Bob Rink

Symmetry and Resonance in Periodic FPU Chains

Abstract: The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This “Birkhoff–Gustavson normal form” retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect this in lower-order resonant Hamiltonian systems. So the periodic FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries.

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Abstract The Fermi–Pasta–Ulam (FPU) lattice with periodic boundary conditions and n particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note. For each k dividing n we find k degree of freedom invariant manifolds. They represent short wavelength solutions composed of k Fourier modes and can be interpreted as embedded lattices with periodic boundary conditions and only k particles. Inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi-periodic solutions and standing and travelling waves. Some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating ‘bushes of normal modes’. The method of this paper is similar to the latter method and much more systematic than the former. We arrive at previously unknown results without any difficult computations. It is shown, moreover, that similar invariant manifolds exist also in the Klein–Gordon (KG) lattice and in the thermodynamic and continuum limits.

Near-integrability of periodic FPU-chains

The FPU-chain with periodic boundary conditions is studied by Birkhoff–Gustavson normalisation. In the cases of up to 6 particles and for β-chains with an odd number of particles the normal forms are integrable, which permits us to apply KAM-theory. This leads to the presence of many invariant tori on which the motion is quasi-periodic. Thus we explain the recurrence phenomena and the small size of chaos observed in experiments. Furthermore, we find a certain clustering of modes.

Coupled cell networks: Semigroups, Lie algebras and normal forms

We introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra. This implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself. Networks without the semigroup property will support normal forms with a more general network architecture, but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network. We explain how to compute Lie brackets and normal forms of coupled cell networks and we characterize the SN-decomposition that determines the normal form symmetry. This paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid.

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

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions 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 understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.

Coupled Cell Networks and Their Hidden Symmetries

Dynamical systems with a coupled cell network structure can display synchronous solutions, spectral degeneracies, and anomalous bifurcation behavior. We explain these phenomena here for homogeneous networks by showing that every homogeneous network dynamical system admits a semigroup of hidden symmetries. The synchronous solutions lie in the symmetry spaces of this semigroup and the spectral degeneracies of the network are determined by its indecomposable representations. Under a condition on the semigroup representation, we prove that a one-parameter synchrony breaking steady state bifurcation in a coupled cell network must generically occur along an absolutely indecomposable subrepresentation. We conclude with a classification of generic one-parameter bifurcations in monoid networks with two or three cells.

Branching patterns of wave trains in the FPU lattice

We study the existence and branching patterns of wave trains in the one-dimensional infinite Fermi–Pasta–Ulam (FPU) lattice. A wave train Ansatz in this Hamiltonian lattice leads to an advance–delay differential equation on a space of periodic functions, which carries a natural Hamiltonian structure. The existence of wave trains is then studied by means of a Lyapunov–Schmidt reduction, leading to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure. While exploring some of the additional symmetries of the FPU lattice, we use invariant theory to find the bifurcation equations describing the branching patterns of wave trains near p : q resonant waves. We show that at such branching points, a generic nonlinearity selects exactly two two-parameter families of mixed-mode wave trains.

The laminations of a crystal near an anti-continuum limit

The anti-continuum limit of a monotone variational recurrence relation consists of a lattice of uncoupled particles in a periodic background. This limit supports many trivial equilibrium states that persist as solutions of the model with small coupling. We investigate when a persisting solution generates a so-called lamination and prove that near the anti-continuum limit the collection of laminations of solutions is homeomorphic to the (N - 1)-dimensional simplex, with N the number of distinct local minima of the background potential. This generalizes a result by Baesens and MacKay on twist maps near an anti-integrable limit.

Amplified Hopf Bifurcations in Feed-Forward Networks

In [B. Rink and J. Sanders, Trans. Amer. Math. Soc., to appear] the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like

A dichotomy theorem for minimizers of monotone recurrence relations

\sim|\lambda|^{\frac{1}{2}},\sim|\lambda|^{\frac{1}{6}},\sim|\lambda|^{\frac{1}{18}}