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

Abstract

Abstract:In this paper we apply a new method of analysis of random behavior of chaotic systems based on the Prony decomposition. The generalized Prony spectrum (GPS) is used for quantitative description of a wide class of random functions when information about their probability distribution function is absent. The scaling properties of the random functions that keep their invariant properties on some range of scales help to fit the compressed function based on the Prony's decomposition. In paper [1] the first author (RRN) found the physical interpretation of this decomposition that includes the conventional Fourier decomposition as a partial case. It has been proved also that the GPS can be used for detection of quasi-periodic processes that are appeared usually in the repeated or similar measurements. A triple physical pendulum is used as a chaotic system to obtain a chaotic behavior of displacement angles with one, two and three positive Lyapunov's exponents (LEs). The chaotic behavior of these angles can be expressed in the form of amplitude-frequency response (AFR) that is extracted from the corresponding GPS and can serve as a specific "fingerprint" characterizing the random behavior of the triple-pendulum system studied. This new quantitative presentation of random data opens additional possibilities in classification of chaotic responses and random behaviors of different complex systems. PACS

Quantifying chaos of curvilinear beams via exponents

“…In those charts the zones of existence of regular, chaotic and bifurcation dynamics on the chosen planes of parameters are comparable with respect to their areas/magnitudes. Furthermore, the same observation holds even for a triple pendulum (see [11]). In the latter case we have an experimental rig and again the strongly nonlinear mechanical system exhibits rich non-linear dynamical phenomena.…”

“…The Prony decomposition and the Prony spectrum are used to describe quantitatively a wide class of random functions. In paper [11] a chaotic system exhibited by a triple physical pendulum with one, two and three positive Lyapunov's exponents was studied. Chaotic dynamics of the mentioned lumped mechanical systems was illustrated via amplitude-frequency response being extracted from the corresponding generalized Prony spectrum, and was used as a specific ''fingerprint'' characterizing the random behavior of the triple-pendulum system.…”

Design of a class of fractional‐order hyperchaotic multidirectional multi‐scroll attractors

“…While used as information encryption key, hyperchaotic systems have advantages of larger dynamic storage capacity, higher sensitivity, anti‐decipher, and anti‐interference. Several works on hyperchaotic systems have been done in the literature 28‐30 …”

“…Several works on hyperchaotic systems have been done in the literature. [28][29][30] The study of FO hyperchaos is few; one promising problem is to construct FOHCMS systems that may have potential application in security preserving. A recent result reported in Chen et al 31 gives some rules of generating FOHCMS attractors; SNFS functions are utilized to extend equilibrium points with index 2 in the original system.…”

Integrability analysis of chaotic and hyperchaotic finance systems

“…Over the last years, hyperchaotic generators have been a topic of active search. Except Rössler's system, many hyperchaotic models have been found, both in theoretical and in practical ways; see, for instance, recent papers [21][22][23][24][25][26][27][28], and references therein.…”