Simultaneous occurence of low-dimensional chaos and colored random noise in nonlinear physical systems

Abstract

Recent efforts in the time series analysis of nonlinear data have been focused upon the search for low-dimensional chaos in a wide variety of physical systems, both driven-dissipative and Hamiltonian. Unfortunately many experimental measurements contain a large component of white and/or “colored random noise” (CRN), e.g. their power spectra behave approximately as ƒ −α, for α=const, with random Fourier phases. Herein we discuss how colored random noise may arise primarily from nonlinear, deterministic processes, i.e. how certain chaotic physical systems can exhibit large-scale fractal stochastic (Brownian) motions (essentially CNR) which may either be erroneously interpreted as experimental noise or be confused with the differentiable dynamics itself. We suggest a new physical perspective, together with data analysis procedures, which allow for the separation and simultaneous study of both the (small scale) differentiable dynamics and the (large scale) stochastic (CRN) dynamics in certain chaotic systems. In order to illustrate our perspective we analyze numerical simulations of fluid particle motions for a particular example problem, the Arnol'd-Beltrami-Childress (ABC) flows. The implications of our approach on the analysis of data in a wide variety of laboratory and field investigations are also discussed, including large and meso-scale geophysical fluid dynamical motions and oceanic surface waves.