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arXiv · 1506.02320

Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side

Roland Hildebrand·Lev Lokutsievskiy·Mikhail Zelikin
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Abstract

We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely the chaotic behaviour of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighbourhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us the possibility to calculate the entropy and the Hausdorff dimension of the non-wandering set which appears to have a Cantor-like structure as in Smale's Horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.

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Roland Hildebrand, Lev Lokutsievskiy, Mikhail Zelikin. 2015-06-07. Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side. https://doi.org/10.1007/s10958-017-3221-y

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