Contemporary Mathematics Volume 58, Part ill, 1987 PERTURBATION RESULTS ON THE LONG RUN BEHAVIOR OF NONLINEAR DYNAMICAL SYSTEMS Humberto Carrillo Calvet* ABSTRACT. The validity of the averaging approximations for ordinary differential equations in standard form is discussed. New results on the uniform validity over infinite i'htervals of time are presented. 1. INTRODUCTION. In this paper we will be concerned with differential equa- tions of the form (E) x = e: f(t,x,e:) where e: is a small positive parameter. The function f is going to be con- sidered oscillatory with respect to the variable t in a very general sense that includes periodicity as a particular case. Precisely, we will assume that · n n f: R x R x (O,co) -+ R is continuous, has a continuous partial derivative with respect to x and is almost periodic in t, uniformly with respect to (x,e:) in compact sets. Equations like (E) were first studied by N. Krylov and N. N. Bogolyubov who called them differential equations in &tand~d 6o~. There is a wide spectrum of dynamical systems models where such equations appear. They are very important in Mechanics and in more general systems that present nonlinear oscillation phenomena. For instance, the equation (1) x = f(x) + g(t/e:) with f: !J c: Rn + Rn and g a periodic vector valued function, models an autonomous dynamical system being acted on by a rapidly oscillating external forcing. Clearly the change of time scale '= t/e: reduces equation (1) to the standard form. Also, a system of n weakly coupled harmonic oscillators (2) uk + wk uk = e: fk(ul , ... ,uk,ul , ... ,uk) k=l,2, ... ,n 1980 Mathematics Subject Classification: 34C29, 34Cl5, 34D20, 34El0. * Partially supported by Consejo Nacional de Ciencia y Tecnologfa, Mexico. 1 © 1987 American Mathematical Society 0271-4132/87 $1.00 + $.25 per page
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