Invariant Manifolds for Analytic Difference Equations

We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, and 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 that we consider include not only the classical strong stable and unstable manifolds but also manifolds associated with nonresonant spaces. When the difference equations are the Euler-Lagrange equations of a discrete variational problem, we have sharper results. Note that, if the Legendre condition fails, the Euler-Lagrange equations cannot 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.