The aim of this review article is to present recent advances in the theory of the dynamics of modulated phases in the Frenkel-Kontorova model. This theory is motivated through two specific condensed matter systems: charge-density wave conductors and Josephson junction arrays. The presentation tries to integrate the existing results into the perspective of the equilibrium theory of the model, which is summarized in the beginning. The issues of defectibility, metastability, pinning and synchronization are discussed in connection with the underlying interplay of continuum and discrete descriptions. Special emphasis is placed on the different transitions between dynamical phases; namely depinning transition, unlocking transition and dynamical Aubry transition.

Dissipative dynamics of the Frenkel-Kontorova model

A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems. An initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop. The "Newton descent" partial differential equation that governs this evolution is an infinitesimal step version of the damped Newton-Raphson iteration. The feasibility of the method is demonstrated by its application to the Henon-Heiles system, the circular restricted three-body problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.

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Variational method for finding periodic orbits in a general flow.

: A generalization of the class of monotone twistmaps to maps of S sub 1 x R sub n is proposed. The existence of Birkhoff orbits is studied, and a criterion for positive topological entropy is given. These results are then specialized to the case of monotone twist maps. Finally it is shown that there is a large class of symplectic maps to which the foregoing discussion applies. Keywords: Dynamical systems; Theorems.

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Monotone Recurrence Relations, Their Birkhoff Orbits and Topological Entropy.

Complete proofs are given for some claims of Middleton and of Floria and Mazo about the asymptotic behaviour of chains of balls and springs in a tilted periodic potential and generalizations, under gradient dynamics. AMS classification number: 58F22

Gradient dynamics of tilted Frenkel-Kontorova models

In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$ H(x,y)=h(y)+\epsilon P(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3. $$ Under typical perturbation $\epsilon P$, the system admits "connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\min\alpha$.

/pdf/arnold-diffusion-in-nearly-integrable-hamiltonian-systems-ye2dherol4.pdf

Arnold diffusion in nearly integrable Hamiltonian systems

We study the oscillation properties of periodic orbits of an area preserving twist map. The results are inspired by the similarity between the gradient flow of the associated action-function, and a scalar parabolic PDE in one space dimension. The Conley-Zehnder Morse theory is used to construct orbits with prescribed oscillatory behavior.

/pdf/the-periodic-orbits-of-an-area-preserving-twist-map-22q2aklcfg.pdf

S. B. Angenent

Papers

The periodic orbits of an area preserving twist map