doi: 10.3934/cpaa.2022090
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Limit cycle bifurcations by perturbing piecewise Hamiltonian systems with a nonregular switching line via multiple parameters

School of Mathematical Sciences, Qufu Normal University, Qufu, Jining, 273165, China

*Corresponding author

Received  September 2021 Revised  December 2021 Early access May 2022

Fund Project: This work was supported by Natural Science Foundation of Shandong Province of China(ZR2019MA067)

In this paper, we investigate limit cycle bifurcations by perturbing planar piecewise Hamiltonian systems with a switching line $ \left\{(x,y): y = \pm kx, k\right. $ $ \left.\in(0,+\infty), x\geqslant0\right\} $ via multiple parameters. With the help of Han and Xiong [3], Han and Liu [5] and Xiong [18], we obtain the second and third terms in expansions of the first order Melnikov function. As an application, we consider limit cycle bifurcations of a piecewise near-Hamiltonian system and prove that the system has four limit cycles.

Citation: Yuan Chang, Yuzhen Bai. Limit cycle bifurcations by perturbing piecewise Hamiltonian systems with a nonregular switching line via multiple parameters. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022090
References:
[1]

C. Christopher and C. Li, Limit cycles of differential equations, Springer Science & Business Media, 2007. doi: 10.1007/978-3-7643-8410-4.

[2]

L. P. C. D. CruzD. D. Novaes and J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 266 (2019), 4170-4203.  doi: 10.1016/j.jde.2018.09.032.

[3]

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.

[4]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250296, 30 pp. doi: 10.1142/S0218127412502963.

[5]

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.

[6]

M. Han and V. G. Romanovski, On the number of limit cycles of polynomial Liénard systems, Nonlinear Anal. Real World Appl., 14 (2013), 1655-1668.  doi: 10.1016/j.nonrwa.2012.11.002.

[7]

D. Hilbert, Mathematical problems, Bulletin of the American Mathematical Society, 8 (1902), 437–479. doi: 10.1090/S0002-9904-1902-00923-3.

[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.

[9]

S. J. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. doi: 10.2307/1401807.

[10]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.

[11]

M. F. S. Lima, C. Pessoa and W. F. Pereira, Limit cycles bifurcating from a period annulus in continuous piecewise linear differential systems with three zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750022, 14 pp. doi: 10.1142/S0218127417500225.

[12]

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.

[13]

J. Llibre and A. C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, J. Math. Anal. Appl., 413 (2014), 763-775.  doi: 10.1016/j.jmaa.2013.12.031.

[14]

D. Pi and X. Zhang, The sliding bifurcations in planar piecewise smooth differential systems, J. Dynam. Differ. Equ., 25 (2013), 1001-1026.  doi: 10.1007/s10884-013-9327-0.

[15]

X. Sun and P. Yu, Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree 4, J. Differ. Equ., 267 (2019), 7369-7384.  doi: 10.1016/j.jde.2019.07.023.

[16]

Y. TianM. Han and F. Xu, Bifurcations of small limit cycles in Liénard systems with cubic restoring terms, J. Differ. Equ., 267 (2019), 1561-1580.  doi: 10.1016/j.jde.2019.02.018.

[17]

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.

[18]

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.

[19]

Y. Xiong, Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters, Nonlinear Anal. Real World Appl., 41 (2018), 384-400.  doi: 10.1016/j.nonrwa.2017.10.020.

[20]

Y. Xiong and C. Wang, Limit cycle bifurcations of planar piecewise differential systems with three zones, Nonlinear Anal. Real World Appl., 61 (2021), Paper No. 103333, 18 pp. doi: 10.1016/j.nonrwa.2021.103333.

[21]

P. YangY. Yang and J. Yu, Up to second order Melnikov functions for general piecewise Hamiltonian systems with nonregular separation line, J. Differ. Equ., 285 (2021), 583-606.  doi: 10.1016/j.jde.2021.03.020.

show all references

References:
[1]

C. Christopher and C. Li, Limit cycles of differential equations, Springer Science & Business Media, 2007. doi: 10.1007/978-3-7643-8410-4.

[2]

L. P. C. D. CruzD. D. Novaes and J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 266 (2019), 4170-4203.  doi: 10.1016/j.jde.2018.09.032.

[3]

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.

[4]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250296, 30 pp. doi: 10.1142/S0218127412502963.

[5]

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.

[6]

M. Han and V. G. Romanovski, On the number of limit cycles of polynomial Liénard systems, Nonlinear Anal. Real World Appl., 14 (2013), 1655-1668.  doi: 10.1016/j.nonrwa.2012.11.002.

[7]

D. Hilbert, Mathematical problems, Bulletin of the American Mathematical Society, 8 (1902), 437–479. doi: 10.1090/S0002-9904-1902-00923-3.

[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.

[9]

S. J. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. doi: 10.2307/1401807.

[10]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.

[11]

M. F. S. Lima, C. Pessoa and W. F. Pereira, Limit cycles bifurcating from a period annulus in continuous piecewise linear differential systems with three zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750022, 14 pp. doi: 10.1142/S0218127417500225.

[12]

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.

[13]

J. Llibre and A. C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, J. Math. Anal. Appl., 413 (2014), 763-775.  doi: 10.1016/j.jmaa.2013.12.031.

[14]

D. Pi and X. Zhang, The sliding bifurcations in planar piecewise smooth differential systems, J. Dynam. Differ. Equ., 25 (2013), 1001-1026.  doi: 10.1007/s10884-013-9327-0.

[15]

X. Sun and P. Yu, Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree 4, J. Differ. Equ., 267 (2019), 7369-7384.  doi: 10.1016/j.jde.2019.07.023.

[16]

Y. TianM. Han and F. Xu, Bifurcations of small limit cycles in Liénard systems with cubic restoring terms, J. Differ. Equ., 267 (2019), 1561-1580.  doi: 10.1016/j.jde.2019.02.018.

[17]

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.

[18]

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.

[19]

Y. Xiong, Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters, Nonlinear Anal. Real World Appl., 41 (2018), 384-400.  doi: 10.1016/j.nonrwa.2017.10.020.

[20]

Y. Xiong and C. Wang, Limit cycle bifurcations of planar piecewise differential systems with three zones, Nonlinear Anal. Real World Appl., 61 (2021), Paper No. 103333, 18 pp. doi: 10.1016/j.nonrwa.2021.103333.

[21]

P. YangY. Yang and J. Yu, Up to second order Melnikov functions for general piecewise Hamiltonian systems with nonregular separation line, J. Differ. Equ., 285 (2021), 583-606.  doi: 10.1016/j.jde.2021.03.020.

Figure 1.  A periodic orbit of the system (1.1) with $ \varepsilon = 0 $
Figure 2.  A periodic orbit of the system (1.7) with $\lambda=\varepsilon=0$
Figure 3.  The periodic orbits of the system (1.7) with $ \varepsilon = 0 $
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