Phase-space linearization for non-linear oscillators: Deterministic and stochastic systems

A new and efficient semi-analytical phase-space linearization (PSL) scheme for a class of non-linear oscillators is developed in this paper. The metho d is based on replacement of the non-linear vector held by a set of linear ones, each valid over a short segment of the evolving trajectory or over su fficiently small interval of time. Based on this concept, a few explicit an d implicit integration schemes are first proposed and applied to a class of low-dimensional non-linear dynamical systems to accurately determine their response trajectories. This approach of local linearization is further ext ended to non-linear oscillators excited by formal derivatives of one or a c ombination of Gauss-Markov processes. Since the present methodology reduces the non-linear operator by a set of linear operators, it is also demonstra ted that the principles of linear random vibration may be suitable exploite d to arrive at a faster and semi-analytical Monte-Carlo scheme for computin g the response statistics, both in stationary and non-stationary regimes. L imited examples are presented and compared with exact solutions whenever av ailable, to illustrate the efficiency and versatility of the proposed schem es. (C) 2000 Academic Press.