Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems

Fei Luo (), Yundong Li and Yi Xiang
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Fei Luo: College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China
Yundong Li: College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China
Yi Xiang: College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China

Mathematics, 2024, vol. 12, issue 5, 1-24

Abstract: In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x -axis. By applying the generalized polar coordinates to the parabolic subsystem and computing the Lyapunov constants, we obtain 11 weak center conditions and 9 weak focus conditions at ( 0 , 0 ) . Under these conditions, we prove that a planar piecewise smooth cubic system with a focus-parabolic-type critical point can bifurcate at least nine limit cycles. So far, our result is a new lower bound of the cyclicity of the piecewise smooth focus-parabolic type cubic system.

Keywords: piecewise smooth cubic system; Lyapunov constants; limit cycles; focus-parabolic-type critical point (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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