Abstract
We consider a smooth diffeomorphism possessing a hyperbolic fixed point with a transversal homoclinic orbit. The aim of this paper is to investigate multivalued perturbations of the form where G is a compact convex-valued Lipschitz map from into itself and parameter is small. Our main result is a multivalued, parametrized version of the well-known Birkhoff–Smale homoclinic theorem asserting the existence of deterministic chaos with the corresponding Smale horseshoes. A similar result is given for multivalued, time-periodic perturbations of time-periodic ordinary differential equations. For autonomous ordinary differential equations, the resulting multivalued, parametrized homoclinic bifurcation problem is also discussed at some length.
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