Limit Cycles and Homoclinic Orbits and Their Bifurcation of the Bogdanov-Takens System

HUANG Cheng-biao; LIU Jia

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HUANG Cheng-biao, LIU Jia. Limit Cycles and Homoclinic Orbits and Their Bifurcation of the Bogdanov-Takens System[J]. Applied Mathematics and Mechanics, 2008, 29(9): 1083-1088.

Citation:

HUANG Cheng-biao, LIU Jia. Limit Cycles and Homoclinic Orbits and Their Bifurcation of the Bogdanov-Takens System[J]. Applied Mathematics and Mechanics, 2008, 29(9): 1083-1088.

Citation:

HUANG Cheng-biao, LIU Jia. Limit Cycles and Homoclinic Orbits and Their Bifurcation of the Bogdanov-Takens System[J]. Applied Mathematics and Mechanics, 2008, 29(9): 1083-1088.

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Limit Cycles and Homoclinic Orbits and Their Bifurcation of the Bogdanov-Takens System

Department of Applied Mechanics & Engineering, Sun Yat-sen University, Guangzhou 510275, P. R. China

Received Date: 2007-09-25
Rev Recd Date: 2008-08-06
Publish Date: 2008-09-15

Abstract

Abstract

The quantitative analysis of limit cycles and homoclinic orbits and the bifurcation curve for the Bogdanov-Takens system were discussed. The parameter incremental method for approximate analytical-expressions of these problems was given. These analytical-expressions of the limit cycle and homoclinic orbit were shown as the generalized harmonic function by employing a time transformation. Some curves for the parameters and the stability characteristic exponent of limit cycle versus amplitude were drawn. And some of the limit cycles and homoclinic orbits phase portraits were plotted. And the relationship curves of parameterand μ and λ with amplitude a and the bifurcation diagrams about the parameter were given too. The numerical accuracy of calculation results was good.
- Bogdanov-Takens system,
- limit cycle,
- homoclinic orbit,
- bifurcation diagrams,
- analytical-expressions

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References(10)

References

[1]	Perko L M. A global analysis of the Bogdanov-Takens system[J].SIAM J Appl Math,1992,52(4):1172-1192. doi: 10.1137/0152069
[2]	Bogdanov R I. Bifurcation of the limit cycle of a family of plane vector fields[J].Selecta Math Soviet,1981，1:373-387.
[3]	Bogdanov R I. Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues[J].Selecta Math Soviet,1981，1:389-421.
[4]	Takens F. Forced oscillations and bifurcations[J].Applications of Global Analysis I, Comm Math Inst Rijksuniversitat Utrecht, 1974，3:1-59.
[5]	Kuznetsov Y.Elements of Applied Bifurcation Theory[M].Vol 112.New York:Springer-Verlag, 1995.
[6]	WANG Duo, LI Jing,HUANG Min-hai,et al.Unique normal form of Bogdanov-Takens singularities[J].Journal of Differential Equations,2000，163(1):223-238. doi: 10.1006/jdeq.1999.3739
[7]	Iliev Iliya D. On the limit cycles available from polynomial perturbations of the Bogdanov-Takens Hamiltonian[J].Israel Journal of Mathematics,2000，115(1):269-284. doi: 10.1007/BF02810590
[8]	岳喜顺.后继函数法Bogdanov-Takens系统的二次扰动[J]. 应用数学学报,2006，29(5):838-847.
[9]	丰建文. Bogdanov-Takens系统的三次齐扰动[J].数学杂志,2004，24(5):565-569.
[10]	Chan H S Y,Chung K W,Xu Z.A perturbation-incremental method for strongly non-linear oscillators[J].Internat J Non-Linear Mech,1996，31(1):59-72. doi: 10.1016/0020-7462(95)00043-7