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

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.

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.

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.

Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors

2014 ◽  
Vol 47 (3) ◽  
pp. 5445-5454 ◽  
Author(s):  
N.V. Kuznetsov ◽  
G.A. Leonov
Keyword(s):  
Dynamical Systems ◽  
Hidden Attractors ◽  
Coexisting Attractors

Perpetual points and hidden attractors in dynamical systems

2015 ◽  
Vol 379 (40-41) ◽  
pp. 2591-2596 ◽  
Author(s):  
Dawid Dudkowski ◽  
Awadhesh Prasad ◽  
Tomasz Kapitaniak
Keyword(s):  
Dynamical Systems ◽  
Hidden Attractors ◽  
Perpetual Points

Hidden Attractors: new horizons in exploring dynamical systems

2021 ◽  
Vol 10 (4) ◽  
pp. 671-673
Author(s):  
Maaita Jamal-Odysseas
Keyword(s):  
Dynamical Systems ◽  
Hidden Attractors ◽  
New Horizons
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