Bifurcation from Homoclinic to Periodic Solutions by an Inclination Lemma with Pointwise Estimate
Bifurcation from homoclinic to periodic orbits in two dimensions has been known for a long time [1,4]. L.P. Šil’nikov  obtained the first result for arbitrary finite dimension. His idea was to consider a point on the homoclinic trajectory as fixed point of a suitably constructed map so that continuation by the implicit function theorem yields fixed points which define periodic solutions. The difficulty involved is to show smoothness of Šil’nikov’s map. This requires a careful investigation of trajectories close to a hyperbolic equilibrium. The underlying vectorfields have to be at least C2-smooth .
KeywordsPeriodic Solution Stable Manifold Pointwise Estimate Saddle Connection Unit Eigenvector
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- A.A. Andronov, C.E. Chaikin. Theory of oscillations. Princeton Univ. Press, Princeton,N.J., 1949Google Scholar
- M. Blazquez. Bifurcation from a homoclinic orbit in parabolic differential equations. Preprint, Brown Univ., Providence, R.I., 1985Google Scholar
- S.N. Chow, B. Deng. Homoclinic and heteroclinic bifurcation in Banach spaces. Preprint, Mich. State Univ., East Lansing, Mi., 1986Google Scholar
- S.N. Chow, J.K. Hale. Methods of bifurcation theory. Springer, New York et al., 1982Google Scholar
- J.K. Hale, X.B. Lin. Symbolic dynamics and nonlinear semi-flows. Preprint LCDS 84–8, Brown Univ., Providence, R.I., 1 984Google Scholar
- D. Henry. Invariant manifolds. Notes, Math. Inst., Univ. of Sao Paulo, 1983Google Scholar
- E. Reyzl. Diplomarbeit. Math. Inst., Univ. München, 1986Google Scholar
- L.P. Sil’nikov. On the generation of periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type. Mat. Sb. 77 (119), 1968; Engl. transi. in: Math. USSR–Sbornik 6 (1968), 427–438Google Scholar
- H.O. Walther. Inclination lemmas with dominated convergence. Research report 85–03, Sem. f. Angew. Math., ETH Zürich, 1985 (submitted)Google Scholar
- H.O. Walther. Bifurcation from a saddle connection in functional differential equations: An approach with inclination lemmas. Preprint, Math. Inst., Univ. München, 1986 (submitted)Google Scholar