead, the solutions seemed to bequasi-periodic.

The observations of Fermi, Pasta, and Ulam greatly stimulated work on nonlineardynamical systems. A popular explanation nowadays for the quasi-periodic behaviourof the FPU system is based on the Kolmogorov-Arnol’d-Moser theorem. As is wellknown [2], the solutions of an n−1 degree of freedom Liouville integrable Hamiltonian

482 B. Rink

system are constrained to move on (n − 1)-dimensional tori and are not at all ergodicbut periodic or quasi-periodic. The KAM theorem states that most invariant tori of suchan integrable system persist under small Hamiltonian perturbations, if the unperturbedintegrable system satisfies a certain nondegeneracy condition known as the Kolmogorovcondition. This condition requires that the Liouville tori of the integrable system can beparametrised locally by their frequencies. Hence quasi-periodic motion can also oftenbe observed in nonintegrable systems and it is tempting to state that the KAM theoremexplains the observations of Fermi, Pasta, and Ulam—and several authors have claimedthis. It has for a long time been completely unclear though, how the FPU system can beviewed as a perturbation of a nondegenerate integrable system. At least the harmonicoscillator is very degenerate: Its frequencies are equal on each invariant torus. This gapin the theory was recently mentioned again in the review article of Ford [10] and thebook of Weissert [22].

Nishida [14] and Sanders [18] were the first to study the Birkhoff normal form ofHamiltonian (1.5). The Birkhoff normal form of a Hamiltonian system is constructed bya symplectic near-identity transformation which “simplifies” the lower-order polynomialterms of the Hamiltonian. The FPU system is related to its Birkhoff normal form in thefollowing two ways:

1. As the normal form equations approximate the original equations in the low energydomain, using Gronwall’s lemma one proves that low energy solutions of the Birkhoffnormal form approximate the low energy solutions of the original FPU chain on along time scale. For a proof see [21].

2. In the low energy domain the original Hamiltonian is a small perturbation of theBirkhoff normal form. So if the normal form is nondegenerately integrable, the KAMtheorem proves that many of the invariant Liouville tori from the normal form surviveas KAM tori in the low energy domain of the original FPU lattice.

Nishida and Sanders hoped to prove that the Birkhoff normal form constitutes a non-degenerate integrable approximation of the original FPU system. But to compute thenormal form, Nishida and Sanders were forced to make very restrictive nonresonanceassumptions, so they were unable to provide the desired proof.

In [16] the Birkhoff normal form of the periodic FPU Hamiltonian (1.1) was finallycomputed correctly, and it was proven to be a Liouville integrable approximation of(1.1) which in many cases is also nondegenerate. This integrability result is remarkable;it is caused by special resonances and discrete symmetries in the Hamiltonian (1.1). Butapart from the nondegenerate integrability, many properties of the normal form remainunstudied, and hence the normal form of (1.1) is the subject of this paper.

It turns out that the structure of the integrable normal form depends strongly on theparity of the number of particles n in the chain. If n is odd, then all the integrals of theBirkhoff normal form are quadratic functions of the phase space variables (q, p). Thesequadratics form a set of global action variables which can be augmented to a set of globalaction-angle variables. This implies that the foliation of the phase space in Liouville toriis trivial and the equations of motion can be solved explicitly.

The situation is not so simple though if the number of particles n is even. In thiscase some of the integrals of the normal form are quartic functions of the phase spacevariables, although there are still also quadratic integrals. The first important remark will


be that these integrals are uncoupled in the following sense: The phase space is the directsum of symplectic invariant subspaces:

T ∗R

n−1 =⊕

0≤ j≤ n4

Aj .

The subspaces

Aj ={(q, p) ∈ T ∗

Rn−1 | qk = pk = 0 ∀ k /∈

{j, n − j,

n

2− j,

n

2+ j

}}describe the interaction of the modes with wave numbers j and n

2 − j . The subspaceA0 = {(q, p) ∈ T ∗

Rn−1 | qj = pj = 0 ∀ j �= n

2 } is the subspace of the n2 -th normal

mode. It has one degree of freedom. A n4

has two degrees of freedom (if n is divisible by4) and for any other j , Aj has four degrees of freedom.

It turns out that each of the integrals of the normal form is defined on only one ofthese subspaces. In other words, one of the integrals is a function on A0, two integralsare functions on A n

4(if n is divisible by 4), and for each 1 ≤ j < n

4 , four integrals arefunctions on Aj . This implies that the foliation of T ∗

Rn−1 by level sets of the integrals

is simply the Cartesian product of the foliations of the various Aj .Of particular interest are the foliations of the subspaces Aj with four degrees of

freedom. These occur for even n ≥ 6. We study these foliations using purely geometricarguments based on invariant theory and the method of singular reduction; see [6]. Thisgeometric approach reveals all the integrable structure that is present in the Birkhoffnormal form.

First, we perform a regular Marsden-Weinstein reduction of the T 2-action induced bytwo of the integrals of the normal form. Thus the system on Aj reduces to a two degreeof freedom reduced Hamiltonian system. This reduced system contains four relativeequilibria of which we study the stability by means of linearisation. We observe thatthey can undergo a Hamiltonian Hopf bifurcation if one varies the energies of the waves.To be able to study this bifurcation in more detail, we perform a singular reduction.As the singular reduced phase space has only one degree of freedom, we can alsoinvestigate the global structure of the normal form. We see that there can be homoclinicor heteroclinic connections between the relative equilibria and we explicitly derive underwhat conditions these connections exist. The singular fibers over the homoclinic andheteroclinic connections are pinched tori: whiskered tori with coinciding stable andunstable manifolds.

It is well known [8, 19] that the presence of a pinched torus results in nontrivialmonodromy: The fibration of the phase space Aj (or T ∗

Rn−1) in Liouville tori is not

trivial in the sense that the Liouville tori do not form a trivial torus bundle over the set ofregular values of the integrals. Instead, we know how the Liouville tori are glued togetherglobally, for instance on an energy level set. Nontrivial monodromy is an importantobstruction to the existence of global action-angle variables; see [8].

On the other hand, our analysis yields interesting dynamical information. The relativeequilibria (in the reconstructed system these are the whiskered tori) can be interpreted aswaves in the periodic FPU lattice that travel clockwise and counter-clockwise. Thus itturns out that in the normal form there are homoclinic and heteroclinic connections (thewhiskers) between these traveling waves. Although the pinched tori themselves might

484 B. Rink

not really be present in the original FPU system, we expect from [20] that many of theLiouville tori close to a pinched torus survive as KAM tori in the full FPU lattice. TheseKAM tori constitute a large collection of interesting new solutions of the periodic FPUchain, showing a relaxation oscillation between traveling waves in opposite directions.We indeed detect this relaxation oscillation numerically in the original FPU lattice (1.1).This type of solution has the remarkable property that it displays an interesting interactionof the normal modes with wave numbers j and n

2− j without transferring energy betweenmodes with different wave numbers.

2. The Birkhoff Normal Form Is Integrable

We wish to simplify the FPU Hamiltonian (1.5) by studying its Birkhoff normal form:

H(q, p) := limT→∞

1

2T

∫ T

−TH(et X H2 (q, p)) dt. (2.1)

The flow of the linear vector field X H2 is simply quasi-periodic with frequencies ωj .H is calculated as the average of H over the flow of the linearised system. There

are several other methods to calculate normal forms, like for instance the well-knownLie-series method, described in [4] and [16]. One has the following result for the normalform:

Theorem 2.1. (Birkhoff normal form theorem) There is an open neighbourhood 0 ∈U ⊂ T ∗

Rn−1 and a symplectic near-identity transformation �: U → T ∗

Rn−1 with the

properties that �∗(dq ∧ d p) = dq ∧ d p, �(0) = 0, D�(0) = Id, D2�(0) = 0, and

�∗H = H ◦� = H +O(‖(q, p)‖5).

In other words, H and H are symplectically equivalent modulo a small perturbation oforder five. Studying H means neglecting the perturbation term O(‖(q, p)‖5). Of coursethis means that we make an approximation error. But this error is very small in the lowenergy domain, that is for small ‖(q, p)‖.

Using Gronwall’s lemma, it is easy to show that the low energy solutions of theBirkhoff normal form (2.1) approximate the low energy solutions of the original Hamil-tonian system (1.5) on a long time scale. One readily proves for instance the followingresult, which is formulated a bit heuristically here:

Proposition 2.2. Let x(t) and y(t) be solution curves of X H and X H respectively suchthat ‖x(0)‖ and ‖y(0)‖ are of order ε (0 < ε � 1) and x(0) = y(0). Then ‖x(t)− y(t)‖is of order ε3 on the time scale 1/ε.

See also [21] for more approximation results.If the Birkhoff normal form H is nondegenerately integrable, then the original system

can in the low energy domain be regarded as a small perturbation of a nondegenerateintegrable system. Using the KAM theorem we can then conclude that many low energysolutions of the original system (1.5) are quasi-periodic. On the other hand, this is only


true for finite n, and the domain where the approximation is valid probably shrinks whenn grows.Before presenting H , let us introduce some notation that will be convenient later on. For1 ≤ j < n

2 , define the “Hopf variables”

uj := qj pn− j − qn− j pj ,

vj := qj qn− j + pj pn− j , (2.2)

wj := 1

2(q2

j + p2j − q2

n− j − p2n− j ),

together with

Hj := 1

2(q2

j + p2j + q2

n− j + p2n− j ). (2.3)

And if n is even,

H n2

:= 1

2(q2

n2+ p2

n2). (2.4)

These quantities satisfy the following relations:

u2j + v2

j + w2j = H2

j , 1 ≤ j <n

2. (2.5)

In terms of the Hopf variables, the following expression for H was found in [16]:

H =∑

1≤ j≤ n2

ωjHj + 1

n

∑1≤k<l< n

2

ωkωl

4HkHl +

∑1≤k< n

2

ω2k

32(3H2

k − u2k) +

1

4H2

n2

+ 1

2H n

2

∑1≤k< n

2

ωkHk + 1

8

∑1≤k< n

4

ωkω n2−k(vkv n

2−k − wkw n2−k) + 1

16(v2

n4− w2

n4)

.

(2.6)

In formula (2.6) it is understood that terms with the subscript n2 and n

4 only appear if 2or 4, respectively, divides n.

As was shown in [16], due to discrete symmetries and special eigenvalues of Hamil-tonian (1.1), H remarkably is Liouville integrable.

Proposition 2.3. If the number of particles n is odd, then the Birkhoff normal form (2.6)is Liouville integrable with the quadratic integrals Hj , uj (1 ≤ j ≤ n−1

2 ).

This result was proved in Corollary 9.1 of [16]. The integral Hj has the interpretationof the linear energy of the two modes with wave number j , whereas uj is the angularmomentum of these modes.

486 B. Rink

The integrable system (2.6) with odd n was studied in [16]. The quadratic integralsform a set of global action variables. It can be augmented to a set of global action-anglevariables. The action-angle variables are explicitly calculated in [16] and the solutionsof the normal form can explicitly be written down. The foliation of the phase spaceinto invariant tori is trivial in the sense that the set of Liouville tori is a trivial torusbundle over the set of regular values of the integrals. The normal form turns out to beeven nondegenerate in the sense of the KAM theorem, which proves the abundance ofquasi-periodic solutions in the low energy domain of the original system (1.1). Thus, forodd n the normal form is quite tractable.

But the situation is not so easy if the number of particles in the FPU lattice is even.

Proposition 2.4. If the number of particles n is even, then the Birkhoff normal form(2.6) is Liouville integrable. The integrals are the quadratics Hj (1 ≤ j ≤ n

2 ), Ij :=uj − u n

2− j (1 ≤ j < n4 ), and J := 1

2√

2nv n

4(if n is a multiple of 4) and the quartics

Kj := 132n (4ωjω n

2− j (vjv n2− j − wjw n

2− j )− ω2j u2

j − ω2n2− j u

2n2− j ) (1 ≤ j < n

4 ).

This result was proved in Corollary 10.3 of [16]. The main part of this paper shallbe devoted to studying the dynamics and bifurcations of the integrable normal form(2.6) in the case that n is even. We shall encounter nice phenomena such as whiskeredtori with heteroclinic and homoclinic connections. They shall have the interpretation ofdirection-reversing traveling waves, and they constitute a class of solutions displayingan interesting interaction of Fourier modes with different wave numbers in the absenceof energy transfer.

We also study how the level sets of integrals of Proposition 2.4 foliate the phase spaceT ∗

Rn−1. It turns out that the foliation in Liouville tori is not trivial if n ≥ 6 is even.

There is monodromy, and global action-angle variables cannot exist for the integrablenormal form.

3. Phase Space Splitting and Regular Reduction

Let us use the shorthand notation = n2 − j .

We want to study how the level sets of the integrals of Proposition 2.4 foliate the phasespace T ∗

Rn−1. Therefore it is very useful to note that these integrals are uncoupled in the

following sense: The integral H n2

depends only on the phase space variables (q n2, p n

2).

The integrals H n4

and J depend only on the variables (q n4, q 3n

4, p n

4, p 3n

4). The integrals

Hj ,H , Ij andKj depend only on the variables (qj , q , qn− j , qn− , pj , p , pn− j , pn− ).So we have the following:

Theorem 3.1. The phase space of the Birkhoff normal form (2.6) is the direct sum ofinvariant symplectic subspaces

T ∗R

n−1 =⊕

0≤ j≤ n4

Aj ,


in which

Aj = {(q, p) ∈ T ∗R

n−1 | qk = pk = 0 ∀ k /∈ { j, n − j, , n − }}.Moreover, the foliation of T ∗

Rn−1 is the Cartesian product of the foliations of the Aj .

Therefore it is sufficient to know how H n2

foliates A0∼= T ∗

R, how H n4

and J foliateA n

4∼= T ∗

R2, and how Hj ,H , Ij and Kj foliate Aj

∼= T ∗R

4. The foliation of T ∗R

n−1

by all integrals is the Cartesian product of these foliations.

3.1. The Foliation of A0

It is clear that H n2= 1

2 (q2n2+ p2

n2) foliates A0

∼= T ∗R in circles which have their center

at the origin. In other words, the Fourier mode with the highest wave number ( n2 waves

in the lattice) and the highest vibrational frequency (ω n2= 2) has constant energy in the

normal form.

3.2. The Foliation of A n4

The system onA n4∼= T ∗

R2 with integralsH n

4andJ describes the interaction of the n

4 -thand 3n

4 -th normal mode. These modes have equal wave number n4 and equal vibrational

frequency√

2, but are out of phase, as one describes a cosine wave in the FPU latticeand the other a sine wave.

The foliation of A n4

can be described in the following standard way. It is clear that

imH n4= R≥0 and that for h n

4∈ R≥0, the inverse image H−1

n4

(h n4) = S3√

2h n4

. Here

S3√2h n

4

⊂ A n4

is the three-dimensional sphere with radius√

2h n4. The flow of XH n

4

induces an S1-symmetry on S3√2h n

4

given by

φ n

4,

q n4

q 3n4

p n4

p 3n4

�→

q n4

cos φ n4+ p n

4sin φ n

4

q 3n4

cos φ n4+ p 3n

4sin φ n

4

p n4

cos φ n4− q n

4sin φ n

4

p 3n4

cos φ n4− q 3n

4sin φ n

4

. (3.1)

The orbits of this circle action have dimension one if h n4

> 0 and dimension zerootherwise. The reduced phase space S3√

2h n4

/S1 can be found by applying the Hopf map

F (1): S3√2h n

4

→ S2h n

4

which is defined by

F (1): (q, p) �→ (u n4, v n

4, w n

4);

see [6]. The fibers of F (1) are exactly the orbits of the S1-action, so S2h n

4

constitutes

the reduced phase space. Every Hamiltonian function on A n4

that commutes with H n4

reduces to a Hamiltonian on S2h n

4

because it is constant on the orbits of the S1-action.

In particular, J = 12√

2nv n

4. The foliation of S2

h n4

in level sets of J is trivial: The level

488 B. Rink

n_4

n_4

n_4

v

u

w

Fig. 1. The reduced space S2h n

4

and the

level sets of J .

sets are the circles of constant v n4

and there are two stable relative equilibria. They aredepicted in Figure 1.

Reconstructing this picture by the Hopf map, we get the desired foliation of A n4.

3.3. The Foliations of the Aj

If n ≥ 6 is even, then for each 1 ≤ j < n4 , we have the four commuting integrals

Hj ,H , Ij , and Kj on Aj∼= T ∗

R4. They describe the interaction between the two

modes with wave number j and the two modes with wave number . Note that thesetwo pairs of modes do not interchange energy, since Hj and H are constants. Still itturns out that their interaction is very interesting. We shall see that the foliations of theAj contain singularities.

On Aj , consider the commuting Hamiltonians Hj and H defined in (2.3). Theydefine a mapping Hj, := (Hj ,H ). It is easy to see that im Hj, = (R≥0)

2, whereasfor the level sets we know that H−1

j, (hj , h ) = S3√2hj× S3√

2h

modulo permutation of

coordinates. Here, S3ε indicates the three-dimensional sphere with radius ε. Note that

ε = 0 is not excluded.Because the flows of the Hamiltonian vector fields of Hj and H commute and are

periodic with period 2π , they define a linear symplectic T 2-action on S3√2hj× S3√

2h

given by

(φj

φ

),

qj

q

qn−

qn− j

pj

p

pn−

pn− j

�→

qj cos φj + pj sin φj

q cos φ + p sin φ

qn− cos φ + pn− sin φ

qn− j cos φj + pn− j sin φj

pj cos φj − qj sin φj

p cos φ − p sin φ

pn− cos φ − qn− sin φ

pn− j cos φj − qn− j sin φj

. (3.2)

The orbits of this T 2-action are all tori of dimension #{k ∈ { j, } | hk �= 0}.


We want to study the reduced phase-space (S3√2hj×S3√

2h

)/T 2 . We do this by applying

the reduction map F (2): S3√2hj× S3√

2h

→ S2hj× S2

h defined by

F (2): (q, p) �→ (uj , vj , wj , u , v , w ).

F (2) is the Cartesian product of two Hopf mappings. The fibers of F (2) are exactlythe orbits of the T 2-action, so S2

hj× S2

h constitutes the reduced phase space. Every

Hamiltonian function on Aj that commutes with Hj and H reduces to a Hamiltonianfunction on S2

hj× S2

h because it is constant on the orbits of the T 2-action, in particular

Ij and Kj . The Hamiltonian equations of motion induced by such a Hamiltonian reduceto Hamiltonian equations on the orbit space S2

hj× S2

h . For a Hamiltonian function H ,

these reduced equations read

d

dt

uk

vk

wk

= 2

uk

vk

wk

×

∂uk H

∂vk H∂wk H

,

for k = j, . Thus we obtain a two degree of freedom integrable Hamiltonian systemon the product of two spheres. We will now study this reduced integrable system, whichdescribes the interaction of waves with wave numbers j and .

4. Traveling Waves

Let us now consider the reduced integrable system on S2hj× S2

h with Hamiltonian Kj and

momentum Ij = uj − u . The flow of XIj induces a symplectic S1-action on S2hj× S2

h

given by

t,

uj

vj

wj

u

v

w

�→

uj

vj cos 2t + wj sin 2twj cos 2t − vj sin 2tu

v cos 2t − w sin 2tw cos 2t + v sin 2t

. (4.1)

This action has four isolated fixed points, namely the points (±hj , 0, 0,±h , 0, 0). Butbecause the HamiltonianKj is invariant under the action (4.1), this implies that the deriva-tive of Kj also vanishes at these points. In other words, the points (±hj , 0, 0,±h , 0, 0)

constitute the set of joint critical points of Ij and Kj .Critical points of the reduced system on S2

hj× S2

h are called relative equilibria,

because in the reconstructed system on Aj∼= T ∗

R4 (or T ∗

Rn−1 if you like) their

fibers represent invariant sets; see [1]. It follows from (3.2) that in Aj the critical fiber(F (2))−1(±j h j , 0, 0,± h , 0, 0) is the following parametrised torus:

{(√hj cos φj ,√

h cos φ ,√

h sin φ ,√

hj sin φj ,∓j√

hj sin φj ,∓

√h sin φ ,

±

√h cos φ ,±j

√hj cos φj ) | (φj , φ ) ∈ T 2}. (4.2)

490 B. Rink

It has dimension #{k ∈ { j, } | hk �= 0}. This torus is invariant under the flow of X H ,so one can write the equations induced by H as equations for φ ∈ T 2. Using expression(2.6) it is not hard to compute that they read as follows:

dφj

dt= ±j

(∂ H

∂Hj|H=h −

ω2j h j

16

), (4.3)

dφ

dt= ±

(∂ H

∂H

|H=h −ω2

h

16

). (4.4)

Hence the motion in the critical fibers is uniform. The corresponding solutions have aclear physical interpretation: With (1.3) one can make the transformation back to theoriginal coordinates:

qk =√

2hj

nωjcos

(2π jk

n− φj

)+√

2h

nω

cos

(2πk

n− φ

).

The constants hj and h are supposed to be small, since otherwise the normal formapproximation has no validity. So a solution that lies in a critical fiber is a superpositionof

1. a small amplitude traveling wave with wave number j and speed approximately±jωj .2. a small amplitude traveling wave with wave number and speed approximately± ω .

If ±j = ± , then these waves move in the same direction. Otherwise one moves clock-wise and the other moves counterclockwise.

Traveling waves and superposed traveling waves have previously been studied in theinfinite FPU lattice [11]. But they can obviously also occur in a finite periodic lattice. Weshall study the stability of the traveling wave solutions and homoclinic and heteroclinicconnections between them. We do this of course in the reduced context, that is, weconsider them as critical points in the reduced phase space S2

hj× S2

h .

5. Stability of the Relative Equilibria

We want to determine the stability type of the superposed traveling wave solutions inthe Birkhoff normal form, that is, the stability type of the relative equilibria (±hj , 0, 0,

±h , 0, 0) on the reduced phase space S2h1× S2

h2. We will assume from now on that the

linear energies hj and h are both strictly positive, so that our reduced phase space hasdimension four. (The analysis is trivial if one of these energies is zero.) We perform ourstability analysis in local coordinates on S2

h1× S2

h2near (±hj , 0, 0,±h , 0, 0) by simply

projecting (uj , vj , wj , u , v , w ) �→ (vj , wj , v , w ). Note that these are not Darbouxcoordinates. The critical points themselves are all mapped to (0, 0, 0, 0).

5.1. A Lyapunov Function

One way of proving stability is by pointing out a Lyapunov function. The HamiltonianKj is the first candidate since it is an a priori constant of motion. But it turns out that


Kj is not definite at any of the relative equilibria. Luckily, we have another constantof motion, namely Ij . We now use that at (±j h j , 0, 0,± h , 0, 0) one may write uj =±j

√h2

j − v2j − w2

j = ±j (hj − 12hj

(v2j + w2

j ) + · · ·) and u = ±

√h2

− v2

− w2

=

± (h − 12h

(v2+ w2

)+ · · ·). So

Ij = uj − u = ±j h j ∓ h ∓j1

2hj(v2

j + w2j )±

1

2h

(v2 + w2

)+ · · · ,

which is definite at (0, 0, 0, 0) if and only if ±j �= ± . We conclude that the relativeequilibria ±(hj , 0, 0, −h , 0, 0) are stable.

Near the relative equilibria ±(hj , 0, 0, h , 0, 0) we will try to make linear combi-nations of Kj and Ij that are definite. It is easily computed that in local coordinates(vj , wj , v , w ),

(32nKj ± 2λhj h Ij ) = (ω2j + λh )(v

2j + w2

j )+ (ω2 − λhj )(v

2 + w2

)

+ 4ωjω (vjv − wjw )+ · · ·modulo constants. Using that |vjv −wjw | ≤ ‖(vj , wj )‖ ·‖(v , w )‖, one sees that thisexpression is definite if and only if

det

(ω2

j + λh 2ωjω

2ωjω ω2− λhj

)= −λ2hj h + λ(ω2

h − ω2j h j )− 3ω2

j ω2 > 0.

The preceding inequality has real solutions λ if the discriminant

r := ω4j h2

j − 14ω2j ω

2 hj h + ω4

h2 (5.1)

is positive. So if r > 0 then the relative equilibria ±(hj , 0, 0, h , 0, 0) are stable.

5.2. Linearisation

Because we still don’t know anything about stability if r < 0, an alternative is to study thelinearisation of the vector field XKj at ±(hj , 0, 0, h , 0, 0). Again in local coordinates,it reads

XKj

vj

wj

v

w

= ± 1

8n

0 −ω2j h j 0 2ωjω hj

ω2j h j 0 2ωjω hj 00 2ωjω h 0 −ω2

h

2ωjω h 0 ω2h 0

vj

wj

v

w

+ h.o.t.

(“h.o.t.” stands of course for “higher order terms.”) One calculates that the characteristicpolynomial of the above matrix reads

C(λ) = λ4 + λ2(ω4j h2

j + ω4 h2

− 8ω2j ω

2 hj h )/(8n)2 + 9(ω4

j ω4 h2

j h2 )/(8n)4 ,

so the eigenvalues are the numbers

λ = ± 1

16n

√p ± q

√r ,

492 B. Rink

where

p := 16ω2j ω

2 hj h − 2ω4

j h2j − 2ω4

h2 , q := 2ω2

j h j − 2ω2 h ,

and r is as defined previously in (5.1). Note that C(λ) = C(−λ), so if λ is an eigenvalueof (5.2), then so are −λ, λ, and −λ. The reason is of course that our matrix is conjugateto an infinitesimally symplectic matrix.

The next observation is that if r ≥ 0 or q = 0, then p± q√

r ∈ R so the eigenvaluesare purely real or purely imaginary, dependent on the signs of p ± q

√r . On the other

hand, if r < 0 and q �= 0, then none of the eigenvalues lies on the real or the imaginaryaxis. A simple analysis now leads to the following results:

• If r > 0, then there are four distinct purely imaginary eigenvalues. We have a doublecentre point.

• If r = 0, we find double imaginary eigenvalues. The linearisation matrix is notsemisimple.

• If r < 0, then none of the eigenvalues lies on the imaginary axis. We have a doublefocus equilibrium point (focus-focus point).

This is illustrated in the bifurcation diagram in Figure 2. The set of hj , h for whichr = 0 consists of two half-lines in the positive quadrant. On passing these half-lines,the eigenvalues move as is typical for the Hamiltonian Hopf bifurcation [12]: Two pairsof imaginary eigenvalues come together and split into a quadruple of nonimaginaryeigenvalues, where at the bifurcation value the linearisation matrix has a nilpotent part.So the linear stability of the relative equilibria±(hj , 0, 0, h , 0, 0) changes from neutrallystable to unstable. The linear instability implies of course that the equilibria are unstablealso in the nonlinear system. This concludes the stability analysis of the relative equilibria.In Section 6 we shall investigate the Hamiltonian Hopf bifurcation in more detail aswe will also incorporate the nonlinear terms of the Hamiltonian in our analysis of thebifurcation.

h j~

h j

stable

r > 0stable

r > 0

unstable

r < 0

Fig. 2. Bifurcation diagram of the linear stabilityof the relative equilibria.


5.3. Stability Results

We investigated the relative equilibria and their stability with respect to perturbations inthe initial data. The results are summarized in the following corollary.

Corollary 5.1. The relative equilibria±(hj , 0, 0,−h , 0, 0) are stable. The stability ofthe relative equilibria ±(hj , 0, 0, h , 0, 0) depends on the bifurcation parameter

r = ω4j h2

j − 14ω2j ω

2 hj h + ω4

h2 .

They are stable for r > 0 and unstable for r < 0.

In terms of traveling wave solutions, one may interpret Corollary 5.1 as follows:

The superposition of two traveling waves with wave numbers j and n/2− j is stable ifthe two waves move in opposite directions. A superposition of waves in equal directionscan be both stable and unstable. It is stable only if one of the waves is relatively smallwith respect to the other. Otherwise it is unstable.

6. Singular Reduction

In the next sections we shall try to understand the Hamiltonian Hopf bifurcation of theprevious paragraph geometrically. For this purpose, we shall make a reduction of theS1-symmetry (4.1) on the four-dimensional reduced phase space S2

hj× S2

h to obtain

another reduced phase space of again lower dimension. However, the S1-action (4.1)on S2

hj× S2

h has isotropy: Four of its orbits are not circles, but the equilibrium points

(±hj , 0, 0,±h , 0, 0). Therefore the regular Marsden-Weinstein reduction (see [1]) isnot sufficient, and we have to use the methods of invariant theory and singular reduction;see [6]. From the singularly reduced system on S2

hj×S2

h /S1 we can study the Hamiltonian

Hopf bifurcation geometrically and in more detail. Moreover, in Section 7 we will showthat if the relative equilibria±(hj , 0, 0, h , 0, 0) are unstable, then there are homoclinicand heteroclinic connections connecting them: pinched tori. In the original foliation ofAj ⊂ T ∗

Rn−1 these pinched tori are whiskered tori with coinciding stable and unstable

manifolds.

The flow of XIj induces a symplectic S1-action on S2hj× S2

h given by (4.1). The orbits

of this S1-action are all circles, except for exactly the relative equilibria (±hj , 0, 0,±h ,

0, 0).The following quantities are invariant under this action:

πj := uj , ρj := u , σj := vjv − wjw , τj := vjw + vwj .

In fact, every other invariant can be expressed as a function of πj , ρj , σj , and τj : Theyform a Hilbert basis of the ring of invariant functions. The invariants satisfy the equation

σ 2j + τ 2

j = (h2j − π2

j )(h2 − ρ2

j ).

494 B. Rink

Therefore, the set

Phj ,h := {(πj , ρj , σj , τj ) ∈ R

4 | σ 2j + τ 2

j = (h2j − π2

j )(h2 − ρ2

j ), |πj | ≤ hj , |ρj | ≤ h }

constitutes the orbit space S2hj× S2

h /S1 Note that we did not yet restrict ourselves to a

level of constant Ij ; this will come later.Every function on S2

hj× S2

h that commutes with Ij reduces to a function on Phj ,h

since it is constant on orbits. In particular,

Ij = πj − ρj and Kj := 1

32n(4ωjωσj − ω2

j π2j − ω2

ρ2j ).

The reduction map is the map

F (3): (uj , vj , wj , u , v , w ) �→ (πj , ρj , σj , τj ),

which goes from S2hj× S2

h to Phj ,h

. The reduction map is a submersion everywhere,except of course at the relative equilibria. Unfortunately it is not possible yet to makea drawing of Phj ,h

since it cannot be embedded in R3. But there is an elegant way to

overcome this problem.One sees that both Ij and Kj are independent of τj ; that is, these Hamiltonians are

invariant under the Z2-action generated by the time-reversal symmetry τj �→ −τj . Theorbits of the Z2-action consist of one point if τj = 0 and otherwise of two points. The Z2-action is reduced by simply forgetting about τj : The reduction map is (πj , ρj , σj , τj ) �→(πj , ρj , σj ). The reduced space is the set

Phj ,h /Z2 = {(πj , ρj , σj ) ∈ R

3 | σ 2j ≤ (h2

j − π2j )(h2

− ρ2j ), |πj | ≤ hj , |ρj | ≤ h }.

In Figure 3 we draw Phj ,h /Z2 for hj , h > 0. Phj ,h

/Z2 = S2hj× S2

h /S1×Z2 has the

shape of a solid pillow. The surface of the pillow is everywhere smooth, except at thefour corner points, which are conelike singularities that represent the relative equilibria(±hj , 0, 0,±h , 0, 0).

The level sets of Ij = πj − ρj are two-dimensional planes. The intersection ofsuch a plane with the pillow is a topological disk, a point, or empty. Near the sin-gularities (πj , ρj , σj ) = ±(hj ,−h , 0) the disks are very small, indicating that Ij is

Fig. 3. The solid pillow orreduced space Phj ,h

/Z2 .


Fig 4. Ij near hj + h . Fig 5. Ij near hj − h .

definite at these points as a function on the pillow and hence that the relative equilibria±(hj , 0, 0,−h , 0, 0) are stable; see Figure 4. But near the other two corners of thepillow, the singularities ±(hj , h , 0), the level set of Ij intersects the pillow in a verylarge set—see Figure 5—meaning that Ij is not definite.

Let us consider the level set ofIj that passes exactly through the singularity (hj , h , 0).It is the plane Ij = πj − ρj = hj − h . The intersection of this plane with the pillow isthe topolocal disk

{(πj , ρj , σj ) ∈ R3 | σ 2

j ≤ (h2j − π2

j )(h2 − (πj + h − hj )

2), ρj = πj + h − hj }.If hj = h , then this disk has two singular points. It has one singular point if hj �= h .The intersection of a level set of Kj with this plane is a parabola. The parabola thatcontains the singularity (hj , h , 0) is given by the formulas

σj = a(πj ) := 1

4ωjω

(ω2j π

2j +ω2

(πj+h −hj )2−ω2

j h2j−ω2

h2 ), ρj = πj+h −hj .

We now make a linear approximation to both this parabola and the singular disk at thesingular point (hj , h , 0). So we calculate the derivative da

dπj

∣∣πj=hj = 1

2ωj ω(ω2

j h j +ω2

h ). On the other hand, the conelike singularity of the singular disk is approximated

by

{|σj | ≤ 2√

hj h (hj − πj ), πj ≤ hj , ρj = πj + h − hj }.So the tangent line to the parabola points into the cone exactly if−2

√hj h < 1

2ωj ω(ω2

j h j

+ω2h ) < 2

√hj h , that is, if r < 0. In this case, the critical point (πj , ρj , σj ) =

(hj , h , 0) is clearly unstable, which agrees with our previous analysis. The tangent tothe parabola does not point into the cone if r > 0. Figures 6 and 7 represent the twopossibilities. This is a geometrical explanation for the motion of the eigenvalues. In thecase that r > 0, we now see that Kj , restricted to (Phj ,h

/Z2)∩ I−1j (hj − h ), is extremal

at (hj , h , 0). In other words, (hj , 0, 0, h , 0, 0) is a stable relative equilibrium insidethe singular fiber (S2

hj× S2

h ) ∩ I−1

j (hj − h ). A theorem of Montaldi [13] then states

that (hj , 0, 0, h , 0, 0) is stable in S2hj× S2

h .

We can study the Hamiltonian Hopf bifurcation in more detail by incorporating in ouranalysis also the nonlinear terms of the Hamiltonian. So we must compute the second-order approximation of the parabola a(πj ) and the cone (Phj ,h

/Z2)∩I−1j (hj −h ) at the

496 B. Rink

Fig 6. r > 0. Fig 7. r < 0.

singular point and at the critical value of the parameter r = 0. If the parabola is morecurved than the cone, the bifurcation is completely different from the case in which thecone is more curved than the parabola. The bifurcation is degenerate when the cone andthe parabola are equally curved, that is, when both second-order derivatives are equal.We do not treat that case. After a short calculation, one finds that the character of thebifurcation depends on the parameter

s := 4ω2j ω

2 h2

j + 4ω2j ω

2 h2

+ (6ω2j ω

2 − ω4

j − ω4 )hj h .

It is not very difficult to arrive at the following conclusions:

1. If on one of the half-lines defined by r = 0 one has that s > 0, then the HamiltonianHopf bifurcation on this half line is such that a stable relative equilibrium (=periodicsolution) emerges from the critical point as the critical point becomes unstable. Thisrelative equilibrium is indicated as a dot in Figure 8.

2. If on one of the half-lines defined by r = 0 one has that s < 0, then the HamiltonianHopf bifurcation on this half line is such that an unstable relative equilibrium (=peri-odic orbit) is annihilated by the critical point as the critical point becomes unstable.The relative equilibrium is indicated with a dot in Figure 9.

We have considered an entirely geometric way to study the Hamiltonian Hopf bifurcationand the stability change of the relative equilibrium (hj , 0, 0, h , 0, 0). The equilibrium(−hj , 0, 0,−h , 0, 0) can of course be handled in the same way.

��

Fig 8. 0 < r � 1 and s < 0. Fig 9. −1 � r < 0 and s > 0.


7. Pinched Tori and Monodromy

We have studied how the level sets of Kj foliate the singular reduced phase space(Phj ,h

/Z2) ∩ I−1j (hj − h ) locally near the singularities, and thus we could analyse the

details of the Hamiltonian Hopf bifurcation at these singular points. In this section wewill give some remarks on the global geometry of both the reduced phase space and thephase space of the original normal form.

We shall see that in the reduced system on S2hj× S2

h there are homoclinic and hete-

roclinic connections between the relative equilibria ±(hj , 0, 0, h , 0, 0) if r < 0. Theseconnections are pinched tori. In the system on T ∗

Rn−1 the pinched tori are reconstructed

as whiskered tori with high-dimensional coinciding stable and unstable manifolds. Thepresence of pinched tori results in nontrivial monodromy: The regular tori (Liouvilletori) do not form a trivial torus bundle.

We reach these conclusions by simply drawing a picture of (Phj ,h /Z2)∩I−1

j (hj −h )

and its foliation into the level sets of Kj . Recall that (Phj ,h /Z2) ∩ I−1

j (hj − h ) is atopological disk that contains one singular point if hj �= h and two singular points ifhj = h . Each point inside the disk represents two three-dimensional tori in Aj . Theregular points on the boundary of the disk each represent one three-dimensional torus.The singular points, which also lie on the boundary of the disk, represent a “singular”two-dimensional torus. This singular torus has the interpretation of a superposition oftraveling waves in the FPU lattice.

Let us first consider the case that hj = h . In that case the reduced space is boundedby parabolas

(Phj ,hj /Z2) ∩ I−1j (0) = {(πj , ρj , σj ) ∈ R

3 | |σj | ≤ h2j − π2

j , ρj = πj }.It has two singular points which lie on the same level set of Kj . This level set is alsoa parabola. Furthermore, note that r = (ω4

j + ω4− 14ω2

j ω2)h2

j = 16(1 − ω2j ω

2)h2

j ,which is negative if and only if ωjω > 1 if and only if n/12 < j < n/4. But thenthe reduced space simply looks like that shown in Figure 10. We observe that there isa heteroclinic connection between the two singular points. In the reconstructed systemon S2

hj× S2

h this corresponds to a doubly pinched torus: Two focus-focus singular

points of which the stable and unstable manifolds coincide. In the original phase spaceAj

∼= T ∗R

4 ⊂ T ∗R

n−1, this doubly pinched torus is again reconstructed as a heteroclinicconnection between two-dimensional tori. They are connected by their “whiskers” whichhave dimension four and are both diffeomorphic to R× T 3.

Fig. 10. The heteroclinic connection in the re-duced context.

498 B. Rink

The heteroclinic connection has the following interpretation. For large negative valuesof time, the solutions on the heteroclinic connection look like a superposition of travelingwaves in one direction. As time runs, these waves stop traveling and change direction.For large positive time values, the waves travel in the opposite direction.

If one starts close to the heteroclinic connection—that is, with a motion that is nearlythe superposition of two traveling waves with equal energy and in equal direction but notexactly on the heteroclinic connection—then after a certain time both traveling waves willcome to a halt and turn around until the motion looks very much like the superpositionof two traveling waves, the directions and energies of which are again almost equal,although the direction is opposite to the direction in the beginning. This “relaxationoscillation” continues and the superposed waves keep changing direction.

In the case that ωjω < 1, i.e., for 1 ≤ j < n/12, we have that r > 0 so the singularpoints are stable and there are no pinched tori. The case j = n/12 is degenerate: r isexactly zero and the parabola coincides with the boundary of the reduced space. We omitthe analysis of this last case.

Let us now briefly also consider the situation where hj �= h and r < 0. The singular levelset then generically looks as in Figure 11. We see that there is a homoclinic connectionthat connects the singular point to itself. In the reconstructed system on S2

hj× S2

h this

corresponds to a singly pinched torus: The stable and unstable manifold of the focus-focussingular point coincide. In Aj ⊂ T ∗

Rn−1 the pinched torus is a homoclinic connection

of a two-dimensional torus to itself. The two-dimensional torus again represents thesuperposition of traveling waves in the same direction with wave number j and , whichnow do not have equal energy.

This completely describes the interaction between traveling waves with wave numberj and traveling waves with wave number n/2− j . The traveling waves do not interchangeany energy. Still, their influence on one another is such that traveling waves can drasticallychange their momenta and thus their directions.

7.1. Monodromy

We observed that the foliation of the normal form has pinched tori. As was shown in [19],the presence of a pinched torus implies that the regular Liouville tori in S2

hj× S2

h do not

constitute a trivial torus bundle. Instead, they have monodromy and the monodromy map

Fig. 11. One of the homoclinicconnections in the reduced con-text.


is known. When we reconstruct, we see that the regular tori in T ∗R

n−1 cannot form atrivial bundle either. Nontrivial monodromy is an important obstruction to the existenceof global action-angle variables as was shown in [8].

Recall that a pinched torus is present in the normal form if n ≥ 6 is even. Weconclude that if n ≥ 6 is even, then the integrable normal form (2.6) does not admitglobal action-angle variables.

We have now described how the level sets of the integrals of the Birkhoff normalform (2.6) globally foliate the phase space. In the next section we shall investigate whathappens in the original FPU lattice with Hamiltonian (1.5).

8. Numerical Comparison for 16 and 32 Particles

We have analytically proven the existence of whiskered tori with coinciding stable andunstable manifolds in the Birkhoff normal form of the periodic FPU lattice. These objectshave the interpretation of direction-reversing traveling waves. It is natural to ask whetherthese homo- and heteroclinic connections can also be found in the original FPU lattice,and the answer to this question is most likely no: The original FPU system is a nonin-tegrable perturbation of its normal form, and hence the stable and unstable manifoldsof the original system will generically have either no intersection at all or transversalintersections. In the latter case, the angle of intersection will be quite small, so this maylead to small-scale chaos. A Melnikov-type analysis would be necessary to investigatethis. On the other hand, it was proven in [20] that near a pinched torus the Kolmogorovcondition holds. The Kolmogorov condition is the nondegeneracy condition that has tobe fulfilled for application of the KAM theorem. Hence one expects that many of theLiouville tori in the normal form lying close to the homo- or heteroclinic connection willsurvive as KAM tori in the original system. And therefore we expect to see many so-lutions of the original system which exhibit the relaxation oscillation between travelingwaves in various directions that was described in the previous section. I tried to detectthis relaxation oscillation in the original system by doing some basic numerical integra-tions of the solutions of the FPU system. All the numerical results in this section areobtained from the original Hamiltonian (1.1), and they are compared with the analyticalpredictions made from the normal form.

Let us first study the periodic FPU lattice (1.1) with n = 16 particles, a numberchosen by Fermi, Pasta, and Ulam themselves. We shall investigate low energy solutionsthat start out as a superposition of two traveling waves with wave numbers j = 3 and = 16

2 − 3 = 5. Note that n12 = 4

3 < j = 3 < n4 = 4, so if we give both waves equal

energy and equal direction, then we start very close to the heteroclinic connection of thenormal form and we expect the solution to reverse its direction once every while.

Let us choose the initial conditions q(0) = (0, 0, .25, 0, .25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

and p(0) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, .25, 0, .25, 0, 0), such that H3(0) = H5(0) =u3(0) = u5(0) = .0625. The total energy is indeed quite low: H = 0.1742. The angularmomenta u3(t) and u5(t) measure the direction of the waves. The normal form predictsthat u3 and u5 remain equal as I3 = u3−u5 is a constant of motion for the normal form.Moreover, we expect that both u3 and u5 exhibit a relaxation oscillation between theextremal values .0625 and −.0625 since this is exactly what happens on the Liouville

500 B. Rink

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

time

angu

lar

mom

enta

Fig. 12. Superposed traveling waves with wavenumbers 3 and 5 in the original FPU lattice with16 particles interact as predicted by the normalform: The two angular momenta exhibit a relax-ation oscillation between their maximal and min-imal values.

tori of the normal form. The true values of u3 and u5 in the course of time are plottedin Figure 12. Note that we have plotted both u3(t) and u5(t) but that we see only onecurve: It turns out that in the original system the angular momenta u3 and u5 remainexactly equal all the time at this energy level, just as was predicted by the normal form.Furthermore, u3(t) and u5(t) vary between their maximal and minimal values at thegiven energy. The influence of the heteroclinic connection of the normal form is clearlyvisible here: The angular momenta seem to stick to their maximal and minimal value forquite some time, before moving off again.

For comparison, I also investigated how waves with wave numbers 2 and 5 interact inthe system with 16 particles. Note that 2 �= 5 so that in the normal form there is no inter-action at all. Let us see numerically what happens in the original system. I chose q(0) =(0, .25, 0, 0, .25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) and p(0) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, .25, 0,

0, .25, 0), such that H2(0) = H5(0) = u2(0) = u5(0) = .0625. The total energy isH = 0.1528. The values of u2 and u5 are depicted on the same time scale in Figure 13.Exactly as was analytically predicted by the normal form, u2 and u5 remain constant inthe original system: The waves do not change their directions.

I also studied how the waves with wave numbers 3 and 5 interact at a higher energylevel, where one does not expect that the solutions of the original system follow the pre-dictions of the normal form. So we start with q(0) = (0, 0, .5, 0, .5, 0, 0, 0, 0, 0, 0, 0, 0,

0, 0) and p(0) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, .5, 0, .5, 0, 0), such that H2(0) = H5(0) =u3(0) = u5(0) = .25 and the total energy is quite high: H = 0.7048. The angular mo-menta u3(t) and u5(t) are shown in Figure 14. Note that the results are still reasonablywell in agreement with the normal form predictions: The angular momenta u3 and u5 arealmost equal all the time and exhibit a relaxation oscillation. Even at this high energylevel the normal form still constitutes a very good approximation! This is a remarkableobservation.


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Fig. 13. As predicted from the normal form,waves with wave numbers 2 and 5 do not in-teract in the original system.

Finally, I numerically integrated the original FPU chain with 32 particles for initialconditions qj (0) = pj (0) = 0 for all j except q7(0) = q9(0) = p23(0) = p25(0) = .25such that H7(0) = H9(0) = u7(0) = u9(0) = .0625 and H = 0.1762. The normalform predicts that the traveling waves with wave number 7 and 9 change direction, andwe plot both angular momenta u7 and u9 in Figure 15. The numerical results of theoriginal system are again in agreement with the normal form. But at a higher energy thenormal form is no longer a good approximation when n = 32. If one chooses the initialconditions qj (0) = pj (0) = 0 for all j except q7(0) = q9(0) = p23(0) = p25(0) = .5,then H7(0) = H9(0) = u7(0) = u9(0) = .25 and H = 0.7091. This time, the numericsshow that u7 and u9 do not remain equal. Neither do they oscillate between the extremalvalues .25 and −.25. See the result in Figure 16.

Thus, the domain of validity of the normal form shrinks if n grows. This is to beexpected, because when n is very large, near-resonances start spoiling the validity ofthe normal form. Still, it turns out that the normal form predicts the behaviour of the

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Fig. 14. Also at a higher energy level, the nor-mal form is a good approximation.

502 B. Rink

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Fig. 15. At low energy, the chain with 32 par-ticles behaves exactly as was predicted by thenormal form.

original FPU chain (1.1) surprisingly well, sometimes even at energies where one wouldnot expect this anymore.

9. Conclusions and Discussion

We studied the Birkhoff normal form of the periodic FPU lattice with periodic bound-ary conditions and quartic nonlinearities. In the domain of the phase space where thesolutions have low energy, this normal form constitutes an approximation of the originalFPU lattice: The solutions of both systems stay close on a long time scale. Because ofsymmetries and nonresonance properties, the normal form is Liouville integrable; see[16]. This enables us to give a very detailed analysis of the normal form equations. In thenondegenerate case, most Liouville tori of the normal form survive as KAM tori in the

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Fig. 16. For n = 32 the solutions of the originalsystem do not follow the normal form predictionsat higher energy.


low energy domain of the original system. We studied how the level sets of the integralsfoliate the phase space of the normal form.

Interesting phenomena occur when the number of particles n is even. In that casethe integrals of the normal form are quadratic and quartic functions of the phase spacevariables. Using the purely geometric methods of regular and singular reduction [6], weshowed that the foliation has singular elements, whiskered tori with coinciding stableand unstable manifolds, also called pinched tori.

It is well known [8], [19] that pinched tori imply monodromy: The Liouville tori ofthe normal form do not form a trivial torus bundle over the set of regular values of theintegrals. This is important information about how the Liouville tori are glued togetherglobally, for instance on an energy level set. Among others, monodromy is an importanttopological obstruction to the existence of global action-angle variables. Because theBirkhoff normal form approximates the FPU system especially well in the low energydomain, we expect that the KAM tori on the low energy level sets are “glued together”similarly.

At the same time, our study unravels interesting dynamical information. We are ableto determine how waves with different wave numbers interact in the normal form. Itturns out that waves with wave number j can only interact with waves of which thewave number is n/2 − j . And even though these waves do not interchange any energy,their interaction is far from trivial. The pinched tori that were mentioned before arehomoclinic and heteroclinic connections between solutions which are a superpositionof traveling waves with these wave numbers. Thus it can happen that these superposedtraveling waves change their direction.

The homoclinic and heteroclinic connections which exist in the normal form are mostprobably not present in the original FPU system: Under perturbations the stable andunstable manifolds will generically not intersect or at least intersect transversally. Theangle of intersection will be quite small then, so this may lead to small-scale chaos. AMelnikov-type analysis is necessary to investigate this. We chose not to perform thatanalysis in this paper and instead to focus on the regular dynamics. From [20] one expectsfor instance that many of the Liouville tori in the normal form that lie close to the homo-or heteroclinic connection will survive as KAM tori in the original system. Therefore,many solutions of the original system should exhibit a relaxation oscillation betweentraveling waves in various directions.

We indeed find this relaxation oscillation numerically in the original FPU lattices(1.1) with 16 and 32 particles. They form a class of interesting new solutions of theperiodic FPU chain.

Surprisingly, the FPU system follows the normal form predictions even at rather highenergy levels, where the Birkhoff normal form is a very questionable approximation.It would be interesting to study how robust the Liouville tori near a pinched torus areunder Hamiltonian perturbations. Maybe the KAM theorem can produce extrastrongconclusions for systems with monodromy, which enables us to understand the validityof the normal form at high energy.

Finally, the reader should be aware of other wave-reversal phenomena that have beenobserved in the literature. I especially refer to [3] which studies the “boomeron,” asoliton that comes back. Thus, we have found yet another interesting link between theFPU lattice and certain integrable wave equations.

504 B. Rink

Acknowledgments

The author thanks Richard Cushman and Ferdinand Verhulst for many valuable discus-sions and remarks. Odo Diekmann, Hans Duistermaat, Darryl Holm, and Claudia Wulffgave some nice comments and references.

References

[1] R. Abraham and J. E. Marsden. Foundations of Mechanics. Benjamin/Cummings PublishingCo., Reading, MA, 1987.

[2] V. I. Arnol’d and A. Avez. Ergodic Problems of Classical Mechanics. Benjamin, New York,Amsterdam, 1968.

[3] F. Calogero and A. Degasperis. Coupled nonlinear evolution equations solvable via theinverse spectral transform, and solitons that come back: The boomeron, Lett. Nuovo Cimento16 (1976) 425–433.

[4] R. C. Churchill, M. Kummer, and D. L. Rod. On averaging, reduction, and symmetry inHamiltonian systems, J. Diff. Eq. 49 (1983) 359–414.

[5] R. H. Cushman. A survey of normalisation techniques. In “Dynamics Reported,” new series,vol. 1, ed. C. K. Jones et al. Springer-Verlag, Berlin, 1992.

[6] R. H. Cushman and L. M. Bates. Global aspects of classical integrable systems. Birkhauser,Basel, 1997.

[7] R. H. Cushman and D. A. Sadovskii. Monodromy in the hydrogen atom in crossed fields,Physica D 142 (2000) 166–196.

[8] J. J. Duistermaat. On global action-angle variables, Commun. Pure Appl. Math. 33 (1980)687–706.

[9] E. Fermi, J. Pasta, and S. Ulam. Los Alamos Report LA-1940. In E. Fermi, Collected Papers2 (1955) 977–988.

[10] J. Ford. The Fermi Pasta Ulam problem: Paradox turns discovery, Phys. Rep. 213 (1992)271–310.

[11] G. Iooss. Traveling waves in the Fermi Pasta Ulam lattice, Nonlinearity 13 (2000) 849–866.[12] J. C. van der Meer. The Hamiltonian Hopf Bifurcation. Springer-Verlag, Berlin, 1985.[13] J. Montaldi. Persistence and stability of relative equilibria, Nonlinearity 10 (1997) 449–466.[14] T. Nishida. A note on an existence of conditionally periodic oscillation in a one-dimensional

anharmonic lattice, Mem. Fac. Eng. Univ. Kyoto 33 (1971) 27–34.[15] B. Rink and F. Verhulst. Near-integrability of periodic FPU-chains, Physica A 285 (2000)

467–482.[16] B. Rink. Symmetry and resonance in periodic FPU chains, Commun. Math. Phys. 218 (2001)

665–685.[17] D. A. Sadovskii and B. I. Zhilinskii. Monodromy, diabolic points, and angular momentum

coupling, Phys. Lett. A 256 (1999) 235–244.[18] J. A. Sanders. On the theory of nonlinear resonance, thesis, University of Utrecht, 1979.[19] N. Tien Zung. A note on focus-focus singularities. Diff. Geom. Appl. 7 (1997) 123–130.[20] N. Tien Zung. Kolmogorov condition for integrable systems with focus-focus singularities,

Phys. Lett. A 215 (1996) 40–44.[21] F. Verhulst. Symmetry and integrability in Hamiltonian normal forms. In Symmetry and

Perturbation Theory, ed. D. Bambusi and G. Gaeta, Quaderni GNFM, Firenze, 1996.[22] T. W. Weissert. The Genesis of Simulation in Dynamics; Pursuing the FPU Problem. Springer-

Verlag, New York, 1997.

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