Localization of hidden Chuaʼs attractors

Alexeeva

2015

Entropy

In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu-Morioka system.

Section: Theoremmentioning

confidence: 99%

Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System

Alexeeva

2015

Entropy

“…Later in studying applied systems, hidden chaotic attractors Leonov et al, 2010c;Bragin et al, 2011;Leonov et al, 2011c;Leonov et al, , 2012aLeonov & Kuznetsov, 2013a, 2013b were also discovered, further the efforts will be focused on the construction of effective methods for the analysis of hidden periodic oscillations and chaotic attractors.…”

Section: Is Stable In the Large (Ie A Zero Solution Of System (98) mentioning

confidence: 99%

“…However, there are chaotic attractors of another type: hidden chaotic attractors, for which the possibility of such simple computational approach turn out to be highly limited. In 2010, for the first time, a hidden chaotic attractor was discovered Bragin et al, 2011;Kuznetsov et al, 2011a;Leonov et al, 2011cLeonov et al, , 2012a] in Chua's circuit, described by three-dimensional dynamical system. Note that Chua himself, in analyzing various cases of attractors' existence in Chua's circuit [Chua & Lin, 1990] did not admit the existence of hidden attractor (discovered later) in the circuit.…”

Section: Hidden Attractor In Chua's Circuitsmentioning

confidence: 99%

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Int. J. Bifurcation Chaos

2013

741

409

From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors.At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes.Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit.This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods. †

“…In studies by (Kuznetsov et al, , 2011Bragin et al, 2011;Leonov et al, 2011d;Kuznetsov et al, 2011;Leonov et al, 2011c;Vagaitsev, 2012;Kuznetsov et al, 2013;Kuznetsov, 2013b,c,a, 2014), chaotic hidden oscillations, acting as hidden attractors, were discovered in the Chua circuit.…”

mentioning

confidence: 99%

“…Several studies have been devoted to these frictional oscillations (Hensen, 2002;Hensen and van de Molengraft, 2002;Juloski et al, 2005;Mallon, 2003;Mallon et al, 2006; Olsson, 1996; Olsson andAstrom, 1996, 2001;Putra, 2004;Putra andNijmeijer, 2003, 2004;van de Wouw et al, 2005;Al-Bender et al, 2004;Batista and Carlson, 1998), as wear and damage to various mechanical systems are often caused by these oscillations. Under certain parameters, the so called hidden oscillations (Leonov et al, 2011d;Bragin et al, 2011;Leonov and Kuznetsov, 2013b) may emerge -oscillations in which the basin of attraction does not intersect with small neighborhoods of equilibrium states. Numerical simulation of complex nonlinear dynamical systems is now possible thanks to the development of computer modeling technology.…”

mentioning

confidence: 99%

Drilling Systems: Stability and Hidden Oscillations

Kiseleva

Discontinuity and Complexity in Nonlinear Physical Systems

et al. 2013

Esitetään Jyväskylän yliopiston informaatioteknologian tiedekunnan suostumuksella julkisesti tarkastettavaksi yliopiston Agora-rakennuksen Delta-salissa joulukuun 16. päivänä 2013 kello 12.Academic dissertation to be publicly discussed, by permission of the Faculty of Information Technology of the University of Jyväskylä, in building Agora, auditorium Delta, on December 16, 2013 at 12 o'clock noon. UNIVERSITY OF JYVÄSKYLÄ JYVÄSKYLÄ 2013Oscillations This work is devoted to the study of electromechanical models of induction motorpowered drilling systems. This is a current issue, as drilling equipment failures cause significant time and expenditure losses for drilling companies. Although there are many papers devoted to the investigation of these systems, equipment failures still occur frequently in the drilling industry. In this study, we continue investigations begun by researchers from the Eindhoven University of Technology, who introduced an experimental model of a drilling system. The model consists of two discs connected to each other by a steel string that experiences purely torsional deformation. The upper disc is connected to the driving part. The lower disc, representing the bottom end of the drill-string, experiences friction torque caused mainly by interaction with the shale. The key idea of the present study that distinguishes it from the previous model was the introduction of more complex equations of the driving part, in particular, considerations of the induction motor.Towards this end, two new mathematical models are considered. The first is a simplified one, its prototype being an electric hand drill. In this case it is assumed that the drill string is absolutely rigid and the friction torque acting on the lower part of the drill-string has asymmetric characteristics of the Coulomb type. The qualitative analysis of this model made it possible to obtain parameters on permissible loads (i.e., permissible values of the friction torque) in case the system remained in operational mode after the shale's type changes. Using computer modeling, analysis of sudden load appearance was also performed.The second mathematical model focuses on the torsional deformation of the drill during operation. For the friction torque with Coulomb type asymmetric characteristic, local analysis of the system is provided. The author carried out computer modeling of the friction model created by the researches at the University of Eindhoven. In this model, we found an interesting effect represented by hidden stick-slip oscillations.The results of the study have been published in 11 papers.