Computation of nonautonomous invariant and inertial manifolds

Abstract

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 continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable, stable and center-stable manifolds, as well as their unstable counterparts.Our results are illustrated using a test example and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler-Bubnov-Galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold.

Computation of integral manifolds for Caratheodory differential equations

“…The proof is conceptually similar to the analogous result in the framework of nonautonomous difference equations (see Pötzsche & Rasmussen (2008) and references therein), where complications due to the measurable time-dependence of (2.2) can be treated as in Aulbach & Wanner (1996). Hence, we present only an outline of the proof.…”

“…• Figure 5 demonstrates that Algorithm 3.3 with an adaptive integration scheme quad is robust and yields convergence for large values of ξ (also in comparison to the corresponding discrete examples discussed in Pötzsche & Rasmussen (2008)). Here, as predicted in Theorem 3.2, the error grows linearly with P + (τ)ξ , whereas the number of evaluations grows logarithmically.…”

“…All these nonautonomous sets allow dynamical interpretations as in the discrete situation of difference equations (cf. Pötzsche & Rasmussen (2008)). …”

“…For numerical purposes, one has to truncate the integral over an infinite interval by a finite, but sufficiently large one. Recently, this approach has already been examined in the context of nonautonomous difference equations or temporal discretizations of differential equations (see Pötzsche & Rasmussen (2008)). In this reference, we formulated a numerical scheme based on the treatment of a nonlinear algebraic equation of finite dimension.…”

Global error analysis and inertial manifold reduction

“…[24,10,39,38,34,11,30] and [24,10,39,38,34,11,7,30], respectively. Recently, the theory of inertial manifolds has been generalized to non-autonomous dynamical systems [40,4,27,36], and recently, to random dynamical systems [35], and [3] (and the references therein).…”