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In this paper the theory of bifurcations in piecewise smooth ﬂows is critically surveyed. The focus is on results that hold in arbitrarily (but ﬁnitely) many dimensions, highlighting signiﬁcant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are deﬁned for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classiﬁcation. Codimension-two bifurcations are discussed with suggestions for further study.
Bibliographical noteSponsorship: Preprint document submitted for publication in the journal Physica D