Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems

Xingwu Chen; Jaume Llibre; Weinian Zhang

doi:10.3934/dcdsb.2017203

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2017, Volume 22, Issue 10: 3953-3965. Doi: 10.3934/dcdsb.2017203

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Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.
Department de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
3.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Weinian Zhang
* Corresponding author: Weinian Zhang

Received: October 31, 2016

Revised: May 31, 2017

Published: September 2017

The first author is supported by NSFC grant #11471228. The second author is partially supported by the MINECO grants MTM2016-77278-P and MTM2013-40998-P, an AGAUR grant number 2014SGR-568, the grant FP7-PEOPLE-2012-IRSES 318999, and from the recruitment program of high-end foreign experts of China. The third author is supported by NSFC grants #11231001, #11221101.

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Abstract

It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in $\mathbb{R}^2$ is $. In contrast here we consider discontinuous differential systems in $\mathbb{R}^2$ defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of $\mathbb{R}^2$, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of $\mathbb{R}^2$. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.

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Mathematics Subject Classification: Primary:37G15;Secondary:37D45.

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Table 1. The numbers of positive simple zeros of the averaged function $f_i(r)$ for $i=1, 2, 3.$

$\#Z_+(f_1)$	condition for $f_1\equiv 0$	$\#Z_+(f_2)$	condition for $f_2\equiv 0$	$\#Z_+(f_3)$
$1$	$C_1$	$2$	$C_{11}$	$3$
			$C_{12}$	$2$
			$C_{13}$	$3$

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Table 2. The numbers of positive simple zeros of the averaged functions $f_4(r)$ and $f_5(r)$

condition for $f_3\equiv 0$	$\#Z_+(f_4)$	condition for $f_4\equiv 0$	$\#Z_+(f_5)$
$C_{111}$	$3$	$C_{1111}$	$3$
$C_{111}$	$3$	$C_{1112}$	$4$
$C_{112}$	$3$	$C_{1121}$	$4$
$C_{112}$	$3$	$C_{1122}$	$4$
$C_{113}$	$2$	$C_{1131}$	$4$
$C_{114}$	$4$	$C_{1141}$	$4$
$C_{114}$	$4$	$C_{1142}$	$5$
$C_{121}$	$3$	$C_{1211}$	$3$
		$C_{1212}$	$4$
		$C_{1213}$	$3$
$C_{122}$	$2$	$C_{1221}$	$2$
		$C_{1222}$	$3$
		$C_{1223}$	$3$
$C_{123}$	$3$	$C_{1231}$	$3$
		$C_{1232}$	$3$
		$C_{1233}$	$2$
$C_{131}$	$3$	$C_{1311}$	$3$
		$C_{1312}$	$2$
		$C_{1313}$	$3$
$C_{132}$	$3$	$C_{1321}$	$3$
		$C_{1322}$	$3$
		$C_{1323}$	$2$
$C_{133}$	$3$	$C_{1331}$	$3$

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Table 3. The number of positive simple zeros of the averaged function $f_6(r)$

$condition for f_4\equiv 0$	$condition for f_5\equiv 0$	$\#Z_+(f_6)$
$C_{1111}$	$C_{11111}$	$3$
$C_{1111}$	$C_{11112}$	$4$
$C_{1112}$	$C_{11121}$	$4$
$C_{1121}$	$C_{11211}$	$4$
$C_{1122}$	$C_{11221}$	$4$
$C_{1131}$	$C_{11311}$	$4$
$C_{1141}$	$C_{11411}$	$4$
$C_{1211}$	$C_{12111}$	$3$
	$C_{12112}$	$3$
	$C_{12113}$	$3$
$C_{1212}$	$C_{12121}$	$3$
	$C_{12122}$	$3$
	$C_{12123}$	$4$
$C_{1213}$	$C_{12131}$	$3$
	$C_{12132}$	$4$
	$C_{12133}$	$3$
$C_{1221}$	$C_{12211}$	$2$
	$C_{12212}$	$3$
	$C_{12213}$	$3$
$C_{1222}$	$C_{12221}$	$3$
	$C_{12222}$	$3$
	$C_{12223}$	$2$
$C_{1223}$	$C_{12231}$	$3$
	$C_{12232}$	$4$
	$C_{12233}$	$3$
$C_{1231}$	$C_{12311}$	$2$
	$C_{12312}$	$3$
	$C_{12313}$	$3$
$C_{1232}$	$C_{12321}$	$3$
	$C_{12322}$	$3$
	$C_{12323}$	$2$
$C_{1233}$	$C_{12331}$	$3$
$C_{1311}$	$C_{13111}$	$3$
	$C_{13112}$	$3$
	$C_{13113}$	$2$
$C_{13121}$	$C_{13121}$	$3$
$C_{1313}$	$C_{13131}$	$3$
	$C_{13132}$	$3$
	$C_{13133}$	$2$
$C_{1321}$	$C_{13211}$	$3$
	$C_{13212}$	$3$
	$C_{13213}$	$2$
$C_{1322}$	$C_{13221}$	$3$
	$C_{13222}$	$3$
	$C_{13223}$	$2$
$C_{1323}$	$C_{13231}$	$3$
$C_{1331}$	$C_{13311}$	$4$

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