JCD
The computation of convex invariant sets via Newton's method
R. Baier M. Dellnitz M. Hessel-von Molo S. Sertl I. G. Kevrekidis
Journal of Computational Dynamics 2014, 1(1): 39-69 doi: 10.3934/jcd.2014.1.39
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.
keywords: Newton's method in Banach spaces directed sets. Nvariant sets set-valued Newton's method

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