44 research outputs found
Proof of Nishida's conjecture on anharmonic lattices
We prove Nishida's 1971 conjecture stating that almost all low-energetic
motions of the anharmonic Fermi-Pasta-Ulam lattice with fixed endpoints are
quasi-periodic. The proof is based on the formal computations of Nishida, the
KAM theorem, discrete symmetry considerations and an algebraic trick that
considerably simplifies earlier results.Comment: 16 pages, 1 figure; accepted for publication in Comm. Math. Phy
Symmetry and resonance in periodic FPU chains
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 it in lower-order
resonant Hamiltonian systems. So the FPU chain is an exception and its special
features are caused by a combination of special resonances and symmetries.Comment: 21 page
Continuity of the Peierls barrier and robustness of laminations
We study the Peierls barrier for a broad class of monotone variational
problems. These problems arise naturally in solid state physics and from
Hamiltonian twist maps.
We start with the case of a fixed local potential and derive an estimate for
the difference of the periodic Peierls barrier and the Peierls barrier of a
general rotation number in a given point. A similar estimate was obtained by
Mather in the context of twist maps, but our proof is different and applies
more generally. It follows from the estimate that the Peierls barrier is
continuous at irrational points.
Moreover, we show that the Peierls barrier depends continuously on parameters
and hence that the property that a monotone variational problem admits a
lamination of minimizers for a given rotation number, is open in the
C1-topology.Comment: 20 pages, submitted to Ergodic Theory and Dynamical System
Amplified Hopf bifurcations in feed-forward networks
In a previous paper, 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 , , , etc. Such amplified
Hopf branches were previously found by others in a subclass of feed-forward
networks with three cells, first under a normal form assumption and later by
explicit computations. We explain here how these bifurcations arise generically
in a broader class of feed-forward chains of arbitrary length
Center manifolds of coupled cell networks
Dynamical systems with a network structure can display anomalous bifurcations
as a generic phenomenon. As an explanation for this it has been noted that
homogeneous networks can be realized as quotient networks of so-called
fundamental networks. The class of admissible vector fields for these
fundamental networks is equal to the class of equivariant vector fields of the
regular representation of a monoid. Using this insight, we set up a framework
for center manifold reduction in fundamental networks and their quotients. We
then use this machinery to explain the difference in generic bifurcations
between three example networks with identical spectral properties and identical
robust synchrony spaces
Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice
The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and
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 dividing we find degree
of freedom invariant manifolds. They represent short wavelength solutions
composed of Fourier-modes and can be interpreted as embedded lattices with
periodic boundary conditions and only 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
traveling 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
lattice and in the thermodynamic and continuum limits.Comment: 14 pages, 1 figure, accepted for publication in Physica
A dichotomy theorem for minimizers of monotone recurrence relations
Variational monotone recurrence relations arise in solid state physics as
generalizations of the Frenkel-Kontorova model for a ferromagnetic crystal. For
such problems, Aubry-Mather theory establishes the existence of "ground states"
or "global minimizers" of arbitrary rotation number.
A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In
this case, the global minimizers have a special property: they can only cross
once. As a nontrivial consequence, every one of them has the Birkhoff property.
In crystals with a larger range of interaction and for higher order recurrence
relations, the single crossing property does not hold and there can exist
global minimizers that are not Birkhoff.
In this paper we investigate the crossings of global minimizers. Under a
strong twist condition, we prove the following dichotomy: they are either
Birkhoff, and thus very regular, or extremely irregular and nonphysical: they
then grow exponentially and oscillate. For Birkhoff minimizers, we also prove
certain strong ordering properties that are well known for twist maps
The parameterization method for center manifolds
In this paper, we present a generalization of the parameterization method,
introduced by Cabr\'{e}, Fontich and De la Llave, to center manifolds
associated to non-hyperbolic fixed points of discrete dynamical systems. As a
byproduct, we find a new proof for the existence and regularity of center
manifolds. However, in contrast to the classical center manifold theorem, our
parameterization method will simultaneously obtain the center manifold and its
conjugate center dynamical system. Furthermore, we will provide bounds on the
error between approximations of the center manifold and the actual center
manifold, as well as bounds for the error in the conjugate dynamical system
Ghost circles in lattice Aubry-Mather theory
Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice,
arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as
discretization of elliptic PDEs. Mathematically, they are a multidimensional
counterpart of monotone twist maps. They often admit a variational structure,
so that the solutions are the stationary points of a formal action function.
Classical Aubry-Mather theory establishes the existence of a large collection
of solutions of any rotation vector. For irrational rotation vectors this is
the well-known Aubry-Mather set. It consists of global minimizers and it may
have gaps.
In this paper, we study the gradient flow of the formal action function and
we prove that every Aubry-Mather set can be interpolated by a continuous
gradient-flow invariant family, the so-called "ghost circle". The existence of
ghost circles is first proved for rational rotation vectors and Morse action
functions. The main technical result is a compactness theorem for ghost
circles, based on a parabolic Harnack inequality for the gradient flow, which
implies the existence of ghost circles of arbitrary rotation vectors and for
arbitrary actions. As a consequence, we can give a simple proof of the fact
that when an Aubry-Mather set has a gap, then this gap must be parametrized by
minimizers, or contain a non-minimizing solution.Comment: 39 pages, 1 figur