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JCD

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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