NEW APPROACHES FOR THE STUDY OF NONLINEAR OSCILLATORS

This paper presents a power series approach to accurately obtain the d amped separatrices for a class of unforced non-linear oscillators. Thi s in turn delineates the basins of attraction of two or more stable at tractors in the phase space. The method is illustrated by its applicat ions to the unforced Duffing - Holmes' and blacklash oscillators. Next , a novel semi-analytical integration scheme, called the phase space l inearization method (PSL) is developed to obtain stable and unstable p eriodic solutions of forced as well as unforced non-linear oscillators and also the damped separatrices. The performance of the proposed met hod has been tested against periodic solutions of three oscillators, n amely Ueda's, Duffing-Holmes' and Van der Polls oscillators, obtained using a fourth order Runge-Kutta method with a sufficiently small time step. Moreover, the separatrices obtained using the PSL method are co mpared with those obtained via the power series method as developed ea rlier. The issue of accumulation of error in the PSL method as against the fourth order Runge-Kutta scheme is also described numerically thr ough an example of a first order non-linear equation having closed for m solution. (C) 1998 Academic Press Limited.