Almost Sure Stable Oscillations in a Large System of Randomly Coupled Equations

This paper is about limiting (large system) behavior of a set of differential equations with random coefficients. Under certain conditions the behavior of the entire system is well described by a small number of “prototype” equations, and these can be derived, heuristically, by applying a law of large numbers to the original system. An application of this theorem is the specification of a small number of parameters which guarantee that sufficiently large versions of the systems studied will oscillate with a predicted period and wave form.