# American Institute of Mathematical Sciences

December  2017, 22(10): 3953-3965. doi: 10.3934/dcdsb.2017203

## 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

Received  November 2016 Revised  June 2017 Published  August 2017

Fund Project: 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.

Citation: Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203
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##### References:
 [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. Google Scholar [2] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Math. Sbornik, 30 (1952), 181-196 (in Russian); Transl. Amer. Math. Soc. , 1954 (1954), 19pp. Google Scholar [3] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008.  Google Scholar [4] N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$\check{\rm i}$, 1945.  Google Scholar [5] N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934. Google Scholar [6] X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848.  doi: 10.1016/j.camwa.2010.04.019.  Google Scholar [7] X. Chen, V. G. Romanovski and W. Zhang, Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl., 432 (2015), 1058-1076.  doi: 10.1016/j.jmaa.2015.07.036.  Google Scholar [8] X. Chen and W. Zhang, Normal forms of planar switching systems, Disc. Cont. Dyn. Syst., 36 (2016), 6715-6736.  doi: 10.3934/dcds.2016092.  Google Scholar [9] C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007.  Google Scholar [10] B. Coll, A. Gasull and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690.  doi: 10.1006/jmaa.2000.7188.  Google Scholar [11] B. Coll, A. Gasull and R. Prohens, Bifurcation of limit cycles from two families of centers, Dyn. Contin. Disc. Impu. Syst., 12 (2005), 275-287.   Google Scholar [12] W. A. Coppel, A survey on quadratic systems, J. Diff. Equa., 2 (1966), 293-304.  doi: 10.1016/0022-0396(66)90070-2.  Google Scholar [13] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006.   Google Scholar [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar [15] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifurc. Chaos, 13 (2003), 1755-1765.  doi: 10.1142/S0218127403007618.  Google Scholar [16] J. Giné, M. Grau and J. Llibre, Averaging theory at any order for computing periodic orbits, Physica D, 250 (2013), 58-65.  doi: 10.1016/j.physd.2013.01.015.  Google Scholar [17] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Diff. Equa., 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.  Google Scholar [18] Yu. Ilyashenko, Centennial history of Hilbert's $16$th problem, Bull. (New Series) Amer. Math. Soc., 39 (2002), 301-354.  doi: 10.1090/S0273-0979-02-00946-1.  Google Scholar [19] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 19 (1911), 1446-1457.   Google Scholar [20] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 20 (1912), 1354-1365.   Google Scholar [21] M. Kunze, Non-Smooth Dynamical Systems Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.  Google Scholar [22] J. Li, Hilbert's $16$th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.  doi: 10.1142/S0218127403006352.  Google Scholar [23] J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583.  doi: 10.1088/0951-7715/27/3/563.  Google Scholar [24] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. math., 139 (2015), 229-244.  doi: 10.1016/j.bulsci.2014.08.011.  Google Scholar [25] O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D, 241 (2012), 1826-1844.  doi: 10.1016/j.physd.2012.08.002.  Google Scholar [26] P. Patou, Sur le mouvement d'un systéme soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139.   Google Scholar [27] D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010 Google Scholar
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 $f_2\equiv 0$ $\#Z_+(f_3)$ $1$ $C_1$ $2$ $C_{11}$ $3$ $C_{12}$ $2$ $C_{13}$ $3$
 $\#Z_+(f_1)$ condition for $f_1\equiv 0$ $\#Z_+(f_2)$ condition for $f_2\equiv 0$ $\#Z_+(f_3)$ $1$ $C_1$ $2$ $C_{11}$ $3$ $C_{12}$ $2$ $C_{13}$ $3$
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 $f_4\equiv 0$ $\#Z_+(f_5)$ $C_{111}$ $3$ $C_{1111}$ $3$ $C_{1112}$ $4$ $C_{112}$ $3$ $C_{1121}$ $4$ $C_{1122}$ $4$ $C_{113}$ $2$ $C_{1131}$ $4$ $C_{114}$ $4$ $C_{1141}$ $4$ $C_{1142}$ $5$ $C_{121}$ $3$ $C_{1211}$ $3$ $C_{1212}$ $4$ $C_{1213}$ $3$ $C_{122}$ $2$ $C_{1221}$ $2$ $C_{1222}$ $3$ $C_{1223}$ $3$ $C_{123}$ $3$ $C_{1231}$ $3$ $C_{1232}$ $3$ $C_{1233}$ $2$ $C_{131}$ $3$ $C_{1311}$ $3$ $C_{1312}$ $2$ $C_{1313}$ $3$ $C_{132}$ $3$ $C_{1321}$ $3$ $C_{1322}$ $3$ $C_{1323}$ $2$ $C_{133}$ $3$ $C_{1331}$ $3$
 condition for $f_3\equiv 0$ $\#Z_+(f_4)$ condition for $f_4\equiv 0$ $\#Z_+(f_5)$ $C_{111}$ $3$ $C_{1111}$ $3$ $C_{1112}$ $4$ $C_{112}$ $3$ $C_{1121}$ $4$ $C_{1122}$ $4$ $C_{113}$ $2$ $C_{1131}$ $4$ $C_{114}$ $4$ $C_{1141}$ $4$ $C_{1142}$ $5$ $C_{121}$ $3$ $C_{1211}$ $3$ $C_{1212}$ $4$ $C_{1213}$ $3$ $C_{122}$ $2$ $C_{1221}$ $2$ $C_{1222}$ $3$ $C_{1223}$ $3$ $C_{123}$ $3$ $C_{1231}$ $3$ $C_{1232}$ $3$ $C_{1233}$ $2$ $C_{131}$ $3$ $C_{1311}$ $3$ $C_{1312}$ $2$ $C_{1313}$ $3$ $C_{132}$ $3$ $C_{1321}$ $3$ $C_{1322}$ $3$ $C_{1323}$ $2$ $C_{133}$ $3$ $C_{1331}$ $3$
The number of positive simple zeros of the averaged function $f_6(r)$
 $condition for f_4\equiv 0$ $condition for f_5\equiv 0$ $\#Z_+(f_6)$ $C_{1111}$ $C_{11111}$ $3$ $C_{11112}$ $4$ $C_{1112}$ $C_{11121}$ $4$ $C_{1121}$ $C_{11211}$ $4$ $C_{1122}$ $C_{11221}$ $4$ $C_{1131}$ $C_{11311}$ $4$ $C_{1141}$ $C_{11411}$ $4$ $C_{1211}$ $C_{12111}$ $3$ $C_{12112}$ $3$ $C_{12113}$ $3$ $C_{1212}$ $C_{12121}$ $3$ $C_{12122}$ $3$ $C_{12123}$ $4$ $C_{1213}$ $C_{12131}$ $3$ $C_{12132}$ $4$ $C_{12133}$ $3$ $C_{1221}$ $C_{12211}$ $2$ $C_{12212}$ $3$ $C_{12213}$ $3$ $C_{1222}$ $C_{12221}$ $3$ $C_{12222}$ $3$ $C_{12223}$ $2$ $C_{1223}$ $C_{12231}$ $3$ $C_{12232}$ $4$ $C_{12233}$ $3$ $C_{1231}$ $C_{12311}$ $2$ $C_{12312}$ $3$ $C_{12313}$ $3$ $C_{1232}$ $C_{12321}$ $3$ $C_{12322}$ $3$ $C_{12323}$ $2$ $C_{1233}$ $C_{12331}$ $3$ $C_{1311}$ $C_{13111}$ $3$ $C_{13112}$ $3$ $C_{13113}$ $2$ $C_{13121}$ $C_{13121}$ $3$ $C_{1313}$ $C_{13131}$ $3$ $C_{13132}$ $3$ $C_{13133}$ $2$ $C_{1321}$ $C_{13211}$ $3$ $C_{13212}$ $3$ $C_{13213}$ $2$ $C_{1322}$ $C_{13221}$ $3$ $C_{13222}$ $3$ $C_{13223}$ $2$ $C_{1323}$ $C_{13231}$ $3$ $C_{1331}$ $C_{13311}$ $4$
 $condition for f_4\equiv 0$ $condition for f_5\equiv 0$ $\#Z_+(f_6)$ $C_{1111}$ $C_{11111}$ $3$ $C_{11112}$ $4$ $C_{1112}$ $C_{11121}$ $4$ $C_{1121}$ $C_{11211}$ $4$ $C_{1122}$ $C_{11221}$ $4$ $C_{1131}$ $C_{11311}$ $4$ $C_{1141}$ $C_{11411}$ $4$ $C_{1211}$ $C_{12111}$ $3$ $C_{12112}$ $3$ $C_{12113}$ $3$ $C_{1212}$ $C_{12121}$ $3$ $C_{12122}$ $3$ $C_{12123}$ $4$ $C_{1213}$ $C_{12131}$ $3$ $C_{12132}$ $4$ $C_{12133}$ $3$ $C_{1221}$ $C_{12211}$ $2$ $C_{12212}$ $3$ $C_{12213}$ $3$ $C_{1222}$ $C_{12221}$ $3$ $C_{12222}$ $3$ $C_{12223}$ $2$ $C_{1223}$ $C_{12231}$ $3$ $C_{12232}$ $4$ $C_{12233}$ $3$ $C_{1231}$ $C_{12311}$ $2$ $C_{12312}$ $3$ $C_{12313}$ $3$ $C_{1232}$ $C_{12321}$ $3$ $C_{12322}$ $3$ $C_{12323}$ $2$ $C_{1233}$ $C_{12331}$ $3$ $C_{1311}$ $C_{13111}$ $3$ $C_{13112}$ $3$ $C_{13113}$ $2$ $C_{13121}$ $C_{13121}$ $3$ $C_{1313}$ $C_{13131}$ $3$ $C_{13132}$ $3$ $C_{13133}$ $2$ $C_{1321}$ $C_{13211}$ $3$ $C_{13212}$ $3$ $C_{13213}$ $2$ $C_{1322}$ $C_{13221}$ $3$ $C_{13222}$ $3$ $C_{13223}$ $2$ $C_{1323}$ $C_{13231}$ $3$ $C_{1331}$ $C_{13311}$ $4$
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