When a set of non linear differential equations is investigated, most of the time there is no analytical solution and only numerial integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which these time variable is discrete. When only a numerical solution is researched, a fourth-order Runge-Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of differential equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small time step. In some cases, it may be of a great interest to possess difference equations with the same type of solutions as for the differential equations when the time step is sufficiently large. Non standard schemes as introduced by Mickens  allow to obtain more robust difference equations. In this paper, using such non standard scheme, we propose some difference equations as discrete analogues of the Rossler system for which it is shown that the dynamics is less dependent on the time step size than when a non standard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the Rossler system as long as the time step is less than the threshold value associated with the Nyquist criterion.
Document Object Identifier (DOI)
Letellier, C., Elaydi, S., Aguirre, L. A., & Alaoui, A. (2004). Difference equations versus differential equations, a possible equivalence for the Rössler system? Physica D: Nonlinear Phenomena, 195, 29-49. doi: 10.1016/j.physd.2004.02.007
Physica D: Nonlinear Phenomena