Uncountably Many Cases of Filippov's Sewed Focus

The sewed focus is one of the singularities of planar piecewise smooth dynamical systems. Defined by Filippov in his book (Differential Equations with Discontinuous Righthand Sides, Kluwer, 1988), it consists of two invisible tangencies either side of the switching manifold. In the case of analytic focus-like behaviour, Filippov showed that the approach to the singularity is in infinite time. For the case of non-analytic focus-like behaviour, we show that the approach to the singularity can be in finite time. For the non-analytic sewed centre-focus, we show that there are uncountably many different topological types of local dynamics, including cases with infinitely many stable periodic orbits, and show how to create systems with periodic orbits intersecting any bounded symmetric closed set.

Automated summary

The sewed focus is a unique phenomenon in planar piecewise smooth dynamical systems, consisting of two invisible tangencies on either side of the switching manifold. Filippov showed that for analytic focus-like behavior, the approach to the singularity is in infinite time. However, we demonstrate that for non-analytic focus-like behavior, the approach to the singularity can be in finite time. Additionally, we explore the non-analytic sewed centre-focus, uncovering a multitude of different topological types of local dynamics, including cases with infinitely many stable periodic orbits and systems with periodic orbits intersecting any bounded symmetric closed set.