On manifolds of connecting orbits in discretizations of dynamical systems Original Research Article

It is shown that one-step methods, when applied to a one-parametric dynamical system with a homoclinic orbit, exhibit a closed loop of discrete homoclinic orbits. On this loop, the parameter varies periodically while the orbit shifts its index after one revolution. We show that at least two homoclinic tangencies occur on this loop. Our approach works for systems with finite smoothness and also applies to general connecting orbits. It provides an alternative to the interpolation approach by Fiedler and Scheurle (Mem. Amer. Math. Soc. 119 (570) (1996)) and it allows to recover some of their results on exponentially small splittings of separatrices by using some recent backward error analysis for the analytic case.