×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.