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 CMFD, 2015, Volume 56, Pages 5–128 (Mi cmfd268)

Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side

M. I. Zelikina, L. V. Lokutsievskiia, R. Hildebrandb

a Lomonosov Moscow State University, Moscow
b Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

Abstract: In this paper, we consider linear-quadratic deterministic optimal control problems where the controls take values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in a finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely, the chaotic behavior of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighborhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us a possibility of calculating the entropy and the Hausdorff dimension of the nonwandering set, which appears to have a Cantor-like structure as in Smale's horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behavior is generic for piecewise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00784

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English version:
Journal of Mathematical Sciences, 2017, 221:1, 1–136

UDC: 517.9

Citation: M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side”, Optimal control, CMFD, 56, PFUR, M., 2015, 5–128; Journal of Mathematical Sciences, 221:1 (2017), 1–136

Citation in format AMSBIB
\Bibitem{ZelLokHil15} \by M.~I.~Zelikin, L.~V.~Lokutsievskii, R.~Hildebrand \paper Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side \inbook Optimal control \serial CMFD \yr 2015 \vol 56 \pages 5--128 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd268} \transl \jour Journal of Mathematical Sciences \yr 2017 \vol 221 \issue 1 \pages 1--136 \crossref{https://doi.org/10.1007/s10958-017-3221-y} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. L. V. Lokutsievskiy, “On an optimal flow in a class of nilpotent convex problems”, Proc. Steklov Inst. Math., 291 (2015), 146–169
2. M. I. Zelikin, “Fractal theory of Saturn's ring”, Proc. Steklov Inst. Math., 291 (2015), 87–101
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