Explorations of the Phase-Space Linearization Method for Deterministic and Stochastic Nonlinear Dynamical Systems

Abstract

A new numeric-analytic phase-space linearization (PSL) schemefor a class of nonlinear oscillators with continuous vector fields isinvestigated in this study. The essence of the PSL method is to replacethe nonlinear vector field by a set of conditionally linear ones, eachvalid either over a short segment of the evolving trajectory or(equivalently) over a sufficiently small interval of time. This conceptmay be usefully exploited to arrive at certain explicit and implicitintegration schemes for analyses and simulations. The explicit schemes,which are found to have ready extensions to systems under stochasticinputs, are first numerically implemented for a few oft-used nonlineardynamical systems under (deterministic) sinusoidal inputs. An estimateof an upper bound to the local error in terms of the chosen time stepsize is provided. The explicit scheme of local linearization is nextextended to nonlinear oscillators under stochastic excitations, namelywhite noise processes, which are formal derivatives of one or acombination of Gauss–Markov processes. Since the PSL approach is todecompose the nonlinear operator into a set of linear operators, theprinciples of linear random vibration may be suitably exploited toarrive at a faster Monte-Carlo scheme for computing the responsestatistics, both in stationary and nonstationary regimes. A fewexamples, based on Ueda's and Duffing–Holmes' oscillators, arepresented and compared with exact solutions, whenever available, toverify the correctness and versatility of the proposed schemes.