@article{39691,
  author = {R. Baier and M. Dellnitz and M. Hessel-von Molo and S. Sertl and I. G. Kevrekidis},
  title = {The computation of convex invariant sets via Newton{\textquoteright}s method},
  abstract = { <p>
	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 <span class="MathJax"><span class="math" id="MathJax-Span-26"><span style="display: inline-block; position: relative; width: 1.389em; height: 0px; font-size: 90\%;"><span style="position: absolute; top: -1em; left: 0em; clip: rect(-0.002em, 1000em, 1.279em, -0.817em);"><span class="mrow" id="MathJax-Span-27"><span class="msubsup" id="MathJax-Span-28"><span style="display: inline-block; position: relative; width: 1.354em; height: 0px;"><span style="position: absolute; top: -3.222em; left: 0em; clip: rect(2.261em, 1000em, 3.501em, -0.817em);"><span class="texatom" id="MathJax-Span-29"><span class="mrow" id="MathJax-Span-30"><span class="mi" id="MathJax-Span-31" style="font-family: MathJax_AMS;">R</span></span></span></span><span style="position: absolute; top: -2.601em; left: 0.816em;"><span class="mi" id="MathJax-Span-32" style="font-size: 74.1\%; font-family: MathJax_Math; font-style: italic;">n</span></span></span></span></span></span></span></span></span> 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{\textquoteright}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.
</p>
 },
  year = {2014},
  journal = {Journal of Computational Dynamics},
  volume = {1},
  pages = {39-69},
  doi = {10.3934/jcd.2014.1.39},
  language = {eng},
}