New Advances on the Lyapunov Constants of Some Families of Planar Differential Systems

Abstract

This note presents some advances regarding the Lyapunov constants of some families of planar polynomial differential systems, as a first step towards the resolution of the center and cyclicity problems. Firstly, a parallelization approach is computationally implemented to achieve the 14th Lyapunov constant of the complete cubic family. Secondly, a technique based on interpolating some specific quantities so as to reconstruct the structure of the Lyapunov constants is used to study a Kukles system, some fifth-degree homogeneous systems and a quartic system with two invariant lines.

Local cyclicity in low degree planar piecewise polynomial vector fields

“…Another hurdle is the huge size that the Lyapunov constants have, even for analytic vector fields. In fact, the Lyapunov constants for cubic polynomial systems have been obtained very recently in [37] and they are unknown even for piecewise quadratic polynomial systems. In our context, as it can be seen in Section 2, both the size and the number of constants needed to be calculated are doubled.…”