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

Colombo, Alessandro

di Bernardo, M

Hogan, S. J.

Jeffrey, MR

English

A. Colombo

S. J. Hogan

M. R. Jeffrey

DI BERNARDO, MARIO

Archivio della ricerca - Università degli studi di Napoli Federico II

Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems

COLOMBO, ALESSANDRO

M. di Bernardo

Archivio istituzionale della ricerca - Politecnico di Milano

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

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