Chaos-Induced Diffusion

Abstract

Though it is known for a long time that deterministic equations of motion can generate solutions which all have phenomenological properties of stochastic functions, this fact has been appreciated to full extent only very recently. At first the pseudo-stochastic character of such solutions was attributed to the nonlinear coupling of many degrees of freedom (thermal noise). With the advent of more and more powerful computers and the possibility of sufficiently accurate numerical integration it became clear that nonlinear equations with only a few degrees of freedom can also generate such chaotic solutions. This insight led to a deeper understanding of a number of dynamical systems, e.g., in the biological sciences [1], which could not be obtained via phenomenological descriptions including stochastic terms. Hence it seems advisable to investigate in general what phenomena may be generated by such nonlinear deterministic systems. In the following sections I will examine more closely discrete-time one-dimensional diffusion models.