Nilpotent center conditions in cubic switching polynomial Li\'enard systems by higher-order analysis

TL;DR

This paper develops a higher-order Poincaré-Lyapunov method to analyze nilpotent centers and limit cycle bifurcations in cubic switching Liénard systems, providing explicit conditions and establishing a new cyclicity lower bound.

Contribution

It introduces a novel higher-order analysis technique for nilpotent centers in switching polynomial systems and determines explicit center conditions for cubic Liénard systems.

Findings

Explicit nilpotent center conditions derived for switching Liénard systems.

Proved existence of five limit cycles around the nilpotent center.

Established a new lower bound for cyclicity in such systems.

Abstract

The aim of this paper is to investigate two classical problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincar\'e-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincar\'e-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Li\'enard systems. With proper perturbations, explicit center conditions are derived for switching Li\'enard systems at a nilpotent center, which is characterized as global. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Li\'enard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center.