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We consider a linear-quadratic deterministic optimal control problem wherethe control takes values in a two-dimensional simplex. The phase portrait ofthe optimal synthesis contains second-order singular extremals and exhibitsmodes of infinite accumulations of switchings in finite time, so-calledchattering. We prove the presence of an entirely new phenomenon, namely thechaotic behaviour of bounded pieces of optimal trajectories. We find thehyperbolic domains in the neighbourhood of a homoclinic point and estimate thecorresponding contraction-extension coefficients. This gives us the possibilityto calculate the entropy and the Hausdorff dimension of the non-wandering setwhich appears to have a Cantor-like structure as in Smale's Horseshoe. Thedynamics of the system is described by a topological Markov chain. In thesecond part it is shown that this behaviour is generic for piece-wise smoothHamiltonian systems in the vicinity of a junction of three discontinuityhyper-surface strata.

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