A Computable Criterion for the Existence of Homoclinic Tangencies

A practical method for rigorously establishing the existence of generic quadratic tangencies associated with a fixed point of a planar diffeomorphism depending on two parameters is presented. We also determine a smooth curve in the two-dimensional parameter space along which the tangency persists. We illustrate the practicality of our theoretical results on several well-known examples of planar diffeomorphisms. These results can readily be generalized to prove the existence of such tangencies associated with periodic orbits. In lieu of the details of such a generalization, we present an example of a planar diffeomorphism possessing generic quadratic tangencies associated with periodic orbits, some exhibiting phase shifts.