Sharp estimates for the number of limit cycles in discontinuous generalized Liénard equations

2023/07/18 by de Abreu, Tiago M. P., Martins, Ricardo Miranda
#34A36 #34C28 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.09599

Abstract

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations x=y, y=-x-ε ⋅ (f(x)⋅ y +\rm sgn(y)⋅ g(x)). Using the averaging method, we were able to generalize a previous result for Liénard systems. In our generalization, we consider g as a polynomial of degree m. We conclude that for sufficiently small values of |ε|, the number [(n)/(2)]+[(m)/(2)]+1 serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center x=y, y=-x. Furthermore, we demonstrate that it is indeed possible to achieve such a number of limit cycles.

Related

• On the determination of exact number of limit cycles in Lienard Systems
• Limit cycles of a Liénard system with symmetry allowing for discontinuity
• On the number of limit cycles of polynomial Lienard systems
• The number of medium amplitude limit cycles of some generalized Liénard systems
• On the Existence of Exactly N Limit Cycles in Lienard Systems