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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

Xingwu Chen 1, , Jaume Llibre 2, and  Weinian Zhang 3,,

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.

Keywords: Hopf bifurcation, cyclicity, discontinuous differential system, limit cycle.
Mathematics Subject Classification: Primary:37G15;Secondary:37D45.
Citation: Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203
References:
[1]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966.

[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.

[3]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008.

[4]

N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$\check{\rm i}$, 1945.

[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.

[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.

[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.

[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.

[9]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007.

[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.

[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. 

[12]

W. A. Coppel, A survey on quadratic systems, J. Diff. Equa., 2 (1966), 293-304.  doi: 10.1016/0022-0396(66)90070-2.

[13] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006. 
[14]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9.

[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.

[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.

[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.

[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.

[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. 

[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. 

[21]

M. Kunze, Non-Smooth Dynamical Systems Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.

[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.

[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.

[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.

[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.

[26]

P. Patou, Sur le mouvement d'un systéme soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139. 

[27]

D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010

show all references

References:
[1]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966.

[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.

[3]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008.

[4]

N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$\check{\rm i}$, 1945.

[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.

[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.

[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.

[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.

[9]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007.

[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.

[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. 

[12]

W. A. Coppel, A survey on quadratic systems, J. Diff. Equa., 2 (1966), 293-304.  doi: 10.1016/0022-0396(66)90070-2.

[13] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006. 
[14]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9.

[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.

[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.

[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.

[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.

[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. 

[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. 

[21]

M. Kunze, Non-Smooth Dynamical Systems Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.

[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.

[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.

[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.

[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.

[26]

P. Patou, Sur le mouvement d'un systéme soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139. 

[27]

D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010

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

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$\#Z_+(f_1)$condition for $f_1\equiv 0$ $\#Z_+(f_2)$condition for $f_2\equiv 0$ $\#Z_+(f_3)$
$1$ $C_1$ $2$ $C_{11}$ $3$
$C_{12}$ $2$
$C_{13}$ $3$
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_{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$
$C_{122}$ $2$ $C_{1221}$ $2$
$C_{1222}$ $3$
$C_{1223}$ $3$
$C_{123}$ $3$ $C_{1231}$ $3$
$C_{1232}$ $3$
$C_{1233}$ $2$
$C_{131}$ $3$ $C_{1311}$ $3$
$C_{1312}$ $2$
$C_{1313}$ $3$
$C_{132}$ $3$ $C_{1321}$ $3$
$C_{1322}$ $3$
$C_{1323}$ $2$
$C_{133}$ $3$ $C_{1331}$ $3$
Table Options

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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$
$C_{122}$ $2$ $C_{1221}$ $2$
$C_{1222}$ $3$
$C_{1223}$ $3$
$C_{123}$ $3$ $C_{1231}$ $3$
$C_{1232}$ $3$
$C_{1233}$ $2$
$C_{131}$ $3$ $C_{1311}$ $3$
$C_{1312}$ $2$
$C_{1313}$ $3$
$C_{132}$ $3$ $C_{1321}$ $3$
$C_{1322}$ $3$
$C_{1323}$ $2$
$C_{133}$ $3$ $C_{1331}$ $3$
Table 3.  The number of positive simple zeros of the averaged function $f_6(r)$
$condition for f_4\equiv 0$ $condition for f_5\equiv 0$ $\#Z_+(f_6)$
$C_{1111}$ $C_{11111}$ $3$
$C_{11112}$ $4$
$C_{1112}$ $C_{11121}$ $4$
$C_{1121}$ $C_{11211}$ $4$
$C_{1122}$ $C_{11221}$ $4$
$C_{1131}$ $C_{11311}$ $4$
$C_{1141}$ $C_{11411}$ $4$
$C_{1211}$ $C_{12111}$ $3$
$C_{12112}$ $3$
$C_{12113}$ $3$
$C_{1212}$ $C_{12121}$ $3$
$C_{12122}$ $3$
$C_{12123}$ $4$
$C_{1213}$ $C_{12131}$ $3$
$C_{12132}$ $4$
$C_{12133}$ $3$
$C_{1221}$ $C_{12211}$ $2$
$C_{12212}$ $3$
$C_{12213}$ $3$
$C_{1222}$ $C_{12221}$ $3$
$C_{12222}$ $3$
$C_{12223}$ $2$
$C_{1223}$ $C_{12231}$ $3$
$C_{12232}$ $4$
$C_{12233}$ $3$
$C_{1231}$ $C_{12311}$ $2$
$C_{12312}$ $3$
$C_{12313}$ $3$
$C_{1232}$ $C_{12321}$ $3$
$C_{12322}$ $3$
$C_{12323}$ $2$
$C_{1233}$ $C_{12331}$ $3$
$C_{1311}$ $C_{13111}$ $3$
$C_{13112}$ $3$
$C_{13113}$ $2$
$C_{13121}$ $C_{13121}$ $3$
$C_{1313}$ $C_{13131}$ $3$
$C_{13132}$ $3$
$C_{13133}$ $2$
$C_{1321}$ $C_{13211}$ $3$
$C_{13212}$ $3$
$C_{13213}$ $2$
$C_{1322}$ $C_{13221}$ $3$
$C_{13222}$ $3$
$C_{13223}$ $2$
$C_{1323}$ $C_{13231}$ $3$
$C_{1331}$ $C_{13311}$ $4$
Table Options

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