Invariant manifolds with asymptotic phase for nonautonomous difference equations

Reverse-engineering invariant manifolds with asymptotic phase

2 Motivation: decomposition of NHIM-defining vector fields 3 2.1 The structure of the basin of attraction of a NHIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Persistence of this structure under perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Implications of NHIM results for model reduction . . . . . . . . . . . . . . . . . . . . . . . ...

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, 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 in...

Asymptotic Properties of Solutions of Nonautonomous Difference Equations

Asymptotic properties of solutions of difference equation of the form ∆xn = anφn(xσ(n)) + bn are studied. Conditions under which every (every bounded) solution of the equation ∆yn = bn is asymptotically equivalent to some solution of the above equation are obtained. Moreover, the conditions under which every polynomial sequence of degree less than m is asymptotically equivalent to some solution...

Computation of nonautonomous invariant and inertial manifolds

We derive a numerical scheme to compute invariant manifolds for timevariant discrete dynamical systems, i.e., nonautonomous difference equations. Our universally applicable method is based on a truncated Lyapunov-Perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton, and globally using c...

Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds