Higher-order averaging for nonperiodic systems

Abstract

In this paper, a recently developed improved averaging method for periodic nonlinear systems of ordinary differential equations (ODEs) is extended to a wide class of nonperiodic ODE systems of nonlinear type. The paper presents the first systematic treatment of Nth-order averaging for nonperiodic systems of the present degree of generality. Our main results are: (i) a basic existence, uniqueness, and approximation theorem for Nth-order averaging for the latter systems, including explicit, rigorous error bounds expressed in terms of suitable order functions; (ii) a proof that our (N+1)st error estimate is smaller (o(1)) than the corresponding Nth-order estimate when the relevant perturbation parameter ε is sufficiently small (this is not obvious, since our error bounds are not generally proportional to powers of ε); (iii) a proof that (ii) holds when the exact order functions are replaced by appropriate upper bounds which are simpler to calculate in practice. For the example of a linear oscillator with time-dependent friction, it is shown that our second-order averaging estimates are generally better than those given by the approach of Sanders and Verhulst. Nth-order averaging for a large class of C∞ quasiperiodic systems is also discussed, generalizing earlier work of Perko.