Inflated deterministic chaos and Smale's horseshoe

We consider a smooth diffeomorphism f : Rn !Rn possessing a hyperbolic fixed point with a transversal homoclinic orbit. The aim of this paper is to investigate multivalued perturbations of the form x!f ðxÞ þ 1GðxÞ where G is a compact convex-valued Lipschitz map from Rn into itself and parameter 1 . 0 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.