Discrete Dichotomies and Bifurcations from Critical Homoclinic Orbits

Perturbed discrete systems like xnC1 D f xn‘ C gxn; ‘, xn 2 N, n 2 , when the associated unperturbed map ( D 0) is not invertible and has a critical orbit ” n• homoclinic to a hyperbolic fixed point p are studied. By critical we mean that the f 0 n‘ are invertible for any integer n 6D 0 but f 0 0‘ is not invertible. The main goal is to give sufficient conditions for a bifurcation from zero to many homoclinics when the parameter crosses zero. We also give a Melnikov like result assuring the persistence of homoclinics in a complete neighborhood of D 0. This result is similar to the ones obtained for diffeomorphisms and flows.