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A Statistical Analysis of Non-linear Equations Based on a Linear Combination of Generalized Moments
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Publication Date: 2004/12/01
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Nonlinear Problems
linearize, non-linear, statistics, logistic map, MMA,
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A moment matrix analysis (MMA) method can derive macroscopic statistical properties such as moments, response time, and power spectra of non-linear equations without solving the equations. MMA expands a non-linear equation into simultaneous linear equations of moments, and reduces it to a linear equation of their coefficient matrix and a moment vector. We can analyze the statistical properties from the eigenvalues and eigenvectors of the coefficient matrix. This paper presents (1) a systematic procedure to linearize non-linear equations and (2) an expansion of the previous work of MMA to derive the statistical properties of various non-linear equations. The statistical properties of the logistic map were evaluated by using MMA and computer simulation, and it is shown that the proposed systematic procedure was effective and that MMA could accurately approximate the statistical properties of the logistic map even though such a map had strong non-linearity.