Russian Journal of Mathematical Physics

, Volume 20, Issue 1, pp 25–32 | Cite as

Generic fractal structure of finite parts of trajectories of piecewise smooth Hamiltonian systems

  • R. HildebrandEmail author
  • L. V. Lokutsievskiy
  • M. I. Zelikin


Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).


Hamiltonian System Optimal Control Problem Poisson Bracket Topological Entropy Pontryagin Maximum Principle 
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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • R. Hildebrand
    • 1
    Email author
  • L. V. Lokutsievskiy
    • 2
  • M. I. Zelikin
    • 2
  1. 1.Laboratory Jean KuntzmannUniversity Grenoble 1 / CNRSGrenoble cedex 09France
  2. 2.Dept. of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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