Abstract:
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).
This work was supported by the Russian Foundation for Basic Research (grant no. 11-01-00986-a) and by the program
“Mathematical Theory of Control” of the Presidium of the Russian Academy of Sciences.
Received: 03.12.2012
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Language: English
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This publication is cited in the following 4 articles:
M. I. Zelikin, “Fractal theory of Saturn's ring”, Proc. Steklov Inst. Math., 291 (2015), 87–101
M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side”, Journal of Mathematical Sciences, 221:1 (2017), 1–136
L. V. Lokutsievskiy, “On an optimal flow in a class of nilpotent convex problems”, Proc. Steklov Inst. Math., 291 (2015), 146–169
L. V. Lokutsievskii, “The Hamiltonian property of the flow of singular trajectories”, Sb. Math., 205:3 (2014), 432–458