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.

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		University of Bialystok