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

## TL;DR

This paper analyzes bifurcations in two-dimensional Filippov systems, establishing conditions for the creation of limit cycles and pseudo-equilibria when a focus interacts with a switching manifold, using Poincaré maps and analytical methods.

## Contribution

It provides new sufficient and necessary conditions for limit cycle creation and pseudo-equilibria in piecewise-linear Filippov systems during focus-switching bifurcations.

## Key findings

- A simple sufficient condition for a unique limit cycle creation.
- Three nested limit cycles can form if the condition is violated.
- Necessary and sufficient conditions for pseudo-equilibria are identified.

## 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.