# American Institute of Mathematical Sciences

January  2014, 1(1): 39-69. doi: 10.3934/jcd.2014.1.39

## The computation of convex invariant sets via Newton's method

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

Received  July 2012 Revised  February 2014 Published  April 2014

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