Homoclinic and Heteroclinic Orbits for a Liénard System

Carmen Rocsoreanu, Mihaela Sterpu


A 2D dynamical system exhibiting a double-zero bifurcation with symmetry of order two is considered. This bifurcation involves the presence in the parameter space of a curve corresponding either to double homoclinic or to heteroclinic bifurcations. In this paper we derive second order approximations for the homoclinic orbits and for the curve of homoclinic bifurcation values considering the system truncated up to five order terms and parameter-dependent coefficients. These approximations were obtained using the regular perturbation method. These formulae are applied to a Liénard system, which develops a double-zero bifurcation with symmetry of order two for some parameters values. Second order approximations for the heteroclinic orbits of this system are also given. The analytical results are very accurate and they are in good accordance with the numerical ones.


Double-zero bifurcation; Homoclinic orbit; Heteroclinic orbit; Liénard system

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B. Al-Hdaibat,W. Govaerts, Y.A. Kuznetsov and H.G.E. Meijer, Initialization of Homoclinic solutions near Bogdanov–Takens Points: Lindstedt–Poincaré compared with regular perturbation method, SIAM J. Appl. Dynam. Sys. 15 (2016), 952, DOI: 10.1137/15M1017491.

M. Belhaq, F. Lakrad abd A. Fahsi, Predicting Homoclinic bifurcations in planar autonomous systems, Nonlinear Dynam. 18 (1999), 303, DOI: 10.1023/A:1026428718802.

M. Belhaq and F. Lakrad, Prediction of homoclinic bifurcation: the elliptic averaging method, Chaos, Solitons & Fractals 11 (2000), 2251, DOI: 10.1016/S0960-0779(99)00144-7.

Y.Y. Cao, K.W. Chung and J. Xu, A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method, Nonlinear Dynam. 64 (2011), 221, DOI: 10.1007/s11071-011-9990-9.

Y.Y. Chen and S.H. Chen, Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method, Nonlinear Dynam. 58 (2009), 417, DOI: 10.1007/s11071-009-9489-9.

S.N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Chapter 4 Cambridge University Press, Cambridge and New York (1994).

B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia (2002).

B. Feng and R. Hu, A survey on homoclinic and heteroclinic orbits, Applied Mathematics E-Notes 2 (2003), 16.

J. Feng, Q. Zhang, W. Wang and S.Hao, A new approach of asymmetric homoclinic and heteroclinic orbits construction in several typical systems based on the undetermined Padé approximation method, Mathematical Problems in Engineering 2016 (2016), 8585290, DOI: 10.1155/2016/8585290.

A. Gasull, H. Giacomini and J. Torregrosa, Some results on homoclinic and heteroclinic connections in planar systems, Nonlinearity 23 (2010), 2977, DOI: 10.1088/0951-7715/23/12/001.

A.I. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity 11 (1998), 1, DOI: 10.1088/0951-7715/11/6/005.

Y.A. Kuznetsov, H.G.E. Meijer, B. Al-Hdaibat and W. Govaerts, Improved homoclinic predictor for Bogdanov-Takens bifurcation, International Journal of Bifurcation and Chaos 24 (2014), 1450057, DOI: 10.1142/S0218127414500576.

Y.A. Kuznetsov, H.G.E. Meijer, B. Al-Hdaibat and W. Govaerts, Accurate approximation of Homoclinic solutions in Gray-Scott kinetic model, International Journal of Bifurcation and Chaos 25 (2015), 1450057, DOI: 10.1142/S0218127415501254.

C. Rocsoreanu and M. Sterpu, Approximations of the homoclinic orbits near a double-zero bifurcation with symmetry of order two, International Journal of Bifurcation and Chaos 27 (2017), 1750109, DOI: 10.1142/S0218127417501097.

C. Rocsoreanu and M. Sterpu, Approximations of the heteroclinic orbits near a double-zero bifurcation with symmetry of order two. Application to a Liénard equation, International Journal of Bifurcation and Chaos, communicated (2018).

Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, IL (2010).

Q.-C. Zhang, W. Wang and W.-Y. Li, Heteroclinic bifurcation of strongly nonlinear oscillator, Chinese Physics Letters 25 (2008), 1905s.

DOI: http://dx.doi.org/10.26713%2Fjims.v10i4.1048

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