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January  2021, 20(1): 55-75. doi: 10.3934/cpaa.2020257

Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters

Meilan Cai 1, and  Maoan Han 2,1,,

1. 

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

* Corresponding author

Received  March 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: This work was supported by National Nature Science Foundation of China (11931016 and 11771296)

In this paper, we concern with the problem of limit cycle bifurcation for a class of piecewise smooth cubic systems. Using the first order Melnikov function we prove that at least thirteen limit cycles can be bifurcated from periodic solutions surrounding the center.

Keywords: Limit cycle bifurcation, piecewise smooth system, multiple parameters, near-Hamiltonian system, Melnikov function.
Mathematics Subject Classification: Primary: 34C05, 34C07; Secondary: 37G15.
Citation: Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
References:
[1]

L. P. da Cruz, D. D. Novaes and J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 226 (2019), 4170-4203.  doi: 10.1016/j.jde.2018.09.032.  Google Scholar

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[3]

M. Grau, F. Ma$\tilde{n}$osas and J. Villadelprat, A Chebyshew criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.  Google Scholar

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L. F. S. Gouveia and J. Torregrosa, 24 crossing limit cycles in only one nest for piecewise cubic systems, Appl. Math. Lett., 103 (2020), 6pp. doi: 10.1016/j.aml.2019.106189.  Google Scholar

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M. Han, G. Chen and C. Sun, On the number of limit cycles in near-Hamiltonian polynomial systems, Int. J. Bifur. Chaos, 17 (2007), 2033-2047.  doi: 10.1142/S0218127407018208.  Google Scholar

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M. Han and L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815.  doi: 10.11948/2015061.  Google Scholar

[7]

M. Han and Y. Xiong, Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters, Chaos Solitons Fractals, 68 (2014), 20-29.  doi: 10.1016/j.chaos.2014.07.005.  Google Scholar

[8]

S. Huan and X. Yang, On 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.  Google Scholar

[9]

S. Karlin and W. J. Studden, Tchebycheff systems: With Applications in Analysis and Statistics, Pure Appa. math., Interscience Publishers, New York, London, Sydney, 1966.  Google Scholar

[10]

F. Liang and M. Han, Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos Solitons Fractals, 45 (2012), 454-464.  doi: 10.1016/j.chaos.2011.09.013.  Google Scholar

[11]

S. Li and T. Huang, Limit cycles for piecewise smooth perturbations of a cubic polynomial differential center, J. Differ. Equ., 2015 (2015), 1-17.   Google Scholar

[12]

X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Int. J. Bifur. Chaos, 5 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.  Google Scholar

[13]

J. Llibre and J. Itikawa, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math., 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007.  Google Scholar

[14]

C. Li, W. Li, J. Llibre and Z. Zhang, Linear estimation of the number of zeros of Abelian integrals for some cubic isochronous centers, J. Differ. Equ., 180 (2002), 307-333.  doi: 10.1006/jdeq.2001.4064.  Google Scholar

[15]

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

[16]

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

[17]

J. Llibre and G. $\acute{S}$wirszcz, On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Ser. A. Math. Anal., 18 (2011), 203-314.   Google Scholar

[18]

S. Li and Y. Zhao, Limit cycles of perturbed cubic isochronous center via the second order averaging method, Int. J. Bifur. Chaos, 24 (2014), 8pp. doi: 10.1142/S0218127414500357.  Google Scholar

[19]

J. N. Mather, Stability of $C^{\infty}$ Mappings: I. The Division Theorem, Ann. Math., 87 (1968), 89-104.  doi: 10.2307/1970595.  Google Scholar

[20]

D. D. Novaes and J. Torregrose, On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl., 448 (2017), 171-186.  doi: 10.1016/j.jmaa.2016.10.076.  Google Scholar

[21]

J. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, 2$^{nd}$ edition, Applied Mathematical Sciences, Springer New York, 2007.  Google Scholar

[22]

Y. Xiong, Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters, J. Math. Anal. Appl., 421 (2015), 260-275. doi: 10.1016/j.jmaa.2014.07.013.  Google Scholar

[23]

Y. Xiong, M. Han and V. G. Romanovski, The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles, Int. J. Bifur. Chaos, 27 (2017), 14pp. doi: 10.1142/S0218127417501267.  Google Scholar

show all references

References:
[1]

L. P. da Cruz, D. D. Novaes and J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 226 (2019), 4170-4203.  doi: 10.1016/j.jde.2018.09.032.  Google Scholar

[2]

J. Gin$\acute{e}$ and J. Llibre, Limit cycles of cubic polynomial vector fields via the averaging theory, Nonlinear Anal., 66 (2007), 1707-1721.  doi: 10.1016/j.na.2006.02.016.  Google Scholar

[3]

M. Grau, F. Ma$\tilde{n}$osas and J. Villadelprat, A Chebyshew criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.  Google Scholar

[4]

L. F. S. Gouveia and J. Torregrosa, 24 crossing limit cycles in only one nest for piecewise cubic systems, Appl. Math. Lett., 103 (2020), 6pp. doi: 10.1016/j.aml.2019.106189.  Google Scholar

[5]

M. Han, G. Chen and C. Sun, On the number of limit cycles in near-Hamiltonian polynomial systems, Int. J. Bifur. Chaos, 17 (2007), 2033-2047.  doi: 10.1142/S0218127407018208.  Google Scholar

[6]

M. Han and L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815.  doi: 10.11948/2015061.  Google Scholar

[7]

M. Han and Y. Xiong, Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters, Chaos Solitons Fractals, 68 (2014), 20-29.  doi: 10.1016/j.chaos.2014.07.005.  Google Scholar

[8]

S. Huan and X. Yang, On 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.  Google Scholar

[9]

S. Karlin and W. J. Studden, Tchebycheff systems: With Applications in Analysis and Statistics, Pure Appa. math., Interscience Publishers, New York, London, Sydney, 1966.  Google Scholar

[10]

F. Liang and M. Han, Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos Solitons Fractals, 45 (2012), 454-464.  doi: 10.1016/j.chaos.2011.09.013.  Google Scholar

[11]

S. Li and T. Huang, Limit cycles for piecewise smooth perturbations of a cubic polynomial differential center, J. Differ. Equ., 2015 (2015), 1-17.   Google Scholar

[12]

X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Int. J. Bifur. Chaos, 5 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.  Google Scholar

[13]

J. Llibre and J. Itikawa, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math., 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007.  Google Scholar

[14]

C. Li, W. Li, J. Llibre and Z. Zhang, Linear estimation of the number of zeros of Abelian integrals for some cubic isochronous centers, J. Differ. Equ., 180 (2002), 307-333.  doi: 10.1006/jdeq.2001.4064.  Google Scholar

[15]

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

[16]

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

[17]

J. Llibre and G. $\acute{S}$wirszcz, On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Ser. A. Math. Anal., 18 (2011), 203-314.   Google Scholar

[18]

S. Li and Y. Zhao, Limit cycles of perturbed cubic isochronous center via the second order averaging method, Int. J. Bifur. Chaos, 24 (2014), 8pp. doi: 10.1142/S0218127414500357.  Google Scholar

[19]

J. N. Mather, Stability of $C^{\infty}$ Mappings: I. The Division Theorem, Ann. Math., 87 (1968), 89-104.  doi: 10.2307/1970595.  Google Scholar

[20]

D. D. Novaes and J. Torregrose, On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl., 448 (2017), 171-186.  doi: 10.1016/j.jmaa.2016.10.076.  Google Scholar

[21]

J. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, 2$^{nd}$ edition, Applied Mathematical Sciences, Springer New York, 2007.  Google Scholar

[22]

Y. Xiong, Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters, J. Math. Anal. Appl., 421 (2015), 260-275. doi: 10.1016/j.jmaa.2014.07.013.  Google Scholar

[23]

Y. Xiong, M. Han and V. G. Romanovski, The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles, Int. J. Bifur. Chaos, 27 (2017), 14pp. doi: 10.1142/S0218127417501267.  Google Scholar

Figure 1.1.  Phase portrait of $ (1.3)|_{\varepsilon = \lambda = 0} $
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Figure 2.1.  The periodic orbit $ L_{\lambda} $ of (2.1)$ |_{\varepsilon = 0} $
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Figure 3.1.  The sketches of functions $ W_{i}(h) $, $ i = 1, 2, \cdots, 8 $
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