The computation of convex invariant sets via Newton's method

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

Research output: Contribution to journalArticle

5 Scopus citations

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 ℝn 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.

Original languageEnglish (US)
Pages (from-to)39-69
Number of pages31
JournalJournal of Computational Dynamics
Volume1
Issue number1
DOIs
StatePublished - Jun 1 2014

All Science Journal Classification (ASJC) codes

  • Computational Mechanics
  • Computational Mathematics

Keywords

  • Directed sets
  • Invariant sets
  • Newton's method in Banach spaces
  • Set-valued Newton's method

Fingerprint Dive into the research topics of 'The computation of convex invariant sets via Newton's method'. Together they form a unique fingerprint.

  • Cite this

    Baier, R., Dellnitz, M., Hessel-von Molo, M., Sertl, S., & Kevrekidis, I. G. (2014). The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 1(1), 39-69. https://doi.org/10.3934/jcd.2014.1.39