October  2021, 26(10): 5581-5599. doi: 10.3934/dcdsb.2020368

Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve

1. 

School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui, 241000, China

a. 

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

b. 

Department of Mathematics Shanghai Normal University Shanghai, 200234, China

* Corresponding author: M. Han

Received  June 2019 Revised  July 2020 Published  October 2021 Early access  December 2020

Fund Project: H. Tian is supported by National Natural Science Foundation of China (No.12001012), Natural Science Foundation of Anhui Province (No. 2008085QA10) and Scientific Research Foundation for Scholars of Anhui Normal University. M. Han is supported by National Natural Science Foundation of China (Nos. 11931016 and 11771296)

This paper deals with the number of limit cycles for planar piecewise smooth near-Hamiltonian or near-integrable systems with a switching curve. The main task is to establish a so-called first order Melnikov function which plays a crucial role in the study of the number of limit cycles bifurcated from a periodic annulus. We use the function to study Hopf bifurcation when the periodic annulus has an elementary center as its boundary. As applications, using the first order Melnikov function, we consider the number of limit cycles bifurcated from the periodic annulus of a linear center under piecewise linear polynomial perturbations with three kinds of quadratic switching curves. And we obtain three limit cycles for each case.

Citation: Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5581-5599. doi: 10.3934/dcdsb.2020368
References:
[1] S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 2001. 
[2]

E. A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff Publishing, Groningen, 1970.

[3]

D. de Carvalho Braga and L. F. Mello, Arbitrary number of limit cycles for planar discontinuous piecewise linear differential systems with two zones, Electron. J. Differential Equations, 2015 (2015), 1-12. 

[4]

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

[5]

J.-P. FrancoiseH. JiD. Xiao and J. Yu, Global dynamics of a piecewise smooth system for brain Lactate metabolism, Qual. Theory Dyn. Syst., 18 (2019), 315-332.  doi: 10.1007/s12346-018-0286-z.

[6]

M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[7]

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.

[8]

M. Han and J. Yang, The Maximum Number of Zeros of Functions with Parameters and Application to Differential Equations, J. Nonlinear Model. Anal., 3 (2021), 13-34.  doi: 10.12150/jnma.2021.13.

[9]

S.-M. Huan and X.-S. 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.

[10]

S. Karlin and W. J. Studden, Tchebycheff Systems: With Application in Analysis and Statistics, Interscience Publisher, 1966.

[11]

V. Křivan, On the Gause predator-prey model with a refuge: a fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.

[12]

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.

[13]

F. LiangM. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.  doi: 10.1016/j.na.2012.03.022.

[14]

F. LiangV. G. Romanovski and D. Zhang, Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non-regular separation line, Chaos Solitons Fractals, 111 (2018), 18-34.  doi: 10.1016/j.chaos.2018.04.002.

[15]

X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.

[16]

Y, Liu, F. Li and P. Dang, Bifurcation analysis in a class of piecewise nonlinear systems with a nonsmooth heteroclinic loop, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850026. doi: 10.1142/S0218127418500268.

[17]

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

[18]

Y. Tian and P. Yu, Center conditions in a switching Bautin system, J. Differential Equations, 259 (2015), 1203-1226.  doi: 10.1016/j.jde.2015.02.044.

[19]

Y. Xiong, M. Han and V. G. Romanovski, The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750126. doi: 10.1142/S0218127417501267.

[20]

C. Zou and J. Yang, Piecewise linear differential system with a center-saddle type singularity, J. Math. Anal. Appl., 459 (2018), 453-463.  doi: 10.1016/j.jmaa.2017.10.043.

show all references

References:
[1] S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 2001. 
[2]

E. A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff Publishing, Groningen, 1970.

[3]

D. de Carvalho Braga and L. F. Mello, Arbitrary number of limit cycles for planar discontinuous piecewise linear differential systems with two zones, Electron. J. Differential Equations, 2015 (2015), 1-12. 

[4]

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

[5]

J.-P. FrancoiseH. JiD. Xiao and J. Yu, Global dynamics of a piecewise smooth system for brain Lactate metabolism, Qual. Theory Dyn. Syst., 18 (2019), 315-332.  doi: 10.1007/s12346-018-0286-z.

[6]

M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[7]

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.

[8]

M. Han and J. Yang, The Maximum Number of Zeros of Functions with Parameters and Application to Differential Equations, J. Nonlinear Model. Anal., 3 (2021), 13-34.  doi: 10.12150/jnma.2021.13.

[9]

S.-M. Huan and X.-S. 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.

[10]

S. Karlin and W. J. Studden, Tchebycheff Systems: With Application in Analysis and Statistics, Interscience Publisher, 1966.

[11]

V. Křivan, On the Gause predator-prey model with a refuge: a fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.

[12]

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.

[13]

F. LiangM. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.  doi: 10.1016/j.na.2012.03.022.

[14]

F. LiangV. G. Romanovski and D. Zhang, Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non-regular separation line, Chaos Solitons Fractals, 111 (2018), 18-34.  doi: 10.1016/j.chaos.2018.04.002.

[15]

X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.

[16]

Y, Liu, F. Li and P. Dang, Bifurcation analysis in a class of piecewise nonlinear systems with a nonsmooth heteroclinic loop, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850026. doi: 10.1142/S0218127418500268.

[17]

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

[18]

Y. Tian and P. Yu, Center conditions in a switching Bautin system, J. Differential Equations, 259 (2015), 1203-1226.  doi: 10.1016/j.jde.2015.02.044.

[19]

Y. Xiong, M. Han and V. G. Romanovski, The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750126. doi: 10.1142/S0218127417501267.

[20]

C. Zou and J. Yang, Piecewise linear differential system with a center-saddle type singularity, J. Math. Anal. Appl., 459 (2018), 453-463.  doi: 10.1016/j.jmaa.2017.10.043.

Figure 1.  The orbit $ \widehat{AA_\epsilon} $ of system (4)
Figure 2.  The orbit $ \widehat{AA_\epsilon} $ of system (31)
Figure 3.  Periodic orbits and switching curve of system (34)$ |_{\epsilon = 0} $
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