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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

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