Generic Fractal Structure of Finite Parts of Trajectories of Piecewise Smooth Hamiltonian Systemsстатья
Статья опубликована в журнале из списка RSCI Web of Science
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Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 15 января 2015 г.
Местоположение издательства:Road Town, United Kingdom
Первая страница:25
Последняя страница:32
Аннотация:Piece-wise smooth Hamiltonian systems with tangent discontinuity is explored. It is discovered the new phenomenon: the generic chaotic behavior of finite parts of trajectories. The method consists in consideration of the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It appears 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 Pontryagin Maximum Principle. The corresponded optimization problem is explored and the topological structure of its optimal trajectories (optimal synthesis) is designed. The synthesis contains the countable number of periodic solutions on the quotient space by the scale group, and the Cantor-like set of non-wandering points (NW) having fractal Hausdorff dimension. The dynamic of the system is described by a topological Markov chain. The entropy and estimations for the Hausdorff and box dimension of (NW) is calculated.