Work

On the Lyapunov Constants of Planar Piecewise Smooth Systems Separated by an Analytical Curve

Year: 2025

Type: article

Abstract: In this paper, we consider the computations of Lyapunov constants of a class of planar piecewise analytical systems defined in two zones separated by an analytical curve $ y=\phi(x) $ with $ \phi(0)=0 $. Assume that the origin $ (0, 0) $ is a pseudo-focus of the system. We propose an extension of the classical polar coordinates for the subsystem with focus contact, and an extension of the $ (R, \theta, 1, 2) $-generalized polar coordinates for the subsystem with parabolic contact. Then we present the method on how to calculate the relevant Lyapunov constants. As applications, we present three planar piecewise quadratic systems. The first one is of parabolic-parabolic type separated by $ y=\sin^2 x $ which has four limit cycles bifurcated from $ (0, 0) $. The second one is of focus-parabolic type separated by $ y=e^x-1 $ which has five limit cycles bifurcated from $ (0, 0) $. The last one is of focus-focus type separated by $ y=\sin x $ which has five limit cycles bifurcated from $ (0, 0) $.

Source: Journal of Applied Analysis & Computation

DOI: 10.11948/20240503

Authors: Qing Zhang, Zhengdong Du

Institutions: Sichuan University, National Natural Science Foundation of China

Language: English

Topic: Advanced Differential Equations and Dynamical Systems

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: National Natural Science Foundation of China

Award: G2018000577