Singular Perturbation Solutions of Noisy Systems

Abstract

Recent work on singular perturbation solutions that persist in the presence of noise is described. Two different settings are considered: small deviation theory in quasi-static problems, where there are small amplitude but highly irregular perturbations, and averaging problems where there are ergodic stochastic perturbations. In the first case, it is shown that quasi-static approximations can be valid when the underlying problem experiences small deviation perturbations in problems that are stable under persistent disturbances. In the second, averaging principles are described for certain dynanical systems in Hilbert spaces that include applications to a wide variety of initial-boundary value problems for partial differential equations and for Volterra integral equations. These methods are applied here to four problems arising in applications. Key words, singular perturbation methods, stochastic integral equations AMS subject classifications. 34E15, 35D35, 60H15, 60H20 The persistence of a mathematical design when it is placed in a noisy environment gives some evidence about the stability or robustness to be expected in the realized system. For example, if a designed system has the form of a system of differential equations whose solutions describe the process, study of the process resulting when the design system is perturbed by random data is useful. These noisy perturbations can take various forms. We consider two here. The first is unstructured, small amplitude noise in systems that have a certain kind of stability, and the second allows ergodic stochastic deviations from which we can obtain useful averaged systems. The novelty here lies in several directions. First, the persistence of a quasi-static manifold establishes the usefulness of quasi-static state approximations in systems that are not even structurally stable. The application described here is a perturbation of Lorenz’s system, and it investigates the reliability of closure assumptions. The result is useful also in studies of a variety of problems that possess some gradient structure. Second, our work on dynamical systems in Hilbert space that are perturbed by ergodic jump processes, while close to the work of others, presents a clean set of results similar to the law of large numbers and to the central limit theorem that facilitate numerical simulation. Finally, our work on Volterra integral equations enables us to study a variety of renewal processes and dynamical systems with random time delays. Careful mathematical treatments of these problems and extensive citations to the large literature on them are presented elsewhere [1]-[4]. Singular perturbation methods are often used to obtain approximate solutions to problems that involve a variety of time’ and space scales. We consider this in two important cases: quasi-static state problems and averaging problems. First, we consider quasi-static state approximations for ideal systems and determine how they will respond to noisy perturbations. Specifically, we give conditions under which the quasi-static state approximation will be valid even in the presence of small amplitude, but highly irregular perturbations. The results are based on a direct Received by the editors June 26, 1993; accepted for publication (in revised form) May 27, 1994. This research was supported in part by National Science Foundation grant DMS9206677. Department of Mathematics and Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824 (FCtt0PPElg0msu.edu). 544 D ow nl oa de d 12 /3 1/ 12 to 1 28 .1 48 .2 52 .3 5. R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp :// w w w .s ia m .o rg /jo ur na ls /o js a. ph p SINGULAR PERTURBATION SOLUTIONS OF NOISY SYSTEMS 545 modification of the concept of stability under persistent disturbances due to Malkin [1]. The essential idea here is that a proper minimum of a potential function corresponds to a stable equilibrium of an associated gradient system. If the gradient system is perturbed by small amplitude noise, the potential function does not change much, and the minimization process described by the gradient system moves the system to near the old minimum. If the old minimum were the quasi-static state approximation to some problem, then we see that it remains a meaningful approximation for the perturbed system. Second, we consider systems that are subjected to highly oscillatory ergodic stochastic perturbations. Specifically, we give conditions under which an averaging result similar to Bogoliuboff’s averaging principle can be used. Some of the averaging results involve dynamical systems in abstract spaces. These are motivated by the following two examples; the first comes from studies of bacterial growth in fluctuating environments [2], and the second from diffusion in random media [3]: 0-7 +V" b x,-,e co u -0, where b is a random field and where the data D and b are smooth functions and y is a jump Markov process that is ergodic in some probability space Y. The first equation includes the Lie differential associated with a nonlinear stochastic system (namely, its characteristic equations) and with random advection. The second equation describes diffusion in a medium in which the diffusivity and the drift are themselves random variables. We will return to these two examples later to describe details of their analysis. Finally, we consider Volterra integral equations of the form

DOI: 10.1137/S0036139993269229

Cite this paper

@article{Hoppensteadt1995SingularPS, title={Singular Perturbation Solutions of Noisy Systems}, author={Frank C. Hoppensteadt}, journal={SIAM Journal of Applied Mathematics}, year={1995}, volume={55}, pages={544-551} }