Linear stochastic models of nonlinear dynamical systems
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
We investigate in this work the validity of linear stochastic models for nonlinear dynamical systems. We exploit as our basic tool a previously proposed Rayleigh-Ritz approximation for the effective action of nonlinear dynamical systems started from random initial conditions. The present paper discusses only the case where the PDF-Ansatz employed in the variational calculation is "Markovian", i.e. is determined completely by the present values of the moment-averages. In this case we show that
... e Rayleigh-Ritz effective action of the complete set of moment-functions that are employed in the closure has a quadratic part which is always formally an Onsager-Machlup action. Thus, subject to satisfaction of the requisite realizability conditions on the noise covariance, a linear Langevin model will exist which reproduces exactly the joint 2-time correlations of the moment-functions. We compare our method with the closely related formalism of principal oscillation patterns (POP), which, in the approach of C. Penland, is a method to derive such a linear Langevin model empirically from time-series data for the moment-functions. The predictive capability of the POP analysis, compared with the Rayleigh-Ritz result, is limited to the regime of small fluctuations around the most probable future pattern. Finally, we shall discuss a thermodynamics of statistical moments which should hold for all dynamical systems with stable invariant probability measures and which follows within the Rayleigh-Ritz formalism.