Bifurcation of limit cycles from a Liénard system with asymmetric figure eight-loop case

Hong Li

doi:10.3934/dcdss.2022033

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2023, Volume 16, Issue 3&4: 474-483. Doi: 10.3934/dcdss.2022033

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Bifurcation of limit cycles from a Liénard system with asymmetric figure eight-loop case

Hong Li^,

Department of Mathematics, Jiujiang University, Jiujiang 332005, China

^*Corresponding author: Hong Li
^*Corresponding author: Hong Li

Dedicated to the 80th birthday of Professor Jibin Li

Received: December 2021

Early access: February 2022

Published: March 2023

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Abstract

In this paper, we consider the number of limit cycles of the Li$ \acute{e} $nard system of the form $ \dot{x} = y, \ \dot{y} = -x(x^2+bx-1)+\varepsilon f_{m}(x)y $, where $ b>0 $, $ f_{m}(x) = \sum_{i = 0}^m a_{i}x^{i} $ is a polynomial of $ x $ with degree not greater than $ m $ and $ 0<\varepsilon \ll 1 $. By studying the number of isolated zeros of the corresponding Abelian integral $ I(h) = \oint_{L_{h}}f_{m}(x)ydx, $ we obtain the upper bound of the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for $ \varepsilon = 0 $.

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Mathematics Subject Classification: 34C05, 34C07, 34C08.

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Figure 1. Phase portraits of Eq. (3) on the $ (x, y) $ plane

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