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

This article is cited in 2 scientific papers (total in 2 papers)

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
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  crossref  isi  elib  elib
    2. M. I. Zelikin, “Fractal theory of Saturn's ring”, Proc. Steklov Inst. Math., 291 (2015), 87–101  mathnet  crossref  crossref  isi  elib
  • Современная математика. Фундаментальные направления
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