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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)].

##### MSC:
 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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##### References:
 [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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