## Abstract

If a family of piecewise smooth systems depending on a real parameter is defined on two different regions of the plane separated by a switching surface, then a boundary equilibrium bifurcation occurs if a stationary point of one of the systems intersects the switching surface at a critical value of the parameter. We derive the leading order terms of a normal form for boundary equilibrium bifurcations of planar systems. This makes it straightforward to derive a complete classification of the bifurcations that can occur. We are thus able to confirm classic results of Filippov [Differential Equations with Discontinuous Right Hand Sides (Kluwer, Dordrecht, 1988)] using different and more transparent methods, and explain why the ‘missing’ cases of Hogan et al. [Piecewise Smooth Dynamical Systems: The Case of the Missing Boundary Equilibrium Bifurcations (University of Bristol, 2015)] are the only cases omitted in more recent work.

The dynamics of a piecewise smooth system are determined by different equations according to which regions of phase space the solutions pass through. In each region, the evolution is defined by a smooth dynamical system, but the defining system changes as solutions cross boundaries between regions (switching surfaces). Piecewise smooth systems arise naturally in mechanics, biology, control theory, and electronics. Even so, the conceptual framework for understanding changes in dynamics as parameters is varied, i.e., bifurcation theory, is still being developed. The simple cases for planar systems in which an equilibrium of one system intersects a switching surface at a critical parameter value were described by Filippov in his seminal book.5 More recent accounts have often been incomplete, and Hogan et al. identify two ‘missing’ cases.7 In this paper, a lowest order normal form is derived that makes the systematic classification of cases much easier to analyze and demonstrates that Filippov's list is, indeed, complete. An important codimension two case that arises repeatedly in the analysis is also described.

The dynamics of a piecewise smooth system are determined by different equations according to which regions of phase space the solutions pass through. In each region, the evolution is defined by a smooth dynamical system, but the defining system changes as solutions cross boundaries between regions (switching surfaces). Piecewise smooth systems arise naturally in mechanics, biology, control theory, and electronics. Even so, the conceptual framework for understanding changes in dynamics as parameters is varied, i.e., bifurcation theory, is still being developed. The simple cases for planar systems in which an equilibrium of one system intersects a switching surface at a critical parameter value were described by Filippov in his seminal book.5 More recent accounts have often been incomplete, and Hogan et al. identify two ‘missing’ cases.7 In this paper, a lowest order normal form is derived that makes the systematic classification of cases much easier to analyze and demonstrates that Filippov's list is, indeed, complete. An important codimension two case that arises repeatedly in the analysis is also described.

Original language | Undefined |
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Journal | Chaos: An Interdisciplinary Journal of Nonlinear Science |

DOIs | |

Publication status | Published - 21 Jan 2016 |