Persistence of attractors for one‐step discretization of ordinary differential equations

We consider numerical one‐step approximations of ordinary differential equations and present two results on the persistence of attractors appearing in the numerical system. First, we show that the upper limit of a sequence of numerical attractors for a sequence of vanishing time-steps is an attractor for the approximated system if and only if for all these time‐steps the numerical one‐step schemes admit attracting sets which approximate this upper limit set and attract with a uniform rate. Second, we show that if these numerical attractors themselves attract with a uniform rate, then they converge to some set if and only if this set is an attractor for the approximated system. In this case, we can also give an estimate for the rate of convergence depending on the rate of attraction and on the order of the numerical scheme.