On Bykov Cycles in Filippov Systems

Abstract

We introduce two types of Bykov cycles for Filippov systems: crossing Bykov cycles, which retain the main dynamical features of their smooth counterparts, and sliding Bykov cycles, whose dynamics intrinsically involves the switching manifold. We construct explicit piecewise linear examples of both configurations. For crossing Bykov cycles, we prove the existence of hyperbolic invariant sets carrying symbolic dynamics, as in the classical smooth setting. For sliding Bykov cycles, under suitable open conditions, we obtain a countable symbolic dynamics on an invariant set; we prove that this invariant set has several interesting properties: the Hausdorff dimension of it and of its closure are equal, and it lies strictly between zero and one, and so both sets have zero Lebesgue measure; its closure is a Cantor set; and we also obtain conformal and invariant ergodic measures for the associated return dynamics. Finally, we show that convolution regularization of a suitable piecewise constant vector field produces a spatial cusp whose versal unfolding contains classical Bykov cycles. Thus, Bykov-type dynamics arises in the non-smooth setting both intrinsically, through crossing and sliding mechanisms, and through smooth regularization.