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

  • * Corresponding author: Weinian Zhang

    * 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.
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  • 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.


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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 $f_2\equiv 0$ $\#Z_+(f_3)$
    $1$ $C_1$ $2$ $C_{11}$ $3$
    $C_{12}$ $2$
    $C_{13}$ $3$
     | Show Table
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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 $f_4\equiv 0$ $\#Z_+(f_5)$
    $C_{111}$ $3$ $C_{1111}$ $3$
    $C_{112}$ $3$ $C_{1121}$ $4$
    $C_{113}$ $2$ $C_{1131}$ $4$
    $C_{114}$ $4$ $C_{1141}$ $4$
    $C_{1142}$ $5$
    $C_{121}$ $3$ $C_{1211}$ $3$
    $C_{1212}$ $4$
    $C_{122}$ $2$ $C_{1221}$ $2$
    $C_{1222}$ $3$
    $C_{1223}$ $3$
    $C_{123}$ $3$ $C_{1231}$ $3$
    $C_{1232}$ $3$
    $C_{1233}$ $2$
    $C_{131}$ $3$ $C_{1311}$ $3$
    $C_{1312}$ $2$
    $C_{1313}$ $3$
    $C_{132}$ $3$ $C_{1321}$ $3$
    $C_{1322}$ $3$
    $C_{1323}$ $2$
    $C_{133}$ $3$ $C_{1331}$ $3$
     | Show Table
    DownLoad: CSV

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

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