November  2017, 16(6): 2321-2336. doi: 10.3934/cpaa.2017114

Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp

1. 

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

2. 

School of Information Engineering, Zhengzhou Institute of Finance and Economics, Zhengzhou 450000, China

* Corresponding author

Received  August 2016 Revised  February 2017 Published  July 2017

Fund Project: The first author is supported by NSFC(11601250), the Science and Technology Pillar Program of Ningxia(KJ[2015]26(4)), the Visual Learning Young Researcher of Ningxia and the Key Program of Ningxia Normal University(NXSFZD1708), the second author is supported by the Key Program of Higher Education of Henan(16A110038, 17B110003), and the third author is supported by NSFC(11361046).

In this paper we study the limit cycle bifurcation of a piecewise smooth Hamiltonian system. By using the Melnikov function of piecewise smooth near-Hamiltonian systems, we obtain that at most $12n+7$ limit cycles can bifurcate from the period annulus up to the first order in $\varepsilon$.

Citation: 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 and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
References:
[1]

M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4612-0873-0.

[2]

M. Akhmet and M. Turan, Bifurcation of discontinuous limit cycles of the Van der Pol equation, Math. Comput. Simulat., 95 (2014), 39-54.  doi: 10.2307/2152750.

[3]

M. di Bernardo, C. Budd, A. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-4612-0873-0.

[4]

B. CollA. Gasull and R. Prohens, Degenerate Hopf bifurcation in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690.  doi: 10.1006/jmaa.2000.7188.

[5]

A. Filippov, Differential Equation with Discontinuous Righthand Sides, Kluwer Academic Pub. , Dordrecht, The Netherlands, 1988. doi: 10.1007/978-1-4612-0873-0.

[6]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[7]

C. Henry, Differential equations with discontinuous righthand side for planning procedure, J. Econom. Theory, 4 (1972), 541-551.  doi: 10.1016/0022-0531(72)90138-X.

[8]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (1972), 340-353.  doi: 10.1016/j.jmaa.2013.08.064.

[9]

Y. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc., 39 (2002), 301-354.  doi: 10.1090/S0273-0979-02-00946-1.

[10]

V. Krivan, On the Gause predator-prey model with a refuge: a fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.

[11]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-1-4612-0873-0.

[12]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.

[13]

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.  doi: 10.1016/j.jmaa.2015.03.074.

[14]

F. LiangM. Han and V. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.  doi: 10.1016/j.na.2012.03.022.

[15]

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.  doi: 10.1142/S021812741002654X.

[16]

J. LlibreA. Mereu and D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.

[17]

N. Minorski, Nonlinear Oscillations, Van Nostrand, New York, 1962. doi: 10.1007/978-1-4612-0873-0.

[18]

M. Teixeira, Perturbation Theory for Non-smooth Systems, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4612-0873-0.

[19]

Y. Xiong and M. Han, Limit cycle bifurcations in a class of perturbed piecewise smooth systems, Appl. Math. Comput., 242 (2014), 47-64.  doi: 10.1016/j.amc.2014.05.035.

[20]

J. Yang and L. Zhao, Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle, Nonlinear Anal.: Real World Appl., 27 (2016), 350-365.  doi: 10.1016/j.nonrwa.2015.08.005.

show all references

References:
[1]

M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4612-0873-0.

[2]

M. Akhmet and M. Turan, Bifurcation of discontinuous limit cycles of the Van der Pol equation, Math. Comput. Simulat., 95 (2014), 39-54.  doi: 10.2307/2152750.

[3]

M. di Bernardo, C. Budd, A. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-4612-0873-0.

[4]

B. CollA. Gasull and R. Prohens, Degenerate Hopf bifurcation in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690.  doi: 10.1006/jmaa.2000.7188.

[5]

A. Filippov, Differential Equation with Discontinuous Righthand Sides, Kluwer Academic Pub. , Dordrecht, The Netherlands, 1988. doi: 10.1007/978-1-4612-0873-0.

[6]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[7]

C. Henry, Differential equations with discontinuous righthand side for planning procedure, J. Econom. Theory, 4 (1972), 541-551.  doi: 10.1016/0022-0531(72)90138-X.

[8]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (1972), 340-353.  doi: 10.1016/j.jmaa.2013.08.064.

[9]

Y. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc., 39 (2002), 301-354.  doi: 10.1090/S0273-0979-02-00946-1.

[10]

V. Krivan, On the Gause predator-prey model with a refuge: a fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.

[11]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-1-4612-0873-0.

[12]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.

[13]

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.  doi: 10.1016/j.jmaa.2015.03.074.

[14]

F. LiangM. Han and V. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.  doi: 10.1016/j.na.2012.03.022.

[15]

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.  doi: 10.1142/S021812741002654X.

[16]

J. LlibreA. Mereu and D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.

[17]

N. Minorski, Nonlinear Oscillations, Van Nostrand, New York, 1962. doi: 10.1007/978-1-4612-0873-0.

[18]

M. Teixeira, Perturbation Theory for Non-smooth Systems, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4612-0873-0.

[19]

Y. Xiong and M. Han, Limit cycle bifurcations in a class of perturbed piecewise smooth systems, Appl. Math. Comput., 242 (2014), 47-64.  doi: 10.1016/j.amc.2014.05.035.

[20]

J. Yang and L. Zhao, Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle, Nonlinear Anal.: Real World Appl., 27 (2016), 350-365.  doi: 10.1016/j.nonrwa.2015.08.005.

Figure 1.  The closed orbits of system (1.4)$|_{\varepsilon=0}$
Figure 2.  Phase portrait of system (1.12)$|_{\varepsilon=0}$
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