Reproduction of Dynamical Systems with the Method of Principal Component Analysis

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[Abstract] Differential equations for Henon-Heiles' non-linear dynamical system are reproduced by using the principal component analysis (PCA) in twice. The adopted method is concretely proved to be capable of the empirical construction of dynamical systems from observed data sets. In this paper data sets are given by numerical integration of Henon-Heiles' original differential equations. In the case of the non-chaotic motion, the accuracy of three figures is obtained concerning the reproduced coefficients for the resulting differential equations. Then the reproduction is performed using various data sets including the case of chaotic motion. The accuracy of the reproduction seems to be strongly correlative with the energy of chaotic motion. Also, the influence of the overlapped white noise upon the data is investigated. The accuracy of the reproduction is rapidly lost under the condition of the overlapped white noise of the magnitude of 10^<-3> (＝1％ relative noise). The observational data should desirably have at least 3 significant figures (＝0.1％ relative observational errors) for the rigorous reproduction of the differential equations.

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