Can We Work Around Numerical Methods? an Insight

Abstract

Introduction Self Sustaining Oscillators are of interest to researchers given the fact that they provide for excellent test beds to study the nuances and intricacies involved in typical oscillatory systems. Numerous such oscillations have been studied using traditional numerical methods in the past, contrary to the propositions of this article. Majority of study of van der Pol equation like oscillatory systems are directed towards deducing a stability criterion. This paper outlines a non-conventional study of the stability standards of the van der Pol Equation. The equation simulates a typical RLC circuit i.e. as a resistor for higher currents, but as a negative resistor for lower currents. Such a behavior is termed Relaxation Oscillation [Nanjundiah 1958]. The van der Pol equation may be defined as an ODE describing oscillations which amplifies smaller oscillations and dampens large oscillations on the other side. It may be obtained by differentiating the Rayleigh Equation and setting y = y’. Detailed analysis of van der Pol equation and its applicability may be found in [Wiggins 1990, Buonomo 1999, Kneubuehl and Kneubhyl 2001]. One of prevalent forms of the van der Pol equation may be given as [Buonomo 1999, Kneubuehl and Kneubhyl 2001]; y” – μ (1 – y) y’ + y = 0 (0)