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

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.

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 CruzD. 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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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. HanG. 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

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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

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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

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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

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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

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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

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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. LiW. LiJ. 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. LlibreC. 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. LlibreD. 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

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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 CruzD. 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. GrauF. 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. HanG. 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. LiW. LiJ. 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. LlibreC. 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. LlibreD. 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} $
Figure 2.1.  The periodic orbit $ L_{\lambda} $ of (2.1)$ |_{\varepsilon = 0} $
Figure 3.1.  The sketches of functions $ W_{i}(h) $, $ i = 1, 2, \cdots, 8 $
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