Long-term control of oscillations in differential equations

Abstract

Ordinary differential equations arise everywhere in science – Newton’s law in physics, N-body problems in astronomy and in molecular dynamics, engineering problems in robotics, population models in biology, and many more. Since their analytic solution can be obtained only in exceptional situations, one is usually restricted to numerical simulations and/or to qualitative investigations of the flow. This article reviews a recent technique the modulated Fourier expansion which permits to get insight into the long-term behaviour of numerical solutions of multivalue methods as well as of analytic solutions of highly oscillatory differential equations.