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doi: 10.3934/dcdss.2022053
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Limit cycle bifurcations of near-Hamiltonian systems with multiple switching curves and applications

1. 

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

2. 

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

* Corresponding author: Maoan Han

Celebrating the 80th Birthday of Professor Jibin Li

Received  December 2021 Early access March 2022

Fund Project: Supported by National Natural Science Foundation of China (No.11931016)

In the present paper, we are devoted to the study of limit cycle bifurcations in piecewise smooth near-Hamiltonian systems with multiple switching curves, obtaining a formula of the first order Melnikov function in general case. As an application, we give lower bounds of the number of limit cycles for a piecewise smooth near-Hamiltonian system with a closed switching curve passing through the origin under piecewise polynomial perturbations.

Citation: Wenye Liu, Maoan Han. Limit cycle bifurcations of near-Hamiltonian systems with multiple switching curves and applications. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022053
References:
[1]

X. Chen and M. Han, Number of limit cycles from a class of perturbed piecewise polynomial systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150123, 29 pp. doi: 10.1142/S0218127421501236.

[2]

M. GrauF. Ma$\tilde{n}$osas 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.

[3]

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.

[4]

M. Han and S. Liu, Further studies on limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters, J. Appl. Anal. Comput., 10 (2020), 816-829.  doi: 10.11948/20200003.

[5]

M. HanV. G. Romanovski and X. Zhang, Equivalence of the melnikov function method and the averaging method, Qual. Theory Dyn. Syst., 15 (2016), 471-479.  doi: 10.1007/s12346-015-0179-3.

[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.

[7]

M. Han and J. Yang, The maximum number of zeros of functions with parameters and application to differential equations, Journal of Nonlinear Modeling and Analysis, 3 (2021), 13-34. 

[8]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[9]

N. Hu and Z. Du, Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3436-3448.  doi: 10.1016/j.cnsns.2013.05.012.

[10]

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

[11]

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.

[12]

S. LiuM. Han and J. Li, Bifurcation methods of periodic orbits for piecewise smooth systems, J. Differential Equations, 275 (2021), 204-233.  doi: 10.1016/j.jde.2020.11.040.

[13]

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.

[14]

O. Ramirez and A. M. Alves, Bifurcation of limit cycles by perturbing piecewise non-Hamiltonian systems with nonlinear switching manifold, Nonlinear Anal. Real World Appl., 57 (2021), 103188, 14 pp. doi: 10.1016/j.nonrwa.2020.103188.

[15]

H. Tian and M. Han, Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5581-5599.  doi: 10.3934/dcdsb.2020368.

[16]

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

[17]

Y. Xiong and M. Han, Limit cycle bifurcations in discontinuous planar systems with multiple lines, J. Appl. Anal. Comput., 10 (2020), 361-377.  doi: 10.11948/20190274.

[18]

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, 14 pp. doi: 10.1142/S0218127417501267.

[19]

J. Yang, Limit cycles appearing from the perturbation of differential systems with multiple switching curves, Chaos Solitons Fractals, 135 (2020), 109764, 11 pp. doi: 10.1016/j.chaos.2020.109764.

show all references

References:
[1]

X. Chen and M. Han, Number of limit cycles from a class of perturbed piecewise polynomial systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150123, 29 pp. doi: 10.1142/S0218127421501236.

[2]

M. GrauF. Ma$\tilde{n}$osas 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.

[3]

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.

[4]

M. Han and S. Liu, Further studies on limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters, J. Appl. Anal. Comput., 10 (2020), 816-829.  doi: 10.11948/20200003.

[5]

M. HanV. G. Romanovski and X. Zhang, Equivalence of the melnikov function method and the averaging method, Qual. Theory Dyn. Syst., 15 (2016), 471-479.  doi: 10.1007/s12346-015-0179-3.

[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.

[7]

M. Han and J. Yang, The maximum number of zeros of functions with parameters and application to differential equations, Journal of Nonlinear Modeling and Analysis, 3 (2021), 13-34. 

[8]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[9]

N. Hu and Z. Du, Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3436-3448.  doi: 10.1016/j.cnsns.2013.05.012.

[10]

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

[11]

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.

[12]

S. LiuM. Han and J. Li, Bifurcation methods of periodic orbits for piecewise smooth systems, J. Differential Equations, 275 (2021), 204-233.  doi: 10.1016/j.jde.2020.11.040.

[13]

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.

[14]

O. Ramirez and A. M. Alves, Bifurcation of limit cycles by perturbing piecewise non-Hamiltonian systems with nonlinear switching manifold, Nonlinear Anal. Real World Appl., 57 (2021), 103188, 14 pp. doi: 10.1016/j.nonrwa.2020.103188.

[15]

H. Tian and M. Han, Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5581-5599.  doi: 10.3934/dcdsb.2020368.

[16]

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

[17]

Y. Xiong and M. Han, Limit cycle bifurcations in discontinuous planar systems with multiple lines, J. Appl. Anal. Comput., 10 (2020), 361-377.  doi: 10.11948/20190274.

[18]

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, 14 pp. doi: 10.1142/S0218127417501267.

[19]

J. Yang, Limit cycles appearing from the perturbation of differential systems with multiple switching curves, Chaos Solitons Fractals, 135 (2020), 109764, 11 pp. doi: 10.1016/j.chaos.2020.109764.

Figure 1.  The orbit $ \widehat{AB_{\varepsilon}A_{\varepsilon}} $ of system (3)
Figure 2.  The orbit $ \widehat{A_{1}A_{2\varepsilon}A_{3\varepsilon}A_{1\varepsilon}} $ of system (18) for $ m = 3 $
Figure 3.  Periodic orbits and switching curve of system (23)$ |_{\varepsilon = 0} $
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