An illustrative example of the Chirikov Criterion for stochastic instability

Numerical experiments have indicated that as a system moves further and further away from an integrable system a threshold is reached at which the motion of the system suddenly becomes quite ''random''. For practical applications it can be very important to know when this threshold, or stochastic instability will occur. An approximate criterion for calculating this threshold has been developed by Chirikov and by Rosenbluth. We will call this criterion the Chirikov Criterion. The purpose of this paper is pedagogical. I will present the Chirikov Criterion by way of an example for which it can be simply calculated. The particular system that I study can be written as a transformation and studied numerically with ease. I will present some results which show that in this case the Chirikov Criterion is reasonably successful. I will then discuss the limitations and shortcomings of the criterion. 3 figs., 1 tab