The Poincaré bifurcation of a SD oscillator

Yangjian Sun; Changjian Liu

doi:10.3934/dcdsb.2020173

Article Contents

2021, Volume 26, Issue 3: 1565-1577. Doi: 10.3934/dcdsb.2020173

This issue Previous Article Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts Next Article On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms

The Poincaré bifurcation of a SD oscillator

Yangjian Sun^1,2, and
Changjian Liu^1,2, ,

1.
School of Mathematics, Soochow University, 215006, Suzhou, China
2.
School of Mathematics (Zhuhai), Sun Yat-sen University, 519082, Zhuhai, China

^* Corresponding author
^* Corresponding author

Received: October 31, 2019

Revised: December 31, 2019

Early access: May 2020

Published: March 2021

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Abstract

A van der Pol damped SD oscillator, which was proposed by Ruilan Tian, Qingjie Cao and Shaopu Yang (2010, Nonlinear Dynamics, 59, 19-27), is studied. By improving the criterion function of determining the lowest upper bound of the number of zeros of Abelian Integrals, we show that the number of zeros of Abelian integrals of this SD oscillator is two which is sharp.

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Mathematics Subject Classification: 34C23, 34C25.

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Figure 1. The global phase portraits of system (1.6) for $ 0<a<1 $ and $ \epsilon = 0 $

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Figure 2. The phase portraits of system (2.1)

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