The numerical computation of unstable manifolds for infinite dimensional dynamical systems by embedding techniques

TL;DR

This paper extends a set-oriented numerical framework to compute unstable manifolds in infinite dimensional dynamical systems, demonstrated on PDEs and delay differential equations, enabling better analysis of complex systems.

Contribution

It adapts a finite-dimensional unstable manifold computation method to infinite dimensional systems using embedding techniques, expanding its applicability.

Findings

Successfully computed unstable manifolds for PDEs and delay differential equations.

Demonstrated the method on Kuramoto-Sivashinsky and Mackey-Glass equations.

Showed the feasibility of the approach for complex infinite dimensional systems.

Abstract

In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo and Ziessler to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo and Ziessler. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto-Sivashinsky equation as well as for the Mackey-Glass delay differential equation.