带两条平行切换直线多项式微分系统的极限环分支

Abstract

Abstract:

In this paper, the limit cycle bifurcation problem is investigated for a class of polynomial differential system with two parallel switch straight lines. We use the generalized first order Melnikov function and related qualitative theory of knowledge to export the algebraic and corresponding coefficients of expression, then, use coefficient of change to research generalized double homoclinic bifurcation, and get a lower bound of its ring number.

Key words: Discontinuous planar system, Limit cycle, Melnikov function, Homoclinic bifurcations

CLC Number:

O175.1

Xueli Jiang,Xuan Deng,Qiuhao Wen,Yanqin Xiong. Limit Circle Bifurcations of Polynomial Differential System with Two Parallel Switch Straight Lines[J].Acta mathematica scientia,Series A, 2022, 42(2): 353-364.

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References 12

1	di Bernardo M, Budd C J, Champneys A R, Kowalczyk P. Piecewise Smooth Dynamical Systems, Theory and Applications. London: Springer-Verlag, 2008
2	Budd C J. Non-smooth dynamical systems and the grazing bifurcation//Aston P J. Nonlinear Mathematics and Its Applications. Cambridge: Cambridge University Press, 1996: 219-235
3	Filipov A F. Differential equations with discontionuous righthand sides. Netherlands: Kluwer Academic, 1988
4	Hilbert D . Mathematical Problems. Bull Amer Math, 1902, 8: 437- 479 doi: 10.1090/S0002-9904-1902-00923-3
5	Pi D , Zhang X . The sliding bifurcations in planar piecewise sooth differential systems. J Dyn Diff Equa, 2013, 25: 1001- 1026 doi: 10.1007/s10884-013-9327-0
6	Dankowicz H , Nordmark A B . On the origin and bifurcations of stick-slip oscillations. Physica D, 2000, 136: 280- 302 doi: 10.1016/S0167-2789(99)00161-X
7	Feigin M I . Doubling of the oscillation period with C-bifurcations in piecewise-continuous systems. J Appl Math Mech, 1970, 34: 822- 830 doi: 10.1016/0021-8928(70)90064-X
8	Liu X , Han M . Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Internat J Bifur Chaos, 2010, 5: 1- 12
9	Xiong Y , Han M . Limit cycle bifurcations in a class of perturbed piecewise smooth systems. Applied Mathematics and Computation, 2014, 242: 47- 64 doi: 10.1016/j.amc.2014.05.035
10	Liang F , Han M , Romanovskic V G . Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop. Nonlinear Analysis, 2012, 75: 4355- 4374 doi: 10.1016/j.na.2012.03.022
11	Hu N , Du Z . Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems. Commun Nonlinear Sci Numer Simulat, 2013, 18: 3436- 3448 doi: 10.1016/j.cnsns.2013.05.012
12	Xiong Y , Han M . Limit cycle bifurcations in discontinuous planar systems with multiple lines. Journal of Applied Analysis and Computation, 2020, 10: 361- 377 doi: 10.11948/20190274

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Limit Circle Bifurcations of Polynomial Differential System with Two Parallel Switch Straight Lines

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