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

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.

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

Are Perpetual Points Sufficient for Locating Hidden Attractors?

2017 ◽  
Vol 27 (03) ◽  
pp. 1750037 ◽  
Author(s):  
Fahimeh Nazarimehr ◽  
Batool Saedi ◽  
Sajad Jafari ◽  
J. C. Sprott
Keyword(s):  
Nonlinear Dynamics ◽  
Hidden Attractors ◽  
Perpetual Points

Perpetual Points (PPs) have been introduced as an interesting new topic in nonlinear dynamics, and there is a conjecture that these points can be used to find hidden attractors. This note demonstrates some examples where PPs cannot locate their 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.

Limitation of Perpetual Points for Confirming Conservation in Dynamical Systems

2015 ◽  
Vol 25 (13) ◽  
pp. 1550182 ◽  
Author(s):  
Sajad Jafari ◽  
Fahimeh Nazarimehr ◽  
J. C. Sprott ◽  
Seyed Mohammad Reza Hashemi Golpayegani
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Perpetual Points

Perpetual Points (PPs) have been introduced as an interesting new topic in nonlinear dynamics, and there is a hypothesis that these points can determine whether a system is dissipative or not. This paper demonstrates that this hypothesis is not true since there are counterexamples. Furthermore, we explain that it is impossible to determine dissipation of a system based only on the structure of the system and its equations.

scholarly journals Existence of Perpetual Points in Nonlinear Dynamical Systems and Its Applications

2015 ◽  
Vol 25 (02) ◽  
pp. 1530005 ◽  
Author(s):  
Awadhesh Prasad
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Critical Points ◽  
Nonlinear Dynamical Systems ◽  
Transient Dynamics ◽  
Coexisting Attractors ◽  
Nonlinear Dynamical ◽  
New Class ◽  
Perpetual Points ◽  
Bifurcation Behavior

A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains nonzero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection behavior. These points also show the bifurcation behavior as the parameters of the system vary. These perpetual points are useful for locating the hidden oscillating attractors as well as coexisting attractors. Results show that these points are important for a better understanding of transient dynamics in the phase space. The existence of these points confirms whether a system is dissipative or not. Various examples are presented, and the results are discussed analytically as well as numerically.

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