# American Institue of Mathematical Sciences

• Previous Article
Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature
• DCDS-B Home
• This Issue
• Next Article
Invariant measures for complex-valued dissipative dynamical systems and applications
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 & 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, W. Zhang, Isochronicity of centers in a switching Bautin system, J. Differential Equations, 252 (2012), 2877-2899. [4] B. Coll, A. Gasull, 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, 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 (Newton M., 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, 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, 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, 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, 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, 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, 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, W. Zhang, Isochronicity of centers in a switching Bautin system, J. Differential Equations, 252 (2012), 2877-2899. [4] B. Coll, A. Gasull, 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, 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 (Newton M., 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, 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, 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, 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, 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, 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, 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.
 [1] 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 [2] 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 [3] Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995 [4] Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 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 - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 [6] 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 [7] 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 [8] 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 [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] Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081 [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] Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 [13] Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795 [14] Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227 [15] 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 [16] 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 [17] Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453 [18] 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 [19] Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129 [20] Yi-Chiuan Chen. Bernoulli shift for second order recurrence relations near the anti-integrable limit. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 587-598. doi: 10.3934/dcdsb.2005.5.587

2016 Impact Factor: 0.994