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August  2017, 22(6): 2417-2425. doi: 10.3934/dcdsb.2017123

Limit cycle bifurcations for piecewise smooth integrable differential systems

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.

Citation: Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and 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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

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