On manifolds of connecting orbits in discretizations of dynamical systems (Q1863618)

Time discretizations of continuous dynamical systems lead often to subtlephenomena near homoclinic and heteroclinic orbits. A fundamental result of this type was derived by \textit{B. Fiedler} and \textit{J. Scheurle} [Mem. Amer. Math. Soc. 570 (1996; Zbl 0923.34049)]. The authors provide an alternative approach, recover and generalize these results. They show that one-step methods, when applied to a one-parameter dynamical systems 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. At least two homoclinic tangencies occur. This approach is suitable to systems with finite smoothness and general connecting orbits.