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Bifurcation of homoclinics in a nonlinear oscillation. (English) Zbl 0691.34031
Summary: We discuss the bifurcation of homoclinics of the equation \[ (*)\quad x''+g(x)+g_ 1(x)=-\lambda x'+\mu (f(t)+f_ 1(t)), \] where g(x) is such that the unperturbed equation \(x''+g(x)=0\) has homoclinic orbits through zero. We give the bifurcation graph of small parameters \(\mu\) and \(\lambda\), and that of small functions \(g_ 1\) and \(f_ 1\). Then we give a criterion to determine the codimensions of bifurcation manifolds of small functions \(g_ 1\) and \(f_ 1\). Thus we generalize the conclusions of J. K. Hale and A. Spezamiglio [Nonlinear Anal., Theory Methods Appl. 9, 181-192 (1985; Zbl 0563.34040)].

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI
[1] Hale, J. K. & Spezamiglio,A., Perturbation of homoclinincs and subharmonics in Duffing’s equation,Nonlinear Analysis, Theory, Methods, and Applications,9 (1985), 181–192. · Zbl 0563.34040
[2] Chow, S. N., Hale, J. K. & Mallet-Paret, J., A example of bifurcation to homoclinic orbits,J. Diff. Eqns,37(1980), 351 - 373. · Zbl 0439.34035
[3] Guckenheimer, J. & Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer-Verlag, 1983. · Zbl 0515.34001
[4] Chow,S.N. & Hale,J.K., Method of bifurcation theory, Springer-Verlag, 1983.
[5] Abraham,R., Marsden,J.E. & Ratin, T., Manifolds, tensor analysis and applications, Addison-Wesley Publishing company, 1983.
[6] Hale, J.K., Ordinary differential equations, Wiley: New York, 1969. · Zbl 0186.40901
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