Complex normal forms for planar double boundary focus points

Abstract

We consider planar piecewise smooth systems constituted by two vector fields with a straight line as separation boundary between them. It is assumed that the origin, which belongs to the boundary, is an isolated equilibrium of center-focus type for both vector fields. Working in the complex setting, firstly we obtain a general normal form with only one term for each degree. Next, we exploit such a normal form, which turns to be very suitable for computing the Lyapunov constants that characterize the cyclicity of the origin. To illustrate the usefulness of the approach, some significative examples regarding piecewise quadratic Liénard systems are considered. In particular, we show how a piecewise quadratic system with an attractive weak focus from both sides can give rise to a repulsive weak focus.