Chains in 3D Filippov systems: A chaotic phenomenon

Abstract

This work is devoted to the study of global connections between typical generic singularities, named T-singularities, in piecewise smooth dynamical systems. Such a singularity presents the so-called nonsmooth diabolo, which consists on a pair of invariant cones emanating from it.We analyze global features arising from the communication between the branches of a nonsmooth diabolo of a T-singularity and we prove that, under generic conditions, such communication leads to a chaotic behavior of the system. More specifically, we relate crossing orbits of a Filippov system presenting certain crossing self-connections to a T-singularity, with a Smale horseshoe of a first return map associated to the system. The techniques used in this work rely on the detection of transverse intersections between invariant manifolds of a hyperbolic fixed point of saddle type of such a first return map and the analysis of the Smale horseshoe associated to it.From the specific case discussed in our approach, we present a robust chaotic phenomenon for which its counterpart in the smooth case seems to happen only for highly degenerate systems.