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 .
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