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arXiv · 1812.03587

Hopf-like boundary equilibrium bifurcations involving two foci in Filippov systems

David J. W. Simpson
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Abstract

This paper concerns two-dimensional Filippov systems --- ordinary differential equations that are discontinuous on one-dimensional switching manifolds. In the situation that a stable focus transitions to an unstable focus by colliding with a switching manifold as parameters are varied, a simple sufficient condition for a unique local limit cycle to be created is established. If this condition is violated, three nested limit cycles may be created simultaneously. The result is achieved by constructing a Poincar\'e map and generalising analytical arguments that have been employed for continuous systems. Necessary and sufficient conditions for the existence of pseudo-equilibria (equilibria of sliding motion on the switching manifold) are also determined. For simplicity only piecewise-linear systems are considered.

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David J. W. Simpson. 2018-12-10. Hopf-like boundary equilibrium bifurcations involving two foci in Filippov systems. https://arxiv.org/abs/1812.03587

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