Statistical Information of Low-Dimensional Processes Hidden in Random State Data

Abstract

For stochastically excited dissipative dynamical systems, the low-dimensional slowly varying processes act as the essential and simplified description of the apparent high-dimensional fast-varying processes (i.e., state variables). Deriving the statistical information of low-dimensional processes has a great significance, which inflects almost all the statistical information of concerned. This work is devoted to an equation-free, data-driven method, which starts from random state data, automatically extracts the slowly varying processes and automatically identifies its stationary probability density. The independent slowly varying processes are extracted by combining the identification of Lagrangian and Legendre transformations; the probability density is identified by the assumption of exponential form and the comparison with calculated data at lattices; both steps are implemented in the framework of linear regression. This method is universally valid for general nonlinear systems with arbitrary parameter values; for systems with heavy damping and/or strong excitations, it provides sparse results with high precision, while the results from stochastic averaging are incorrect even in function property.