Discrete Dichotomies and Bifurcations from Critical Homoclinic Orbits

AbstractPerturbed discrete systems likexn+1=f(xn)+μg(xn,μ),xn∈RN,n∈Z, when the associated unperturbed map (μ=0) is not invertible and has acriticalorbit {γn} homoclinic to a hyperbolic fixed pointpare studied. Bycriticalwe mean that thef′(γn) are invertible for any integern≠0 butf′(γ0) is not invertible. The main goal is to give sufficient conditions for a bifurcationfrom zero to many homoclinicswhen the parameter crosses zero. We also give a Melnikov like result assuring thepersistence of homoclinicsin a complete neighborhood of μ=0. This result is similar to the ones obtained for diffeomorphisms and flows