The computation of convex invariant sets        via Newton's method

R. Baier; M. Dellnitz; M. Hessel-von Molo; S. Sertl; I. G. Kevrekidis

doi:10.3934/jcd.2014.1.39

Article Contents

2014, Volume 1, Issue 1: 39-69. Doi: 10.3934/jcd.2014.1.39

This issue Previous Article Global invariant manifolds near a Shilnikov homoclinic bifurcation Next Article Continuation and collapse of homoclinic tangles

The computation of convex invariant sets via Newton's method

1.
Chair of Applied Mathematics, University of Bayreuth, 95440 Bayreuth
2.
Chair of Applied Mathematics, University of Paderborn, 33098 Paderborn
3.
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544

Received: July 2012

Revised: February 2014

Published: January 2014

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Abstract

In this paper we present a novel approach to the computation of convex invariant sets of dynamical systems. Employing a Banach space formalism to describe differences of convex compact subsets of $\mathbb{R}^n$ by directed sets, we are able to formulate the property of a convex, compact set to be invariant as a zero-finding problem in this Banach space. We need either the additional restrictive assumption that the image of sets from a subclass of convex compact sets under the dynamics remains convex, or we have to convexify these images. In both cases we can apply Newton's method in Banach spaces to approximate such invariant sets if an appropriate smoothness of a set-valued map holds. The theoretical foundations for realizing this approach are analyzed, and it is illustrated first by analytical and then by numerical examples.

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Mathematics Subject Classification: 65J15, 52A20, 37M99, 26E25, 54C60.

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