Hopf-like boundary equilibrium bifurcations involving two foci in Filippov systems (Q2323824)

The author considers systems of the form $(\dot{x},\dot{y})=F_L(x,y;\mu )$ for $x<0$, $(\dot{x},\dot{y})=F_R(x,y;\mu )$ for $x>0$, where $F_L$ and $F_R$ are smooth vector fields and $\mu$ is a parameter. This is a particular case of two-dimensional Filippov systems, ordinary differential equations that are discontinuous on one-dimensional switching manifolds. If a stable focus becomes an unstable focus by colliding with a switching manifold as parameters are varied, the author proposes a simple sufficient condition for a unique local limit cycle to be created. If this condition fails, then three nested limit cycles might be created simultaneously. In the proof a Poincaré map and generalising analytical arguments applied to continuous systems are used. Necessary and sufficient conditions for the existence of pseudo-equilibria (equilibria of sliding motion on the switching manifold) are also determined. The paper contains examples of piecewise-linear systems.