Transformation theory as applied to the solutions of non-linear differential equations of the second order

Abstract

This paper describes an approach to finding the solutions of second-order non-linear differential equations with a periodic forcing term. The mapping procedure based on the transformation theory of differential equations is found to be useful for this purpose. Иn this theory a fixed point of the mapping corresponds to a periodic solution. The bahaviour of invariant curves of the mapping in the phase plane reveals the global aspect of the steady-state response. An almost periodic solution is represented by an invariant closed curve on which successive images of the mapping keep on moving. Иn this paper the fundamental theory of the method is explained and then applied to obtaining the solutions of non-linear differential equations. Two examples are discussed in particular, i.e. Duffing's equation and van der Pol's equation with a forcing term. This method, when combined with the use of computers, provides an effective means of finding various types of solutions which may occur depending on the parameters of those equations.