Detection of quasi-periodic processes in complex systems: how do we quantitatively describe their properties?

Abstract

It has been shown that in reality at least two general scenarios of data structuring are possible: (a) a self-similar (SS) scenario when the measured data form an SS structure and (b) a quasi-periodic (QP) scenario when the repeated (strongly correlated) data form random sequences that are almost periodic with respect to each other. In the second case it becomes possible to describe their behavior and express a part of their randomness quantitatively in terms of the deterministic amplitude-frequency response belonging to the generalized Prony spectrum. This possibility allows us to re-examine the conventional concept of measurements and opens a new way for the description of a wide set of different data. In particular, it concerns different complex systems when the 'best-fit' model pretending to be the description of the data measured is absent but the barest necessity of description of these data in terms of the reduced number of quantitative parameters exists. The possibilities of the proposed approach and detection algorithm of the QP processes were demonstrated on actual data: spectroscopic data recorded for pure water and acoustic data for a test hole. The suggested methodology allows revising the accepted classification of different incommensurable and self-affine spatial structures and finding accurate interpretation of the generalized Prony spectroscopy that includes the Fourier spectroscopy as a partial case.

Application of the generalized Prony spectrum for extraction of information hidden in chaotic trajectories of triple pendulum

“…See, for example the papers [6][7][8][9], where this decomposition was used for solution of different problems. The principal difference between the previous approaches and new one that was proposed in this paper is associated with the fact that Prony's decomposition has an additional and natural physical meaning that in the first time was stressed in paper [1]. This additional meaning lies in formation of a linear memory between successive (repeated) measurements and this observation attaches an additional importance to the Prony's decomposition in different practical applications.…”

“…In this paper we tried to solve this problem from the opposite end (when the distribution probability function corresponding to some random function is not known but the desired fitting function can be derived from the self-similar (scaling) hypothesis (17) and coincides with the Prony's decomposition corresponding to the simplest case (16). More complicated cases were considered in paper [1]. So, at the given stage one can try to connect the parameters corresponding to chaotic behavior with parameters describing the AFR of the found decomposition (20).…”

“…and form approximately the functional equation The solution of this functional equation was obtained earlier in [1] and can be expressed in the form …”

“…This scaling invariance implies that F ( ) can be the most probable candidate for the fitting of integrated curves having the same scaling property. The function considered in paper [1] can be used also as a competitive hypothesis. But in the given case considered it requires more number of decomposition parameters (K = 100) and because of this reason it is rejected.…”

“…One can remind two principles that will be useful for description of different random functions when information about their distribution probability functions is not available. One of the principles is based on self-similar properties of the long-time series considered [2] but another one, tightly associated with detection of quasi-periodic processes in repeated measurements [1], is useful when it is necessary to present a random function by means of the Prony decomposition. This decomposition includes the conventional Fourier transformation as a partial case.…”