A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems

Abstract

A new implicit local linearization method termed as the locally transversal linearization (LTL) is developed in the present paper for a numeric‐analytical integration of nonlinear dynamical systems under deterministic and/or stochastic excitations. The LTL procedure is developed in such a way that, at a chosen time‐instant, the LTL solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross‐section in the state‐space where the solution vector needs to be obtained. In general, construction of the LTL equation is non‐unique and thus it is generally needed to substitute the conditionally linear LTL‐based solution into the given nonlinear ordinary differential equation (ODE) at a specified time‐instant to ensure a transversal intersection. In the present study, however, it is demonstrated that a transversal intersection at a given time‐instant may as well be ensured by just choosing a few specific forms of the LTL equation, thereby eliminating the need for substitution. This new variant of LTL is particularly suitable for discretization of stochastic differential equations (SDEs) in that one simultaneously avoids dealing with formal derivatives of Wiener processes while satisfying the nonlinear SDE at discrete time‐points. The stochastic LTL procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous (but not necessarily differentiable) vector fields under additive and/or multiplicative excitations. A specific advantage of the LTL method is that for a class of response regime, herein referred to as the phase‐independent regime, it can obtain an accurate description of response irrespective of the choice of the time‐step and without a propagation of local errors. Presently, the method has been numerically illustrated for strong solutions of the hardening Duffing oscillator under deterministic (sinusoidal) or additive stochastic (white noise) excitations or their combination. Limited numerical results on periodic solutions, chaos with or without noise and stochastic resonance are included in this study. The results presented are indicative of the sample path accuracy, consistency, stochastic numerical stability and a partial insensitivity to time‐step sizes of the proposed method.