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Eigen Analysis of Space Embedded Equation in Moment Vector Space for MultiDimensional Chaotic Systems
Hideki SATOH
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E96A
No.2
pp.600608 Publication Date: 2013/02/01 Online ISSN: 17451337
DOI: 10.1587/transfun.E96.A.600 Print ISSN: 09168508 Type of Manuscript: PAPER Category: Nonlinear Problems Keyword: chaos, GCM, MVE, nonlinear, dimension reduction,
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Summary:
Multihighdimensional chaotic systems were reduced to lowdimensional space embedded equations (SEEs), and their macroscopic and statistical properties were investigated using eigen analysis of the moment vector equation (MVE) of the SEE. First, the state space of the target system was discretized into a finite discrete space. Next, an embedding from the discrete space to a lowdimensional discrete space was defined. The SEE of the target system was derived using the embedding. Finally, eigen analysis was applied to the MVE of the SEE to derive the properties of the target system. The geometric increase in the dimension of the MVE with the dimension of the target system was avoided by using the SEE. The pdfs of arbitrary elements in the target nonlinear system were derived without a reduction in accuracy due to dimension reduction. Moreover, since the dynamics of the system were expressed by the eigenvalues of the MVE, it was possible to identify multiple steady states that cannot be done using numerical simulation. This approach can thus be used to analyze the macroscopic and statistical properties of multidimensional chaotic systems.

