Piecewise smooth systems with a pseudo-focus: A normal form approach

Abstract

To cope with higher degenerations, some previous results about computable normal forms are extended for two dimensional systems having a pseudo-focus within its discontinuity line. More precisely, it is shown how the standard theory of normal forms allows the analysis of pseudo-focus points when the contact order with the discontinuity boundary, being even for both vector fields, is different from each side. Once obtained the associated canonical forms, the computation of half-return maps is almost direct, and the characterization of the pseudo-focus becomes an easier task.While the methodology is developed in its entire generality, an illustrating example with contacts of order 2 and 4 is studied in detail.