Attractors near grazing–sliding bifurcations

In this paper we prove, for the first time, that multistability can occur in 3-dimensional Fillipov type flows due to grazing-sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing-sliding bifurcation to the study of appropri- ately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing-sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexsist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three orbit with two sliding segments may coexist.

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