November  2020, 13(11): 3115-3124. doi: 10.3934/dcdss.2020133

Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory

1. 

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

2. 

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

* Corresponding author: Maoan Han

Received  December 2018 Revised  February 2019 Published  November 2019

Fund Project: The second author is supported by National Natural Science Foundation of China (11771296 and 11431008)

In this paper we study the maximal number of limit cycles for a class of piecewise smooth near-Hamiltonian systems under polynomial perturbations. Using the second order averaging method, we obtain the maximal number of limit cycles of two systems respectively. We also present an application.

Citation: Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133
References:
[1]

R. Benterki and J. Llibre, Periodic solutions of the Duffing differential equation revisited via the averaging theory, Journal of Nonlinear Modeling and Analysis, 1 (2019), 11-26.   Google Scholar

[2]

X. L. CenS. M. Li and Y. L. Zhao, On the number of limit cycles for a class of discontinuous quadratic differential systems, Journal of Mathematical Analysis and Applications, 449 (2017), 314-342.  doi: 10.1016/j.jmaa.2016.11.033.  Google Scholar

[3]

M. A. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing, Alpha Science International Ltd., Oxford, 2017.  Google Scholar

[4]

M. A. Han, On the maximum number of periodic solution of piecewise smooth periodic equations by average method, Journal of Applied Analysis and Computation, 7 (2017), 788-794.   Google Scholar

[5]

M. A. HanG. Chen and C. Sun, On the number of limit cycles in near-Hamiltonian polynomial systems, International Journal of Bifurcation and Chaos, 17 (2007), 2033-2047.  doi: 10.1142/S0218127407018208.  Google Scholar

[6]

M. A. HanV. G. Romanovski and X. Zhang, Equivalence of the Melnikov function method and the averaging method, Qualitative Theory of Dynamical Systems, 15 (2016), 471-479.  doi: 10.1007/s12346-015-0179-3.  Google Scholar

[7]

M. A. Han and L. J. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, Journal of Applied Analysis and Computation, 5 (2015), 809-815.   Google Scholar

[8]

M. A. HanL. J. Sheng and X. Zhang, Bifurcation theory for finitely smooth planar autonomous differential systems, Journal of Differential Equations, 264 (2018), 3596-3618.  doi: 10.1016/j.jde.2017.11.025.  Google Scholar

[9]

M. Han and P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, Springer, New York, 2012. doi: 10.1007/978-1-4471-2918-9.  Google Scholar

[10]

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

[11]

X. Liu and M. A. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, International Journal of Bifurcation and Chaos, 20 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.  Google Scholar

[12]

J. Llibre and A. C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, Journal of Mathematical Analysis and Applications, 413 (2014), 763-775.  doi: 10.1016/j.jmaa.2013.12.031.  Google Scholar

[13]

J. LlibreA. C. Mereu and D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, Journal of Differential Equations, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.  Google Scholar

[14]

J. LlibreD. 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 Nonlinear Phenomena, 353/354 (2017), 1-10.  doi: 10.1016/j.physd.2017.05.003.  Google Scholar

[15]

J. LlibreD. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bulletin Des Sciences Mathématiques, 139 (2015), 229-244.  doi: 10.1016/j.bulsci.2014.08.011.  Google Scholar

[16]

S. Y. Sui and L. Q. Zhao, Bifurcation of limit cycles from the center of a family of cubic polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850063, 11 pp. doi: 10.1142/S0218127418500633.  Google Scholar

[17]

H. H. Tian and M. A. Han, Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems, Journal of Differential Equations, 263 (2017), 7448-7474.  doi: 10.1016/j.jde.2017.08.011.  Google Scholar

[18]

Y. Q. Xiong and M. A. Han, Limit cycle bifurcations in a class of perturbed piecewise smooth systems, Applied Mathematics and Computation, 242 (2014), 47-64.  doi: 10.1016/j.amc.2014.05.035.  Google Scholar

[19]

J. H. Yang and L. Q. Zhao, Limit cycle bifurcations for piecewise smooth intergrable differential systems, Discrete and Continuous Dynamical Systems Serise B, 22 (2017), 2417-2425.  doi: 10.3934/dcdsb.2017123.  Google Scholar

show all references

References:
[1]

R. Benterki and J. Llibre, Periodic solutions of the Duffing differential equation revisited via the averaging theory, Journal of Nonlinear Modeling and Analysis, 1 (2019), 11-26.   Google Scholar

[2]

X. L. CenS. M. Li and Y. L. Zhao, On the number of limit cycles for a class of discontinuous quadratic differential systems, Journal of Mathematical Analysis and Applications, 449 (2017), 314-342.  doi: 10.1016/j.jmaa.2016.11.033.  Google Scholar

[3]

M. A. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing, Alpha Science International Ltd., Oxford, 2017.  Google Scholar

[4]

M. A. Han, On the maximum number of periodic solution of piecewise smooth periodic equations by average method, Journal of Applied Analysis and Computation, 7 (2017), 788-794.   Google Scholar

[5]

M. A. HanG. Chen and C. Sun, On the number of limit cycles in near-Hamiltonian polynomial systems, International Journal of Bifurcation and Chaos, 17 (2007), 2033-2047.  doi: 10.1142/S0218127407018208.  Google Scholar

[6]

M. A. HanV. G. Romanovski and X. Zhang, Equivalence of the Melnikov function method and the averaging method, Qualitative Theory of Dynamical Systems, 15 (2016), 471-479.  doi: 10.1007/s12346-015-0179-3.  Google Scholar

[7]

M. A. Han and L. J. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, Journal of Applied Analysis and Computation, 5 (2015), 809-815.   Google Scholar

[8]

M. A. HanL. J. Sheng and X. Zhang, Bifurcation theory for finitely smooth planar autonomous differential systems, Journal of Differential Equations, 264 (2018), 3596-3618.  doi: 10.1016/j.jde.2017.11.025.  Google Scholar

[9]

M. Han and P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, Springer, New York, 2012. doi: 10.1007/978-1-4471-2918-9.  Google Scholar

[10]

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

[11]

X. Liu and M. A. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, International Journal of Bifurcation and Chaos, 20 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.  Google Scholar

[12]

J. Llibre and A. C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, Journal of Mathematical Analysis and Applications, 413 (2014), 763-775.  doi: 10.1016/j.jmaa.2013.12.031.  Google Scholar

[13]

J. LlibreA. C. Mereu and D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, Journal of Differential Equations, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.  Google Scholar

[14]

J. LlibreD. 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 Nonlinear Phenomena, 353/354 (2017), 1-10.  doi: 10.1016/j.physd.2017.05.003.  Google Scholar

[15]

J. LlibreD. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bulletin Des Sciences Mathématiques, 139 (2015), 229-244.  doi: 10.1016/j.bulsci.2014.08.011.  Google Scholar

[16]

S. Y. Sui and L. Q. Zhao, Bifurcation of limit cycles from the center of a family of cubic polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850063, 11 pp. doi: 10.1142/S0218127418500633.  Google Scholar

[17]

H. H. Tian and M. A. Han, Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems, Journal of Differential Equations, 263 (2017), 7448-7474.  doi: 10.1016/j.jde.2017.08.011.  Google Scholar

[18]

Y. Q. Xiong and M. A. Han, Limit cycle bifurcations in a class of perturbed piecewise smooth systems, Applied Mathematics and Computation, 242 (2014), 47-64.  doi: 10.1016/j.amc.2014.05.035.  Google Scholar

[19]

J. H. Yang and L. Q. Zhao, Limit cycle bifurcations for piecewise smooth intergrable differential systems, Discrete and Continuous Dynamical Systems Serise B, 22 (2017), 2417-2425.  doi: 10.3934/dcdsb.2017123.  Google Scholar

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