In the conventional integration of a continuous dynamical system, the interaction between the model and fixed-step algorithms may produce important numerical effects over the resulting discrete representation. Our results indicate that there are remarkably simple scaling laws connecting the relevant parameters of the system to that value of integration step capable of overflowing the calculations. Moreover, we have identified a new type of chaotic numerical instability, which appears as the step size approaches some critical value. This effect is accurately described by means of nonanalytical power laws characteristic of phase transition phenomena. Finally, it is shown that simple nonlocal replacements in the discrete constructions significantly reduce or eliminate some of these numerical instabilities. These discretization effects were tested in several nonlinear dynamical systems of physical importance.