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

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 Poincaré 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.(2.5) Therefore β J = 0 ensures that (x * J (µ), y * J (µ)) is admissible for exactly one sign of µ (i.e. either for µ < 0 or for µ > 0). This paper concerns the case that the equilibria are admissible for different signs of µ, thus β L β R > 0. In this case (1.1) appears to have a single focus that changes stability as the value of µ is varied through 0, much like a Hopf bifurcation. In view

A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems

“…where G(s; ν) = e −νs ̺(s;ν) sin(s) , and ̺ is the auxiliary function (4.22). Theorem 6.2 is proved for piecewise-linear systems in [176] and in general in Appendix B. As shown in [176], there can exist pseudo-equilibria but this only seems to occur over a relatively small fraction of parameter space.…”

“…There exists an admissible unstable focus in ΩL; a visible fold of the left half-system and an invisible fold of the right half-system bound a repelling sliding region. On this region the direction of sliding motion is not indicated because there may be pseudo-equilibria, see [176].…”