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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Is it Possible to Replace the Probability Distribution Function Describing a Random Process by the Prony’s Spectrum (I)

Journal of Applied Nonlinear Dynamics 1(2) (2012) 173--194 | DOI:10.5890/JAND.2012.05.005

Raoul R. Nigmatullin

Department of Theoretical Physics, Kazan Federal University, Kremlevskaya str. 18, 420008 Russian Federation

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The new law that governs by the generation of frequencies ωk (k=1,2,...,K-1)for the strongly-correlated systems (having a memory) has been found. The generalization of the present idea is based on detailed analysis of the previous results obtained in paper [1] that were devoted to new solutions of the Prony’s problem. It was turned out that many complex systems with memory generate new set of frequencies based on frequencies that have been generated in the nearest past. For justification of this relationship we collected different data that confirm this statement. We created also a special mathematical program, which selects (based on some criteria) a desired hypothesis that is chosen from other six similar ones. For all available data considered there is an optimal hypothesis that describes the distribution of frequencies that follows from the recurrence relationship including in itself the neighboring frequencies. The found hypothesis provides the optimal fit of the random smoothed sequence with high accuracy (the relative error less that 10%) including also the fit of the remnant function.The physical interpretation of this law is given also. This “unexpected” discovery found for a wide class of the strongly-correlated systems with memory allows to replace the probability distribution function associated with some process by its Prony’s spectrum. From mathematical point of view it will help to obtain new solutions of the old Prony’s problem and replace also the Fourier spectrum containing usually the excess of artifact frequencies by the informative-significant band of frequencies obtained from new general law that, in turn, was found for the strongly-correlated systems.


This paper was written in the frame of the scientific research program that was accepted by Kazan Federal University for 2012 year “Dielectric spectroscopy and kinetics of complex systems”.


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