The non-linear Van der Pol equation

Abstract

A two-dimensional dynamical system is defined by two coupled first order differential equations of the form : x = P(x,y), y = Q(x,y), d (.):= dt, (II) where P and Q are two functions of the variables x and y parametrised by the time independent variable t. The dynamical system (Tl) appears very often within several branches of science, such as biology, chemistry, astrophysics, mechanics, electronics, fluid mechanics, etc. The most important problem connected with the study of system (II) is the limit ·cycle. Stable limit-cycles are very important in science. They model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing. Of the countless examples that could be given, we mention only a few : The beating of a heart, chemical reactions that oscillate spontaneously, self-excited vibrations in bridges and aeroplane wings, etc. In each case, there is a standard oscillation of some preferred period, and amplitude. If the system is slightly perturbated, it always returns to the standard cycle. Limit-cycles are an inherently non-linear phenomena; they cannot occur in linear systems.