Independent Component Analysis of Spatiotemporal Chaos

Abstract

Two types of spatiotemporal chaos exhibited by ensembles of coupled nonlinear oscillators are analyzed using independent component analysis (ICA). For diffusively coupled complex Ginzburg-Landau oscillators that exhibit smooth amplitude patterns, ICA extracts localized one-humped basis vectors that reflect the characteristic hole structures of the system, and for nonlocally coupled complex Ginzburg-Landau oscillators with fractal amplitude patterns, ICA extracts localized basis vectors with characteristic gap structures. Statistics of the decomposed signals also provide insight into the complex dynamics of the spatiotemporal chaos.KEYWORDS: coupled oscillators, spatiotemporal chaos, signal processing, feature extraction Spatiotemporal chaos (STC) arising from interaction between autonomous elements is ubiquitous in nonequilibrium dissipative systems such as fluid flows and chemical reactions.1-4 It is generally difficult to reduce the complex behavior of strongly coupled systems to the individual dynamics of the component elements, so we must employ collective modes of the system for description. For example, we employ various types of bases in the description of fluids, e.g. Fourier basis, wavelet basis, or eigenfunctions of linearized evolution equations, which represent flow modes such as waves, vortices, and convections. [1][2][3][4][5][6][7] In this case, the observer needs to subjectively choose which basis to use depending on the situation.On the other hand, there also exist methods of constructing basis functions objectively from statistical properties of the system by some criterion, without explicitly fixing them. A representative method is the principal component analysis (PCA) or the Karhunen-Loéve expansion, 5-17 which is a standard method in multivariate analysis. PCA has already been employed in analyzing STC, for example, to truncate its evolution equation or to estimate its degree of freedom.13-17 However, as we show below, when PCA is applied to STC, spatially delocalized bases are typically extracted, which are not appropriate for describing local field structures. Particularly, when the system has translational symmetry, it can be proven that PCA extracts the Fourier basis itself. Therefore, some statistical method that can objectively and compactly capture the complex field structures is desirable.In this letter, we apply independent component analysis (ICA) to STC. ICA is a recently developed method of statistical signal processing, which attempts to decompose observed mixed signals into maximally independent signals. [8][9][10][11][12] It is known that the ICA basis gives an information-theoretically efficient representation of multivariate signals. [8][9][10][11][12] We analyze STC exhibited by two types of coupled nonlinear oscillators, namely, locally (diffusively) coupled complex Ginzburg-Landau oscillators (LCGL), 1-4 and nonlocally coupled complex * Corresponding author: nakao@ton.scphys.kyoto-u.ac.jpGinzburg-Landau oscillators (NCGL), [18][19][20][21][22] and assess the utility of I...

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“…They further developed a new scheme to combine their improved ICA design with state feedback control to achieve chaos synchronization. Asano & Nakao [13] used ICA components to analyse the complex dynamics of two types of spatiotemporal chaos. For diffusively coupled complex Ginzburg-Landau oscillators, which exhibit smooth amplitude patterns, ICA was able to extract localized one-humped basis vectors reflecting the characteristic hole structures of the system.…”