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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. 1. Introduction. This paper continues a study of limit laws for large systems of weakly and randomly coupled equations, begun in (5). In many instances, the correct limiting (large system) behavior of the solutions of these equations can be guessed from an assumption of "local chaos": roughly, the assumption that the solutions of individual equations within the system are mutually independent. A consequence of this assumption will often be a specific conjecture about the limiting behavior of the system. Although the solutions are not, in fact, independent, intuition suggests that they are "nearly" so, and the resulting conjecture is quite often correct. The main result of this paper is about the behavior of a system of nonlinear differential equations consisting of large "homogeneous" subpopulations. The coefficients which couple these equations are random variables. The subpopulations are homogeneous in the sense that these random variables have a common distribution within each subpopulation. I will make an assumption of local chaos and derive its consequence, which is that the behavior of the entire system can be described arbitrarily well, as the number of equations grows to infinity, by the solutions to a small set of deterministic equations, one for each subpopulation. The conjecture, then, is a sort of law of large numbers for the system of differential equations; it is the purpose of this paper to prove that conjecture. If there are two subpopulations, then the associated deterministic equations form a two-dimensional system, and parameters of the full system can be chosen so that this reduced system has a globally stable limit cycle. By the theorem we present, the entire system must demonstrate a stable oscillation, with period and wave form predicted by the two-dimensional equation, provided that the full system is sufficiently large. Computer simulation nicely illustrates this application of the theorem. The theorem is about a system of differential equations containing a relatively

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