Semi-closed-form solutions of the van der Pol oscillator system

Abstract

Second order vector-valued nonlinear differential equations occurring in science and engineering have been considered which generally do not have closed-form solutions. Explicit incremental semi-analytical numerical solution procedures for nonlinear multiple-degree-of-freedom systems have been developed. Higher order equivalent differential equations were formulated and then subsequent values of vectors were updated using explicit Taylor series expansions. As the time-step tends to zero, the values of displacement and velocity are exact in the Taylor series expansions involving as many higher order derivatives as necessary. A typical second order differential equation considered was, the van der Pol oscillator. Further developments consisted of closed-form solutions of the van der Pol equation. What remains to be determined is the closed-form solution of displacement, which is being addressed. Further applications of the semi-analytical procedures to time-dependent systems should also include, time-independent equations that are differentiable in terms of other independent variables, such as partial differential equations that have many independent variables.