American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Invariant measures for complex-valued dissipative dynamical systems and applications
  • DCDS-B Home
  • This Issue
  • Next Article
    Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature
August  2017, 22(6): 2417-2425. doi: 10.3934/dcdsb.2017123

Limit cycle bifurcations for piecewise smooth integrable differential systems

Jihua Yang 1,, and  Liqin Zhao 2,

1. 

School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756000, China
2. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Jihua Yang, E-mail address: jihua1113@163.com

Received  May 2016 Revised  March 2017 Published  March 2017

Fund Project: The first author is supported by NSFC(11671040,11601250), the Visual Learning Young Researcher of Ningxia, the Science and Technology Pillar Program of Ningxia(KJ[2015]26(4)) and the Key Program of Ningxia Normal University(NXSFZD1708); The second author is supported by NSFC(11671040).

In this paper, we study a class of piecewise smooth integrable non-Hamiltonian systems, which has a center. By using the first order Melnikov function, we give an exact number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n.

Keywords: Integrable differential system, limit cycle, Melnikov function.
Mathematics Subject Classification: Primary: 34C07, 34C0.
Citation: Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
References:
[1]

S. Banerjee and G. Verghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 2001. Google Scholar

[2]

M. Bernardo, C. Budd, A. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. Google Scholar

[3]

W. Chen and W. Zhang, Isochronicity of centers in a switching Bautin system, J. Differential Equations, 252 (2012), 2877-2899. Google Scholar

[4]

B. Coll, A. Gasull and R. Prohens, Bifurcation of limit cycles from two families of ceters, Dyn. Contin. Discrete Implus, Syst. Ser. A Math. Anal., 12 (2005), 275-287. Google Scholar

[5]

L. Dieci and C. Elia, Periodic orbits for planar piecewise smooth systems with a line of discontinuity, J. Dyn. Diff. Equat., 26 (2014), 1049-1078. Google Scholar

[6]

D. Hilbert, Mathematical problems (M. Newton, Transl.), Bull. Am. Math., 8 (1902), 437-479. Google Scholar

[7]

Y. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc., 39 (2002), 301-354. Google Scholar

[8]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. Google Scholar

[9]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128. Google Scholar

[10]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcation Chaos, 13 (2003), 47-106. Google Scholar

[11]

S. Li and C. Liu, A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system, J. Math. Anal. Appl., 428 (2015), 1354-1367. Google Scholar

[12]

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. Google Scholar

[13]

F. Liang, M. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374. Google Scholar

[14]

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. Google Scholar

[15]

J. Llibre, A. Mereu and D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations, 258 (2015), 4007-4032. Google Scholar

[16]

Y. Xiong, Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters, J. Math. Anal. Appl., 421 (2015), 260-275. Google Scholar

[17]

Y. Xiong, The number of limit cycles in perturbations of polynomial systems with multiple circles of critical points, J. Math. Anal. Appl., 440 (2016), 220-239. Google Scholar

[18]

J. Yang and L. Zhao, Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle, Nonlinear Analysis: Real World Applications, 27 (2016), 350-265. Google Scholar

show all references

References:
[1]

S. Banerjee and G. Verghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 2001. Google Scholar

[2]

M. Bernardo, C. Budd, A. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. Google Scholar

[3]

W. Chen and W. Zhang, Isochronicity of centers in a switching Bautin system, J. Differential Equations, 252 (2012), 2877-2899. Google Scholar

[4]

B. Coll, A. Gasull and R. Prohens, Bifurcation of limit cycles from two families of ceters, Dyn. Contin. Discrete Implus, Syst. Ser. A Math. Anal., 12 (2005), 275-287. Google Scholar

[5]

L. Dieci and C. Elia, Periodic orbits for planar piecewise smooth systems with a line of discontinuity, J. Dyn. Diff. Equat., 26 (2014), 1049-1078. Google Scholar

[6]

D. Hilbert, Mathematical problems (M. Newton, Transl.), Bull. Am. Math., 8 (1902), 437-479. Google Scholar

[7]

Y. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc., 39 (2002), 301-354. Google Scholar

[8]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. Google Scholar

[9]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128. Google Scholar

[10]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcation Chaos, 13 (2003), 47-106. Google Scholar

[11]

S. Li and C. Liu, A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system, J. Math. Anal. Appl., 428 (2015), 1354-1367. Google Scholar

[12]

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. Google Scholar

[13]

F. Liang, M. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374. Google Scholar

[14]

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. Google Scholar

[15]

J. Llibre, A. Mereu and D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations, 258 (2015), 4007-4032. Google Scholar

[16]

Y. Xiong, Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters, J. Math. Anal. Appl., 421 (2015), 260-275. Google Scholar

[17]

Y. Xiong, The number of limit cycles in perturbations of polynomial systems with multiple circles of critical points, J. Math. Anal. Appl., 440 (2016), 220-239. Google Scholar

[18]

J. Yang and L. Zhao, Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle, Nonlinear Analysis: Real World Applications, 27 (2016), 350-265. Google Scholar

[1]

Hang Zheng, Yonghui Xia. Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function. Discrete & Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021319
[2]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[3]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[4]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[5]

Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103
[6]

Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995
[7]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[8]

Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669
[9]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
[10]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447
[11]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1
[12]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081
[13]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[14]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
[15]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[16]

Octavian G. Mustafa, Yuri V. Rogovchenko. Existence of square integrable solutions of perturbed nonlinear differential equations. Conference Publications, 2003, 2003 (Special) : 647-655. doi: 10.3934/proc.2003.2003.647
[17]

Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021264
[18]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
[19]

Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129
[20]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (246)
  • HTML views (91)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy