Hybrid computer modelling of a stochastic nonlinear dynamic system

Abstract

The need for more accurate analysis of real physical systems has led to the use of stochastic modelling of such systems. Many of these formulations have been the result of an extension of ordinary differential equations to include a white noise excitation. It has been shown (see, e.g. [1–3]) that when the set of stochastic differential equations describing a particular system is linear, there are no difficulties in the interpretation of procedures for solving the equations. However, when the stochastic equations are nonlinear, computational as well as conceptual difficulties are encountered. The above ambiguities have been discussed by Mortensen [2] and Wong and Zakai [4]. While these studies clarify the conceptual aspects (demonstrating that at least two interpretations of the integral exist) they do not consider the practical problem of deciding which of these solutions is to be used when the integration is intended to describe a particular real physical system of interest. Furthermore, in cases where the integration is performed by computer there is the additional question: which calculus does the computer use, Ito or Stratonovich? In this paper, we discuss the problems associated with the application of analog/hybrid computers to obtain a description of a nonlinear stochastic system for a real process. These problems have already arisen in previous work by Bullin [5] and Bullin and Dukler [6] who used the hybrid computer to model turbulent diffusion as a stochastic process. The modern hybrid computer provides a particularly useful tool for stochastic modelling. Electronic noise generators are readily available which produce analog noise of specified statistical characteristics. Control circuitry in a hybrid computer makes it possible to solve any differential equation on the analog side a large number of time in sequence, each realization with a different random input. Through the interface and digital components of the hybrid computer a record is easily kept of each solution, the necessary book-keeping can be executed and statistical computations carried out digitally with high efficiency and speed. In such a modern machine 100,000 realization can be executed in seconds with the result that the statistical reliability of the resulting probability densities and moments of the solution are very high. Therefore, there is a need to understand the nature of the solution produced by the hybrid computer. In this study hybrid computer solutions of two nonlinear stochastic differential equations are compared to analytical solutions determined from both the Ito and Stratonovich calculus. It is shown that an analog computer can be programmed to produce either an Ito or Stratonovich solution or one that falls between the two values. The physically significant solution is shown to depend on the value of the cross correlation between the noise and the integrand.