IWCFTA2012 Keynote Speech I - Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits

2012 ◽  
Author(s):  
G.A. Leonov ◽  
N.V. Kuznetsov
Keyword(s):  
Dynamical Systems ◽  
Chaotic Attractor ◽  
Hidden Attractors ◽  
Keynote Speech ◽  
Hidden Chaotic Attractor

scholarly journals A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 564 ◽  
Author(s):  
Jesus Munoz-Pacheco ◽  
Ernesto Zambrano-Serrano ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Equilibrium System ◽  
Hidden Attractors ◽  
Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

scholarly journals Hidden Attractors in Discrete Dynamical Systems

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 616
Author(s):  
Marek Berezowski ◽  
Marcin Lawnik
Keyword(s):  
Dynamical Systems ◽  
Chaos Theory ◽  
Scientific Literature ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Hidden Attractors ◽  
Negative Effects ◽  
Chemical Reactors ◽  
Points Of View

Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.

Bifurcation structure of chaotic attractor in switched dynamical systems with spike noise

2012 ◽  
Vol 45 (6) ◽  
pp. 795-804 ◽  
Author(s):  
Akihito Matsuo ◽  
Hiroyuki Asahara ◽  
Takuji Kousaka
Keyword(s):  
Dynamical Systems ◽  
Chaotic Attractor ◽  
Bifurcation Structure

scholarly journals HIDDEN ATTRACTORS IN DYNAMICAL SYSTEMS. FROM HIDDEN OSCILLATIONS IN HILBERT–KOLMOGOROV, AIZERMAN, AND KALMAN PROBLEMS TO HIDDEN CHAOTIC ATTRACTOR IN CHUA CIRCUITS

2013 ◽  
Vol 23 (01) ◽  
pp. 1330002 ◽  
Author(s):  
G. A. LEONOV ◽  
N. V. KUZNETSOV
Keyword(s):  
Dynamical Systems ◽  
Numerical Methods ◽  
Automatic Control ◽  
Control Systems ◽  
Computational Procedure ◽  
Phase Locked Loops ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
Computational Point ◽  
Hidden Oscillations

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.

scholarly journals ON CONTROLLING A CHAOTIC SYSTEM

1991 ◽  
Vol 05 (22) ◽  
pp. 1489-1497 ◽  
Author(s):  
HAIM H. BAU
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Boundary Conditions ◽  
Chaotic Attractor ◽  
Bifurcation Structure ◽  
Small Perturbations ◽  
Chaotic Flow ◽  
Active Feedback ◽  
Active Feedback Control ◽  
At Will

The use of active (feedback) control to alter the bifurcation structure of dynamical systems is discussed and illustrated with an example. It is shown that with the use of a feedback controller effecting small perturbations in the boundary conditions, one can stabilize some of the otherwise non-stable orbits embedded in the chaotic attractor. The controller also can be used to destabilize stable flows or, in other words, to induce chaos in otherwise laminar (fully predictable), non-chaotic flow. Finally, the controller can be used to switch at will from one flow pattern to another.

scholarly journals Hidden attractors in dynamical systems

2016 ◽  
Vol 637 ◽  
pp. 1-50 ◽  
Author(s):  
Dawid Dudkowski ◽  
Sajad Jafari ◽  
Tomasz Kapitaniak ◽  
Nikolay V. Kuznetsov ◽  
Gennady A. Leonov ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Hidden Attractors

Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

2021 ◽  
Vol 31 (03) ◽  
pp. 2150047
Author(s):  
Liping Zhang ◽  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Qinsheng Bi
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Complex Dynamics ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Hidden Attractor ◽  
Hidden Attractors

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.

FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS

1994 ◽  
Vol 04 (04) ◽  
pp. 933-941 ◽  
Author(s):  
ANDREW L. KATZ ◽  
EARL H. DOWELL
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Stable And Unstable Manifolds ◽  
Detailed Explanation ◽  
Single Well ◽  
Unstable Manifolds ◽  
Parameter Values

The study of stable and unstable manifolds, and their intersections with each other, is a powerful technique for interpreting complex bifurcations of nonlinear systems. The escape phenomenon in the twin-well Duffing oscillator is one such bifurcation that is elucidated through the analysis of manifold intersections. In this paper, two escape scenarios in the twin-well Duffing oscillator are presented. In each scenario, the relevant manifold structures are examined for parameter values on either side of the escape bifurcation. Included is a description of the role of the hilltop saddle stable manifolds, which are known to separate the single well basins (should single well attractors exist). In each of the two bifurcation scenarios, it is shown through a detailed analysis of Poincaré maps that a homoclinic intersection of the manifolds of a specific period-3 saddle implies the destruction of the single well chaotic attractor. Although the Duffing oscillator is used to illustrate the ideas advanced here, it is thought that the approach will be useful for a variety of dynamical systems.

Sign in / Sign up

Export Citation Format

Share Document