Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters

Meilan Cai; Maoan Han

doi:10.3934/cpaa.2020257

Article Contents

2021, Volume 20, Issue 1: 55-75. Doi: 10.3934/cpaa.2020257

This issue Previous Article A note on Riemann-Liouville fractional Sobolev spaces Next Article Scattering of the focusing energy-critical NLS with inverse square potential in the radial case

Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters

Meilan Cai^1, and
Maoan Han^2,1, ,

1.
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
2.
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

^* Corresponding author
^* Corresponding author

Received: March 2020

Revised: August 2020

Early access: October 2020

Published: January 2021

This work was supported by National Nature Science Foundation of China (11931016 and 11771296).

Abstract / Introduction Full Text(HTML) Figure(3) Related Papers Cited by

Abstract

In this paper, we concern with the problem of limit cycle bifurcation for a class of piecewise smooth cubic systems. Using the first order Melnikov function we prove that at least thirteen limit cycles can be bifurcated from periodic solutions surrounding the center.

Keywords:

Mathematics Subject Classification: Primary: 34C05, 34C07; Secondary: 37G15.

Citation:

Full Text(HTML)

Figure 1.1. Phase portrait of $ (1.3)|_{\varepsilon = \lambda = 0} $

Download: Full-size image PowerPoint slide

Figure 2.1. The periodic orbit $ L_{\lambda} $ of (2.1)$ |_{\varepsilon = 0} $

Download: Full-size image PowerPoint slide

Figure 3.1. The sketches of functions $ W_{i}(h) $, $ i = 1, 2, \cdots, 8 $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	L. P. da Cruz, D. D. Novaes and J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 226 (2019), 4170-4203. doi: 10.1016/j.jde.2018.09.032.
[2]	J. Gin$\acute{e}$ and J. Llibre, Limit cycles of cubic polynomial vector fields via the averaging theory, Nonlinear Anal., 66 (2007), 1707-1721. doi: 10.1016/j.na.2006.02.016.
[3]	M. Grau, F. Ma$\tilde{n}$osas and J. Villadelprat, A Chebyshew criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129. doi: 10.1090/S0002-9947-2010-05007-X.
[4]	L. F. S. Gouveia and J. Torregrosa, 24 crossing limit cycles in only one nest for piecewise cubic systems, Appl. Math. Lett., 103 (2020), 6pp. doi: 10.1016/j.aml.2019.106189.
[5]	M. Han, G. Chen and C. Sun, On the number of limit cycles in near-Hamiltonian polynomial systems, Int. J. Bifur. Chaos, 17 (2007), 2033-2047. doi: 10.1142/S0218127407018208.
[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 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.
[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. Karlin and W. J. Studden, Tchebycheff systems: With Applications in Analysis and Statistics, Pure Appa. math., Interscience Publishers, New York, London, Sydney, 1966.
[10]	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.
[11]	S. Li and T. Huang, Limit cycles for piecewise smooth perturbations of a cubic polynomial differential center, J. Differ. Equ., 2015 (2015), 1-17.
[12]	X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Int. J. Bifur. Chaos, 5 (2010), 1379-1390. doi: 10.1142/S021812741002654X.
[13]	J. Llibre and J. Itikawa, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math., 277 (2015), 171-191. doi: 10.1016/j.cam.2014.09.007.
[14]	C. Li, W. Li, J. Llibre and Z. Zhang, Linear estimation of the number of zeros of Abelian integrals for some cubic isochronous centers, J. Differ. Equ., 180 (2002), 307-333. doi: 10.1006/jdeq.2001.4064.
[15]	J. Llibre, C. Mereu and D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differ. Equ., 258 (2015), 4007-4032. doi: 10.1016/j.jde.2015.01.022.
[16]	J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math., 139 (2015), 229-244. doi: 10.1016/j.bulsci.2014.08.011.
[17]	J. Llibre and G. $\acute{S}$wirszcz, On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Ser. A. Math. Anal., 18 (2011), 203-314.
[18]	S. Li and Y. Zhao, Limit cycles of perturbed cubic isochronous center via the second order averaging method, Int. J. Bifur. Chaos, 24 (2014), 8pp. doi: 10.1142/S0218127414500357.
[19]	J. N. Mather, Stability of $C^{\infty}$ Mappings: I. The Division Theorem, Ann. Math., 87 (1968), 89-104. doi: 10.2307/1970595.
[20]	D. D. Novaes and J. Torregrose, On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl., 448 (2017), 171-186. doi: 10.1016/j.jmaa.2016.10.076.
[21]	J. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, 2$^{nd}$ edition, Applied Mathematical Sciences, Springer New York, 2007.
[22]	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.
[23]	Y. Xiong, M. Han and V. G. Romanovski, The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles, Int. J. Bifur. Chaos, 27 (2017), 14pp. doi: 10.1142/S0218127417501267.