New approaches for the study of non-linear oscillators

Abstract

This paper presents a power series approach to accurately obtain the damped separatrices for a class of unforced non-linear oscillators. This in turn delineates the basins of attraction of two or more stable attractors in the phase space. The method is illustrated by its applications to the unforced Duffing–Holmes and blacklash oscillators. Next, a novel semi-analytical integration scheme, called the phase space linearization method (PSL) is developed to obtain stable and unstable periodic solutions of forced as well as unforced non-linear oscillators and also the damped separatrices. The performance of the proposed method has been tested against periodic solutions of three oscillators, namely Ueda's, Duffing-Holmes and Van der Pol's oscillators, obtained using a fourth order Runge-Kutta method with a sufficiently small time step. Moreover, the separatrices obtained using the PSL method are compared with those obtained via the power series method as developed earlier. The issue of accumulation of error in the PSL method as against the fourth order Runge-Kutta scheme is also described numerically through an example of a first order non-linear equation having closed form solution.