Stabilization of Unstable Periodic and Aperiodic Orbits of Chaotic Systems by Self-Controlling Feedback

Abstract The methods of stabilization of unstable periodic and aperiodic orbits of a strange attractor with the help of a small time-continuous perturbation are discussed. The perturbation is applied to the system in such a way that the desired periopdic or aperiodic orbits remain unperturbed. An experimental application of the methods can be carried out by a purely analogous technique without use of any computer.

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On Fuzzy Control of Chaotic Systems

Journal of Vibration and Control ◽

10.1177/1077546304041541 ◽

2004 ◽

Vol 10 (7) ◽

pp. 979-993 ◽

Cited By ~ 5

Author(s):

Ahmad M. Harb ◽

Issam A. Smadi

Keyword(s):

Mathematical Model ◽

Fuzzy Logic ◽

Fuzzy Control ◽

Power Systems ◽

Strange Attractor ◽

Chaotic Systems ◽

Chaotic Behavior ◽

Secure Communications ◽

Nonlinear Controllers ◽

Linear And Nonlinear

In this paper, we introduce the control of the strange attractor, chaos. Because of the importance of controlling undesirable behavior in systems. researchers are investigating the use of linear and nonlinear controllers, either to remove such oscillations (in power systems) or to match two chaotic systems (in secure communications). The idea of using the fuzzy logic concept for controlling chaotic behavior is presented. There are two good reasons for using fulzy control: first, there is no mathematical model available for the process; secondly. it can satisfy nonlinear control that can be developed empirically. without complicated mathematics. The two systems are well-known models so the first reason is not a big problem. and we can take advantage of the second reason.

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Simple Chaotic Systems with Specific Analytical Solutions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501165 ◽

2019 ◽

Vol 29 (09) ◽

pp. 1950116 ◽

Cited By ~ 3

Author(s):

Zahra Faghani ◽

Fahimeh Nazarimehr ◽

Sajad Jafari ◽

Julien C. Sprott

Keyword(s):

Analytical Solution ◽

Differential Equations ◽

Periodic Orbits ◽

Analytical Solutions ◽

Strange Attractor ◽

Chaotic Systems ◽

Analytic Solutions ◽

Unstable Periodic Orbits ◽

Chaotic Orbit ◽

Dynamical Properties

In this paper, a new structure of chaotic systems is proposed. There are many examples of differential equations with analytic solutions. Chaotic systems cannot be studied with the classical methods. However, in this paper we show that a system that has a simple analytical solution can also have a strange attractor. The main goal of this paper is to show examples of chaotic systems with a simple analytical solution that is unstable so that the chaotic orbit does not track it. We believe the structures presented here are new. Two categories of chaotic systems are described, and their dynamical properties are investigated. The proposed systems have analytic solutions that exist far from the equilibrium. Of course, all strange attractors are dense in unstable periodic orbits, but mostly the equations that describe these orbits are unknown and difficult to calculate. The analytical solutions provide examples where the orbits can be calculated despite their instability.

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Strange attractor existence for non-local operators applied to four-dimensional chaotic systems with two equilibrium points

Chaos An Interdisciplinary Journal of Nonlinear Science ◽

10.1063/1.5085440 ◽

2019 ◽

Vol 29 (2) ◽

pp. 023117 ◽

Cited By ~ 15

Author(s):

Emile F. Doungmo Goufo

Keyword(s):

Strange Attractor ◽

Chaotic Systems ◽

Equilibrium Points ◽

Local Operators ◽

Non Local

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A New System with a Self-Excited Fully-Quadratic Strange Attractor and Its Twin Strange Repeller

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421300470 ◽

2021 ◽

Vol 31 (16) ◽

Author(s):

Arthanari Ramesh ◽

Iqtadar Hussain ◽

Hayder Natiq ◽

Mahtab Mehrabbeik ◽

Sajad Jafari ◽

...

Keyword(s):

Lyapunov Exponents ◽

Strange Attractor ◽

Quadratic Forms ◽

Chaotic Systems ◽

Basin Of Attraction ◽

General Information ◽

Quadratic System ◽

Two Parameters ◽

System Variables ◽

New System

In nonlinear dynamics, the study of chaotic systems has attracted the attention of many researchers around the world due to the exciting and peculiar properties of such systems. In this regard, the present paper introduces a new system with a self-excited strange attractor and its twin strange repeller. The unique characteristic of the presented system is that the system variables are all in their quadratic forms; therefore, the proposed system is called a fully-quadratic system. This paper also elaborates on the study of the bifurcation diagram, the interpretation of Lyapunov exponents, the representation of basin of attraction, and the calculation of connecting curves as the employed method for investigating the system’s dynamics. The investigation of 2D bifurcation diagrams and Lyapunov exponents indicated in this paper can better recognize the system’s dynamics since they are plotted considering simultaneous changes of two parameters. Moreover, the connecting curves of the proposed system are calculated. The system’s connecting curves help identify the system’s different behaviors by providing general information about the nature of the flows.

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