LIMIT CYCLE BIFURCATIONS IN DISCONTINUOUS PLANAR SYSTEMS WITH MULTIPLE LINES

Yanqin Xiong; Maoan Han

doi:10.11948/20190274

2020 Volume 10 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(1): 361-377

Next Article Previous Article

Yanqin Xiong, Maoan Han. LIMIT CYCLE BIFURCATIONS IN DISCONTINUOUS PLANAR SYSTEMS WITH MULTIPLE LINES[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 361-377. doi: 10.11948/20190274

Citation:

Yanqin Xiong, Maoan Han. LIMIT CYCLE BIFURCATIONS IN DISCONTINUOUS PLANAR SYSTEMS WITH MULTIPLE LINES[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 361-377. doi: 10.11948/20190274

LIMIT CYCLE BIFURCATIONS IN DISCONTINUOUS PLANAR SYSTEMS WITH MULTIPLE LINES

Yanqin Xiong¹,
Maoan Han^2, ,

1.
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China
2.
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

Corresponding author: Email address: mahan@ahnu.edu.cn(M. Han)
Fund Project: The work was supported by National Natural Science Foundation of China (11431008, 11771296, 11701289) and Natural Science Foundation of Jiangsu Province(BK20170936)

Abstract

Abstract

In this paper, the limit cycle bifurcation problem is investigated for a class of planar discontinuous perturbed systems with $ n $ parallel switch lines. Under the assumption that the unperturbed system has a family of periodic orbits crossing all of the lines, an explicit expression of the first order Melnikov function along the periodic orbits is presented, which plays an important role in studying the problem of limit cycle bifurcations. As an application of the established method, the maximal number of limit cycles of a discontinuous system is considered.
- Discontinuous planar system /
- limit cycle /
- bifurcation /
- periodic
MSC: 34C05, 34C07, 34C25

FullText(HTML)

References

[1]	U. Akhmetov and D. Aruǧaslan, Bifurcation of a non-smooth planar limit cycle from a vertex, Nonlinear Analysis-TMA, 2009, 71, 2723–2733. doi: 10.1016/j.na.2009.06.031 CrossRef Google Scholar
[2]	U. Akhmetov, On the smoothness of solutions of differential equations with a discontinuous right-hand side, Ukrainian Math. J., 1993, 45, 1785–1792. doi: 10.1007/BF01061348 CrossRef Google Scholar
[3]	J. Budd, A.R. Champneys, and P. Kowalczyk, Piecewise smooth dynamical systems, theory and applications, Springer-Verlag, London, 2008. Google Scholar
[4]	J. Budd, Non-smooth dynamical systems and the grazing bifurcation. In: Nonlinear mathematics and its applications, Guildford, 1995. Cambridge: Cambridge University Press, 1996, 219–235. Google Scholar
[5]	B. Coll, A. Gasull, and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 2001, 253, 671–690. doi: 10.1006/jmaa.2000.7188 CrossRef Google Scholar
[6]	H. Dankowicz and A.B. Nordmark, On the origin and bifurcations of stick-slip oscillations, Physica D, 2000, 136, 280–302. doi: 10.1016/S0167-2789(99)00161-X CrossRef Google Scholar
[7]	D. Euzébio and J.Llibre, On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line, J. Math. Anal. Appl., 2015, 424, 475–486. doi: 10.1016/j.jmaa.2014.10.077 CrossRef Google Scholar
[8]	F. Filipov, Differential equations with discontionuous righthand sides, Kluwer Academic, Netherlands, 1988. Google Scholar
[9]	I. Feigin, 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 CrossRef Google Scholar
[10]	M. Han, L. Sheng, Bifurcation of limit cycles in piecewise smooth system via Melnikov function, Journal of Applied Analysis and Computation, 2015, 5, 809–815. Google Scholar
[11]	M. Han, On the maximal number of periodic solution of piecewise smooth periodic equations by average method, Journal of Applied Analysis and Computation, 2017, 7, 788–794. Google Scholar
[12]	M. Han, V.G. Romanovski, and X. Zhang, Equivalence of the Melnikov Function Method and the Averaging Method, Qual. Theory Dyn. Syst., 2016, 15, 471–479. doi: 10.1007/s12346-015-0179-3 CrossRef Google Scholar
[13]	M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 2010, 248, 2399–2416. doi: 10.1016/j.jde.2009.10.002 CrossRef Google Scholar
[14]	M. Han, H. Yan, J. Yang, and C. Lhotka, On the number of limit cycles of some Liénard systems, Can. Appl. Math. Q., 2009, 17, 61–83. Google Scholar
[15]	M. Han, J. Yang, and P. Yu, Hopf bifurcations for near-Hamiltonian systems, Internat. J. Bifur. Chaos, 2009, 19, 4117–4130. doi: 10.1142/S0218127409025250 CrossRef Google Scholar
[16]	N. Hu and Z. Du, 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 CrossRef Google Scholar
[17]	X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Internat. J. Bifur. Chaos, 2010, 5, 1–12. Google Scholar
[18]	F. Liang, M. Han, and V.G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Analysis-TMA, 2012, 75, 4355–4374. doi: 10.1016/j.na.2012.03.022 CrossRef Google Scholar
[19]	F. Liang and M. Han, Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos, Solition & Fractals, 2012, 45, 150–162. Google Scholar
[20]	F. Liang, M. Han, and X. Zhang, Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems, J. Differential Equations, 2013, 255, 4403–4436. doi: 10.1016/j.jde.2013.08.013 CrossRef Google Scholar
[21]	Y. Li and Z. Du, Applying Battelli-Fêchan's method to transversal heteroclinic bifurction in piecewise smooth systems, Discrete Cont. Dynam. System B, 2019, 24, 6025–6052. Google Scholar
[22]	J. Llibre, D.D. Novaes, and M.A. Teixeira, Averaging methods for studying the periodic orbits of discontinuous differential systems, IMECC Technical Report, 2012, 8. Google Scholar
[23]	J. Llibre and A.C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, J. Math. Anal. Appl., 2014, 413, 763–775. doi: 10.1016/j.jmaa.2013.12.031 CrossRef Google Scholar
[24]	J. Llibre and M.A. Teixeira, Limit cycles for $m$-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys., 2014. DOI10.1007/s00033-013-0393-2. Google Scholar
[25]	I. Leine and H. Nijmeijer, Dynamics and bifurcations of nonsmooth mechanical systems, Berlin, Lecture Notes in Applied and Computational Mechanics, Springer-Verlag, 2004. Google Scholar
[26]	D. Pi and X. Zhang, The sliding bifurcations in planar piecewise Smooth differential Systems, J. Dyn. Diff. Equat., 2013, 25, 1001–1026. doi: 10.1007/s10884-013-9327-0 CrossRef Google Scholar
[27]	A. Tonnelier, On the number of limit cycles in piecewise-linear Liénard systems, Internat. J. Bifur. Chaos, 2005, 15, 1417–1422. doi: 10.1142/S0218127405012624 CrossRef Google Scholar
[28]	A. Tonnelier, The McKean's caricature of the FitzHugh-Nagumo model. I: The space-clamped system, SIAM J. Appl. Math., 2002, 63, 459–484. Google Scholar
[29]	Y. Xiong and M. Han, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system, Abstract and Applied Analysis., 2013, Article ID 575390, 19 pages. Google Scholar
[30]	Y. Xiong and M. Han, 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 CrossRef Google Scholar
[31]	Y. Xiong and M. Han, Stability of a homoclinic loop and limit cycle bifurcations in non-smooth systems, Chaos, Solitons & Fractals, 2015, 78, 107–117. Google Scholar
[32]	Y. Xiong and H. Zhong, The number of limit cycles in a $Z_2$-equivariant Liénard system, Internat. J. Bifur. Chaos, 2013, 23(5), Article ID 1350085, 17 pages. Google Scholar
[33]	W. Xu and C. Li, Limit cycles of some polynomial Liénard systems, J. Math. Anal. Appl., 2012, 389, 367–378. doi: 10.1016/j.jmaa.2011.11.070 CrossRef Google Scholar
[34]	Y. Zou and T. Küpper, Generalized Hopf bifurcation emanated from a corner for piecewise smooth planar systems, Nonlinear Analysis-TMA, 2005, 62, 1–17. doi: 10.1016/j.na.2004.06.004 CrossRef Google Scholar