Bi-center conditions and bifurcation of limit cycles in a class of Z₂-equivariant cubic switching systems with two nilpotent points

2024/03/09 by Ting Chen, Feng Li, Chen, Ting +5
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Quantum chaos and dynamical systems #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.2403.05744

Abstract

In this paper, we generalize the Poincaré-Lyapunov method for systems with linear type centers to study nilpotent centers in switching polynomial systems and use it to investigate the bi-center problem of planar Z₂-equivariant cubic switching systems associated with two symmetric nilpotent singular points. With a properly designed perturbation, 6 explicit bi-center conditions for such polynomial systems are derived. Then, based on the 6 center conditions, by using Bogdanov-Takens bifurcation theory with general perturbations, we prove that there exist at least 20 small-amplitude limit cycles around the nilpotent bi-center for a class of Z₂-equivariant cubic switching systems. This is a new lower bound of cyclicity for such cubic polynomial systems, increased from 12 to 20.

Related

• Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysis
• Limit cycles bifurcating from the quasi-homogeneous polynomial centers of weight-degree 2 under non-smooth perturbations
• Centers and limit cycles of a generalized cubic Riccati system
• Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton system under perturbations of piecewise smooth polynomials
• Limit cycles bifurcating from a degenerate center