Article Contents
Article Contents

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

• * Corresponding author: Weinian Zhang
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.
• 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$. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function. Mathematics Subject Classification: Primary:37G15;Secondary:37D45.  Citation: • Table 1. The numbers of positive simple zeros of the averaged function$f_i(r)$for$i=1, 2, 3.$$\#Z_+(f_1)$condition for$f_1\equiv 0\#Z_+(f_2)$condition for$f_2\equiv 0\#Z_+(f_3)1C_12C_{11}3C_{12}2C_{13}3$Table 2. The numbers of positive simple zeros of the averaged functions$f_4(r)$and$f_5(r)$ condition for$f_3\equiv 0\#Z_+(f_4)$condition for$f_4\equiv 0\#Z_+(f_5)C_{111}3C_{1111}3C_{1112}4C_{112}3C_{1121}4C_{1122}4C_{113}2C_{1131}4C_{114}4C_{1141}4C_{1142}5C_{121}3C_{1211}3C_{1212}4C_{1213}3C_{122}2C_{1221}2C_{1222}3C_{1223}3C_{123}3C_{1231}3C_{1232}3C_{1233}2C_{131}3C_{1311}3C_{1312}2C_{1313}3C_{132}3C_{1321}3C_{1322}3C_{1323}2C_{133}3C_{1331}3$Table 3. The number of positive simple zeros of the averaged function$f_6(r)$$condition for f_4\equiv 0condition for f_5\equiv 0\#Z_+(f_6)C_{1111}C_{11111}3C_{11112}4C_{1112}C_{11121}4C_{1121}C_{11211}4C_{1122}C_{11221}4C_{1131}C_{11311}4C_{1141}C_{11411}4C_{1211}C_{12111}3C_{12112}3C_{12113}3C_{1212}C_{12121}3C_{12122}3C_{12123}4C_{1213}C_{12131}3C_{12132}4C_{12133}3C_{1221}C_{12211}2C_{12212}3C_{12213}3C_{1222}C_{12221}3C_{12222}3C_{12223}2C_{1223}C_{12231}3C_{12232}4C_{12233}3C_{1231}C_{12311}2C_{12312}3C_{12313}3C_{1232}C_{12321}3C_{12322}3C_{12323}2C_{1233}C_{12331}3C_{1311}C_{13111}3C_{13112}3C_{13113}2C_{13121}C_{13121}3C_{1313}C_{13131}3C_{13132}3C_{13133}2C_{1321}C_{13211}3C_{13212}3C_{13213}2C_{1322}C_{13221}3C_{13222}3C_{13223}2C_{1323}C_{13231}3C_{1331}C_{13311}4$•  [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [2] N. N. 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