On the analysis of pseudo-orbits of continuous chaotic nonlinear systems simulated using discretization schemes in a digital computer

This paper reports the existence of more than one pseudo-orbit when simulating continuous nonlinear systems using a digital computer in a set-up different from the ones normally seen in the literature, that is, in a set-up where the step-size is not varied, the discretization scheme is kept the same as well as the initial conditions. Taking advantage of the roundoff error, a simple but effective method to determine a lower bound error and the critical time for the pseudo-orbits is used and the connection to the maximum (positive) Lyapunov exponent is established considering the bit resolution and the computational platform used for the simulations. To illustrate the effectiveness of the method and problems of using discretization schemes for simulating continuous nonlinear systems in a digital computer, the well-known Lorenz equations, the Rossler hyperchaos system, Mackey-Glass equation and the Sprott A system are used. The method can help the user of such schemes to keep track of the reliability of numerical simulations.