Russ. J. Math. Phys., 2013, Volume 20, Issue 1, Pages 25–32
Generic fractal structure of finite parts of trajectories of piecewise smooth hamiltonian systems
R. Hildebranda, L. V. Lokoutsievskiyb, M. I. Zelikinb
a Laboratory Jean Kuntzmann, University Grenoble 1 / CNRS, 51
rue des Mathématiques, BP 53, 38041 Grenoble cedex 09, France
b Dept. of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russia
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).
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