Piecewise Smooth Dynamical Systems Theory: The Case of the Missing Boundary Equilibrium Bifurcations

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• 作者：S. J. Hogan ; M. E. Homer ; M. R. Jeffrey ; R. Szalai
• 关键词：Dynamical systems ; Piecewise smooth ; Bifurcation ; Boundary equilibrium collision
• 刊名：Journal of Nonlinear Science
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：26
• 期：5
• 页码：1161-1173
• 全文大小：423 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467
• 卷排序：26

文摘

We present two codimension-one bifurcations that occur when an equilibrium collides with a discontinuity in a piecewise smooth dynamical system. These simple cases appear to have escaped recent classifications. We present them here to highlight some of the powerful results from Filippov’s book Differential Equations with Discontinuous Righthand Sides (Kluwer, 1988). Filippov classified the so-called boundary equilibrium collisions without providing their unfolding. We show the complete unfolding here, for the first time, in the particularly interesting case of a node changing its stability as it collides with a discontinuity. We provide a prototypical model that can be used to generate all codimension-one boundary equilibrium collisions, and summarize the elements of Filippov’s work that are important in achieving a full classification.