Invariant manifolds for analytic difference equations

TL;DR

This paper develops a modified parameterization method to analyze invariant manifolds in difference equations, including singular limits and non-dynamical cases, with applications to physics models and efficient algorithms.

Contribution

It introduces a novel approach for studying invariant manifolds in difference equations, extending to non-dynamical and singular cases, with practical algorithms and applications.

Findings

Established existence and regularity of invariant manifolds.

Developed efficient algorithms with implementations.

Applied methods to physics models like the Frenkel-Kontorova and Heisenberg models.

Abstract

We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds we consider include not only the classical strong stable and unstable manifolds but also manifolds associated to non-resonant spaces. When the difference equations are the Euler-Lagrange equations of a discrete variational we present sharper results. Note that, if the Legendre condition fails, the Euler-Lagrange equations can not be treated as a dynamical system. If the Legendre condition becomes singular, the dynamical system may be singular while the difference equation remains regular. We present numerical applications to several examples in the physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We also present extensions to finite differentiable difference equations.