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Journal of Mathematical Sciences

, Volume 221, Issue 1, pp 1–136 | Cite as

Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side

  • M. I. ZelikinEmail author
  • L. V. Lokutsievskii
  • R. Hildebrand
Article

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.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • M. I. Zelikin
    • 1
    Email author
  • L. V. Lokutsievskii
    • 1
  • R. Hildebrand
    • 2
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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