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

TL;DR

This paper demonstrates the chaotic behavior of optimal trajectories in a specific control problem with discontinuities, revealing complex dynamics like fractal structures and hyperbolic domains near homoclinic points.

Contribution

It introduces the discovery of chaos in bounded optimal trajectories of a control system with discontinuous Hamiltonian dynamics, including entropy and fractal dimension calculations.

Findings

Chaotic behavior observed in bounded optimal trajectories.

Calculation of entropy and Hausdorff dimension of the non-wandering set.

Chaotic dynamics are shown to be generic near junctions of discontinuity hyper-surfaces.

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.