XXXII Workshop on Geometric Methods in Physics 30.06-6.07.2013
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# Mikhail Zelikin

## STOCHASTIC DYNAMICS OF THE LIE ALGEBRA OF POISSON BRACKETS IN THE VICINITY OF DISCONTINUITY POINTS OF HAMILTONIAN SYSTEMS

The structure of solutions to Hamiltonian systems with continues but non-smooth Hamiltonian $H$ is explored. It is considered solutions passing through a point $x_0$, which belongs to the junction of three domains of smoothness $\Omega_i, \; (i=1,2,3)$ of the Hamiltonian $H$. Let $H_i$ be the restriction of the Hamiltonian $H$ to the smoothness domain $\Omega_i$. The Lie algebra ${\cal L}$ of Poisson brackets with generators $H_i$ is a graded, homogeneous algebra with a scale group ${\goth g}$. The dynamics ${\goth A}$ of ${\cal L}$ along the Hamiltonian system is explored. The system ${\goth A}$ coincides with that of Pontryagin Maximum Principle for a problem $P$ of minimization the mean square deviation from the point $x_0$ of solutions to the system $\ddot x = u$ where $x,u \in \R^2$ and the control $u \subset U$ belongs to the equilateral triangle $U$. We factories this dynamics by the scale group ${\goth g}$. After the resolution of singularity of the Poincar\'e map of the break surface at the point $x_0$ (blow up procedure), one obtains a dynamical system that has a stochastic dynamics defined by the Bernoulli shift on a topological Markov chain. The synthesis of optimal trajectories of the problem $P$ is designed. The set of non-wandering points (NW) has the structure of a Cantor set similar to that of the Smale horseshoe. The Hausdorff dimension and the entropy of NW are calculated.

 Event sponsored by: University of Bialystok

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