Abstract
Bifurcation from homoclinic to periodic orbits in two dimensions has been known for a long time [1,4]. L.P. Šil’nikov [8] 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 [7].
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© 1987 Springer-Verlag Berlin Heidelberg
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Walther, HO. (1987). Bifurcation from Homoclinic to Periodic Solutions by an Inclination Lemma with Pointwise Estimate. In: Chow, SN., Hale, J.K. (eds) Dynamics of Infinite Dimensional Systems. NATO ASI Series, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-86458-2_37
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