November  2021, 20(11): 3887-3909. doi: 10.3934/cpaa.2021136

Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

1. 

School of Economic Mathematics, Southwestern University of Finance and Economics, 611130 Chengdu, Sichuan, China

2. 

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

* Corresponding author

Received  April 2021 Revised  July 2021 Published  November 2021 Early access  August 2021

Fund Project: The first author is supported by the grant CSC #201906240094 from the P.R. China. The second author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d'Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911

In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center $ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $ with $ m\ge0 $ under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree $ n $ with the discontinuity set $ \{(x, y)\in\mathbb{R}^2: xy = 0\} $. Using the averaging theory up to any order $ N $, we give upper bounds for the maximum number of limit cycles in the function of $ m, n, N $. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.

Citation: Tao Li, Jaume Llibre. Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3887-3909. doi: 10.3934/cpaa.2021136
References:
[1]

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

[2]

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

[3]

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

[4]

A. Buic${\rm\breve{a}}$, J. Giné and J. Llibre, Bifurcation of limit cycles from a polynomial degenerate center, Adv. Nonlinear Stud., 10 (2010), 597–609. doi: 10.1515/ans-2010-0305.

[5]

C. A. Buzzi, M. F. S. Lima and J. Torregrosa, Limit cycles via higher order perturbations for some piecewise differential systems, Physica D, 371 (2018), 28–47. doi: 10.1016/j.physd.2018.01.007.

[6]

C. A. Buzzi, J. C. Medrado and J. Torregrosa, Limit cycles in 4-star-symmetric planar piecewise linear systems, J. Differ. Equ., 268 (2020), 2414–2434. doi: 10.1016/j.jde.2019.09.008.

[7]

C. A. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 9 (2013), 3915–3936. doi: 10.3934/dcds.2013.33.3915.

[8]

P. T. Cardin and J. Torregrosa, Limit cycles in planar piecewise linear differential systems with nonregular separation line, Physica D, 337 (2016), 67–82. doi: 10.1016/j.physd.2016.07.008.

[9]

T. de Carvalho, J. Llibre and D. J. Tonon, Limit cycles of discontinuous piecewise polynomial vector fields, J. Math. Anal. Appl., 449 (2017), 572–579. doi: 10.1016/j.jmaa.2016.11.048.

[10]

G. Dong and C. Liu, Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys., 68 (2017), No. 97. doi: 10.1007/s00033-017-0844-2.

[11]

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

[12]

A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles, Rocky. Mountain J. Math., 31 (2001), 1277–1303. doi: 10.1216/rmjm/1021249441.

[13]

M. Han, On the maximum number of periodic solutions of piecewise smooth periodic equations by average method, J. Appl. Anal. Comput., 7 (2017), 788–794. doi: 10.11948/2017049.

[14]

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.

[15]

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

[16]

Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One parameter bifurcations in planar Filippov systems, Int. J. Bifur. Chaos, 13 (2003), 2157–2188. doi: 10.1142/S0218127403007874.

[17]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos, 13 (2003), 47–106. doi: 10.1142/S0218127403006352.

[18]

T. Li and J. Llibre, Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary, Physica D, 419 (2021), 132855. doi: 10.1016/j.physd.2021.132855.

[19]

T. Li and J. Llibre, On the 16th Hilbert problem for discontinuous piecewise polynomial Hamiltonian systems, J. Dyn. Differ. Equ., (2021) 16pp doi: https://doi.org/10.1007/s10884-021-09967-3.

[20]

A. Lins Neto, W. de Melo and C. C. Pugh, On Liénard equations, in: Proc. Symp. Geom. and topol, in: Lectures Notes in Math., vol. 597, Springer-Verlag, 1977, pp. 335–357.

[21]

S. Liu and M. Han, Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3115–3124. doi: 10.3934/dcdss.2020133.

[22]

J. Llibre, D. D. Novaes and C. A. B. Rodrigues, Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones, Physica D, 353-354 (2017), 1–10. doi: 10.1016/j.physd.2017.05.003.

[23]

J. Llibre and Y. Tang, Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1769–1784. doi: 10.3934/dcdsb.2018236.

[24]

J. Llibre and M. A. Teixeira, Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys., 66 (2015), 51–66. doi: 10.1007/s00033-013-0393-2.

[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]

Y. Wang, M. Han and D. Constantinescu, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos, Solitons and Fractals, 83 (2016), 158–177. doi: 10.1016/j.chaos.2015.11.041.

[27]

L. Wei and X. Zhang, Averaging theory of arbitrary order for piecewise smooth differential systems and its application, J. Dyn. Differ. Equ., 30 (2018), 55–79. doi: 10.1007/s10884-016-9534-6.

[28]

Y. Xiong, Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters, Nonlin. Anal. Real World Appl., 41 (2018) 384–400. doi: 10.1016/j.nonrwa.2017.10.020.

[29]

J. Yang, M. Han and W. Huang, On Hopf bifurcations of piecewise Hamiltonian systems, J. Differ. Equ., 250 (2011), 1026–1051. doi: 10.1016/j.jde.2010.06.012.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

A. Buic${\rm\breve{a}}$, J. Giné and J. Llibre, Bifurcation of limit cycles from a polynomial degenerate center, Adv. Nonlinear Stud., 10 (2010), 597–609. doi: 10.1515/ans-2010-0305.

[5]

C. A. Buzzi, M. F. S. Lima and J. Torregrosa, Limit cycles via higher order perturbations for some piecewise differential systems, Physica D, 371 (2018), 28–47. doi: 10.1016/j.physd.2018.01.007.

[6]

C. A. Buzzi, J. C. Medrado and J. Torregrosa, Limit cycles in 4-star-symmetric planar piecewise linear systems, J. Differ. Equ., 268 (2020), 2414–2434. doi: 10.1016/j.jde.2019.09.008.

[7]

C. A. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 9 (2013), 3915–3936. doi: 10.3934/dcds.2013.33.3915.

[8]

P. T. Cardin and J. Torregrosa, Limit cycles in planar piecewise linear differential systems with nonregular separation line, Physica D, 337 (2016), 67–82. doi: 10.1016/j.physd.2016.07.008.

[9]

T. de Carvalho, J. Llibre and D. J. Tonon, Limit cycles of discontinuous piecewise polynomial vector fields, J. Math. Anal. Appl., 449 (2017), 572–579. doi: 10.1016/j.jmaa.2016.11.048.

[10]

G. Dong and C. Liu, Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys., 68 (2017), No. 97. doi: 10.1007/s00033-017-0844-2.

[11]

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

[12]

A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles, Rocky. Mountain J. Math., 31 (2001), 1277–1303. doi: 10.1216/rmjm/1021249441.

[13]

M. Han, On the maximum number of periodic solutions of piecewise smooth periodic equations by average method, J. Appl. Anal. Comput., 7 (2017), 788–794. doi: 10.11948/2017049.

[14]

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.

[15]

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

[16]

Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One parameter bifurcations in planar Filippov systems, Int. J. Bifur. Chaos, 13 (2003), 2157–2188. doi: 10.1142/S0218127403007874.

[17]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos, 13 (2003), 47–106. doi: 10.1142/S0218127403006352.

[18]

T. Li and J. Llibre, Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary, Physica D, 419 (2021), 132855. doi: 10.1016/j.physd.2021.132855.

[19]

T. Li and J. Llibre, On the 16th Hilbert problem for discontinuous piecewise polynomial Hamiltonian systems, J. Dyn. Differ. Equ., (2021) 16pp doi: https://doi.org/10.1007/s10884-021-09967-3.

[20]

A. Lins Neto, W. de Melo and C. C. Pugh, On Liénard equations, in: Proc. Symp. Geom. and topol, in: Lectures Notes in Math., vol. 597, Springer-Verlag, 1977, pp. 335–357.

[21]

S. Liu and M. Han, Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3115–3124. doi: 10.3934/dcdss.2020133.

[22]

J. Llibre, D. D. Novaes and C. A. B. Rodrigues, Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones, Physica D, 353-354 (2017), 1–10. doi: 10.1016/j.physd.2017.05.003.

[23]

J. Llibre and Y. Tang, Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1769–1784. doi: 10.3934/dcdsb.2018236.

[24]

J. Llibre and M. A. Teixeira, Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys., 66 (2015), 51–66. doi: 10.1007/s00033-013-0393-2.

[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]

Y. Wang, M. Han and D. Constantinescu, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos, Solitons and Fractals, 83 (2016), 158–177. doi: 10.1016/j.chaos.2015.11.041.

[27]

L. Wei and X. Zhang, Averaging theory of arbitrary order for piecewise smooth differential systems and its application, J. Dyn. Differ. Equ., 30 (2018), 55–79. doi: 10.1007/s10884-016-9534-6.

[28]

Y. Xiong, Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters, Nonlin. Anal. Real World Appl., 41 (2018) 384–400. doi: 10.1016/j.nonrwa.2017.10.020.

[29]

J. Yang, M. Han and W. Huang, On Hopf bifurcations of piecewise Hamiltonian systems, J. Differ. Equ., 250 (2011), 1026–1051. doi: 10.1016/j.jde.2010.06.012.

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