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April  2019, 24(4): 1769-1784. doi: 10.3934/dcdsb.2018236

Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center

Jaume Llibre 1, and  Yilei Tang 2,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received  November 2017 Revised  February 2018 Published  April 2019 Early access  August 2018

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line.

We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order $ n $ for $ n = 1, 2, 3, 4, 5 $. Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations.

Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi-homogenous polynomials.

Keywords: Periodic solution, limit cycle, discontinuous piecewise differential system, averaging theory.
Mathematics Subject Classification: 34C29, 34C25, 34C05.
Citation: Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
References:
[1]

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

[2]

V. I. Arnold, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11 (1977), 85-92. 

[3]

V. I. Arnold, Ten problems, Adv. Soviet Math., 1 (1990), 1-8. 

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems. Theory and Applications, Appl. Math. Sci. Series, 163, Springer-Verlag London, Ltd., London, 2008.

[5]

I. S. Berezin and N. P. Zhidkov, Computing Methods, Reading, Mass. -London, 1965.

[6]

D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam., 73 (2013), 1283-1288.  doi: 10.1007/s11071-013-0862-3.

[7]

C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936.  doi: 10.3934/dcds.2013.33.3915.

[8]

X. Chen, J. Llibre and W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3953-3965.  doi: 10.3934/dcdsb.2017203.

[9]

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.

[10]

B. Coll, Qualitative Study of Some Vector Fields in the Plane (Ph. Thesis in Catalan), Universitat Autònoma de Barcelona, 1987.

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[12]

E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728.

[13]

B. García, J. Llibre and J. S. Pérez del Río, Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equation, 255 (2013), 3185-3204.  doi: 10.1016/j.jde.2013.07.032.

[14]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equation, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[15]

S. Huan and X. Yang, The number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[16]

J. Itikawa, J. Llibre and D. D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, Revista Matemática Iberoamericana, 33 (2017), 1247-1265.  doi: 10.4171/RMI/970.

[17]

A. G. Khovansky, Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl., 18 (1984), 40-50. 

[18]

I. D. Iliev, The number of limit cycles due to polynomial perturbations of the harmonic oscillator, Math. Proc. Cambridge Philos. Soc., 127 (1999), 317-322.  doi: 10.1017/S0305004199003795.

[19]

J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. of Computational and Applied Mathematic, 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007.

[20]

J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. of Computational and Applied Mathematics, 287 (2015), 98-114.  doi: 10.1016/j.cam.2015.02.046.

[21]

L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 14pp.

[22]

J. Llibre, A. C. Mereu and D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equation, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.

[23]

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.

[24]

J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B Appl. Algorithms, 19 (2012), 325-335. 

[25]

J. Llibre, E. Ponce and C. Valls, Uniqueness and non-uniqueness of limit cycles of piecewise linear differential systems with three zones and no symmetry, J. Nonlinear Science, 25 (2015), 861-887.  doi: 10.1007/s00332-015-9244-y.

[26]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅰ: Simplest case in $ \mathbb{R} ^2 $, Internat. J. Circuit Theory Appl., 19 (1991), 251-307. 

[27]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅱ: simplest symmetric in $ \mathbb R^2 $, Internat. J. Circuit Theory Appl., 20 (1992), 9-46.  doi: 10.1002/cta.4490200103.

[28]

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.

[29]

G. Sansone and R. Conti, Non-Linear Differential Equations, 2$ ^{nd} $ edition, Pergamon Press, New York, 1964.

[30]

D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity, J. Differential Equations, 215 (2005), 357-400.  doi: 10.1016/j.jde.2004.11.001.

[31]

D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. doi: 10.1142/7612.

[32]

A. N. Varchenko, An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles, Funct. Anal. Appl., 18 (1984), 14-25. 

[33]

L. Wei and X. Zhang, Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete Contin. Dyn. Syst., 36 (2016), 2803-2825.  doi: 10.3934/dcds.2016.36.2803.

[34]

Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.

[35]

Y. Zou, T. Kupper and W. J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Science, 16 (2006), 159-177.  doi: 10.1007/s00332-005-0606-8.

show all references

References:
[1]

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

[2]

V. I. Arnold, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11 (1977), 85-92. 

[3]

V. I. Arnold, Ten problems, Adv. Soviet Math., 1 (1990), 1-8. 

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems. Theory and Applications, Appl. Math. Sci. Series, 163, Springer-Verlag London, Ltd., London, 2008.

[5]

I. S. Berezin and N. P. Zhidkov, Computing Methods, Reading, Mass. -London, 1965.

[6]

D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam., 73 (2013), 1283-1288.  doi: 10.1007/s11071-013-0862-3.

[7]

C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936.  doi: 10.3934/dcds.2013.33.3915.

[8]

X. Chen, J. Llibre and W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3953-3965.  doi: 10.3934/dcdsb.2017203.

[9]

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.

[10]

B. Coll, Qualitative Study of Some Vector Fields in the Plane (Ph. Thesis in Catalan), Universitat Autònoma de Barcelona, 1987.

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[12]

E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728.

[13]

B. García, J. Llibre and J. S. Pérez del Río, Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equation, 255 (2013), 3185-3204.  doi: 10.1016/j.jde.2013.07.032.

[14]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equation, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[15]

S. Huan and X. Yang, The number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[16]

J. Itikawa, J. Llibre and D. D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, Revista Matemática Iberoamericana, 33 (2017), 1247-1265.  doi: 10.4171/RMI/970.

[17]

A. G. Khovansky, Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl., 18 (1984), 40-50. 

[18]

I. D. Iliev, The number of limit cycles due to polynomial perturbations of the harmonic oscillator, Math. Proc. Cambridge Philos. Soc., 127 (1999), 317-322.  doi: 10.1017/S0305004199003795.

[19]

J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. of Computational and Applied Mathematic, 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007.

[20]

J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. of Computational and Applied Mathematics, 287 (2015), 98-114.  doi: 10.1016/j.cam.2015.02.046.

[21]

L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 14pp.

[22]

J. Llibre, A. C. Mereu and D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equation, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.

[23]

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.

[24]

J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B Appl. Algorithms, 19 (2012), 325-335. 

[25]

J. Llibre, E. Ponce and C. Valls, Uniqueness and non-uniqueness of limit cycles of piecewise linear differential systems with three zones and no symmetry, J. Nonlinear Science, 25 (2015), 861-887.  doi: 10.1007/s00332-015-9244-y.

[26]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅰ: Simplest case in $ \mathbb{R} ^2 $, Internat. J. Circuit Theory Appl., 19 (1991), 251-307. 

[27]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅱ: simplest symmetric in $ \mathbb R^2 $, Internat. J. Circuit Theory Appl., 20 (1992), 9-46.  doi: 10.1002/cta.4490200103.

[28]

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.

[29]

G. Sansone and R. Conti, Non-Linear Differential Equations, 2$ ^{nd} $ edition, Pergamon Press, New York, 1964.

[30]

D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity, J. Differential Equations, 215 (2005), 357-400.  doi: 10.1016/j.jde.2004.11.001.

[31]

D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. doi: 10.1142/7612.

[32]

A. N. Varchenko, An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles, Funct. Anal. Appl., 18 (1984), 14-25. 

[33]

L. Wei and X. Zhang, Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete Contin. Dyn. Syst., 36 (2016), 2803-2825.  doi: 10.3934/dcds.2016.36.2803.

[34]

Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.

[35]

Y. Zou, T. Kupper and W. J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Science, 16 (2006), 159-177.  doi: 10.1007/s00332-005-0606-8.

Figure 1.  Existence of closed orbits for system (29)
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Figure 2.  Existence of global center for system (29)
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Table 1.  Maximum number of limit cycles bifurcating from the periodic orbits of the linear center using averaging theory of order $n$
Order $n$ $L_1(n)$ $L_2(n)$ $L_3(n)$ $L_2^I(n)$ $L_3^I(n)$
112301
213512
325813
4361124
5381325
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Order $n$ $L_1(n)$ $L_2(n)$ $L_3(n)$ $L_2^I(n)$ $L_3^I(n)$
112301
213512
325813
4361124
5381325
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