Parametric Adaptive Control in Nonlinear Dynamical Systems

1998 ◽  
Vol 08 (11) ◽  
pp. 2215-2223 ◽  
Author(s):  
Jie Wang ◽  
Xiaohong Wang
Keyword(s):  
Adaptive Control ◽  
Chaotic Systems ◽  
Lyapunov Method ◽  
Nonlinear Dynamical Systems ◽  
Asymptotically Stable ◽  
Nonlinear Dynamical ◽  
System Parameters ◽  
Parametric Control ◽  
Nonlinear Chaotic System ◽  
Nonlinear Form

The paper is concerned with parametric adaptive control of continuous time chaotic systems. A method of parametric adaptive control is presented for a nonlinear chaotic system with multi-parameters. First, the system parameters are considered to be linear form in the adaptive control. Secondly, the Lyapunov method is used to prove parametric control equations are global asymptotically stable. Finally, the nonlinear form of the system parameter with uncertain noise is considered. It has been shown that the method in this paper is a very effective one to analyze parametric adaptive control for chaotic systems.

ON ADAPTIVE SYNCHRONIZATION AND CONTROL OF NONLINEAR DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (03) ◽  
pp. 455-471 ◽  
Author(s):  
CHAI WAH WU ◽  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Spread Spectrum ◽  
Communication Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Adaptive Synchronization ◽  
Adaptive Controllers ◽  
Nonlinear Dynamical ◽  
Two Systems ◽  
And Control

In this paper, we study the synchronization of two coupled nonlinear, in particular chaotic, systems which are not identical. We show how adaptive controllers can be used to adjust the parameters of the systems such that the two systems will synchronize. We use a Lyapunov function approach to prove a global result which shows that our choice of controllers will synchronize the two systems. We show how it is related to Huberman-Lumer adaptive control and the LMS adaptive algorithm. We illustrate the applicability of this method using Chua's oscillators as the chaotic systems. We choose parameters for the two systems which are orders of magnitude apart to illustrate the effectiveness of the adaptive controllers. Finally, we discuss the role of adaptive synchronization in the context of secure and spread spectrum communication systems. In particular, we show how several signals can be encoded onto a single scalar chaotic carrier signal.

Coexistence of Chaos with Hyperchaos, Period-3 Doubling Bifurcation, and Transient Chaos in the Hyperchaotic Oscillator with Gyrators

2015 ◽  
Vol 25 (04) ◽  
pp. 1550052 ◽  
Author(s):  
J. Kengne
Keyword(s):  
Integrated Circuit ◽  
Nonlinear Dynamical Systems ◽  
Piecewise Linear ◽  
Phase Portraits ◽  
Piecewise Linear Approximation ◽  
Transient Chaos ◽  
Frequency Spectra ◽  
Nonlinear Dynamical ◽  
System Parameters ◽  
Coexistence Of Attractors

In this paper, the dynamics of the paradigmatic hyperchaotic oscillator with gyrators introduced by Tamasevicius and co-workers (referred to as the TCMNL oscillator hereafter) is considered. This well known hyperchaotic oscillator with active RC realization of inductors is suitable for integrated circuit implementation. Unlike previous literature based on piecewise-linear approximation methods, I derive a new (smooth) mathematical model based on the Shockley diode equation to explore the dynamics of the oscillator. Various tools for detecting chaos including bifurcation diagrams, Lyapunov exponents, frequency spectra, phase portraits and Poincaré sections are exploited to establish the connection between the system parameters and various complex dynamic regimes (e.g. hyperchaos, period-3 doubling bifurcation, coexistence of attractors, transient chaos) of the hyperchaotic oscillator. One of the most interesting and striking features of this oscillator discovered/revealed in this work is the coexistence of a hyperchaotic attractor with a chaotic one over a broad range of system parameters. This phenomenon was not reported previously and therefore represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. A close agreement is observed between theoretical and experimental analyses.

COMPETITIVE MODES AND THEIR APPLICATION

2006 ◽  
Vol 16 (03) ◽  
pp. 497-522 ◽  
Author(s):  
WEIGUANG YAO ◽  
PEI YU ◽  
CHRISTOPHER ESSEX ◽  
MATT DAVISON
Keyword(s):  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Point Of View ◽  
Frequency Function ◽  
Mode Competition ◽  
Integration Scheme ◽  
Nonlinear Dynamical ◽  
Competitive Modes ◽  
Definition Of

We investigate nonlinear dynamical systems from the mode competition point of view, and propose the necessary conditions for a system to be chaotic. We conjecture that a chaotic system has at least two competitive modes (CM's). For a general nonlinear dynamical system, we give a simple, dynamically motivated definition of mode suitable for this concept. Since for most chaotic systems it is difficult to obtain the form of a CM, we focus on the competition between the corresponding modulated frequency components of the CM's. Some direct applications result from the explicit form of the frequency functions. One application is to estimate parameter regimes which may lead to chaos. It is shown that chaos may be found by analyzing the frequency function of the CM's without applying a numerical integration scheme. Another application is to create new chaotic systems using custom-designed CM's. Several new chaotic systems are reported.

scholarly journals Chaotic Systems and Their Recent Implementations on Improving Intelligent Systems

Economics ◽  
2015 ◽  
pp. 1167-1200
Author(s):  
Utku Köse ◽  
Ahmet Arslan
Keyword(s):  
Artificial Intelligence ◽  
Chaos Theory ◽  
Intelligent Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Research Effort ◽  
Nonlinear Dynamical ◽  
Main Research ◽  
Reference Guide

Chaos Theory is a kind of a scientific approach/research effort which is based on examining behaviors of nonlinear dynamical systems which are highly sensitive to their initial conditions. Currently, there are many different scientific studies based on the Chaos Theory and the related solution approaches, methods, or techniques for problems of this theory. Additionally, the theory is used for improving the introduced studies of different fields in order to get more effective, efficient, and accurate results. At this point, this chapter aims to provide a review-based study introducing recent implementations of the Chaos Theory on improving intelligent systems, which can be examined in the context of the Artificial Intelligence field. In this sense, the main research way is directed into the works performed or introduced mostly in years between 2008 and 2013. By providing a review-based study, the readers are enabled to have ideas on Chaos Theory, Artificial Intelligence, and the related works that can be examined within intersection of both fields. At this point, the chapter aims to discuss not only recent works, but also express ideas regarding future directions within the related implementations of chaotic systems to improve intelligent systems. The chapter is generally organized as a reference guide for academics, researchers, and scientists tracking the literature of the related fields: Artificial Intelligence and the Chaos Theory.

scholarly journals Stability of The Equilibrium of Nonlinear Dynamical Systems

2018 ◽  
Vol 71 (1) ◽  
pp. 71-80
Author(s):  
Irada A. Dzhalladova ◽  
Miroslava Růžičková
Keyword(s):  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Fractional Part ◽  
Stability Domain ◽  
Nonlinear Dynamical ◽  
Quadratic Lyapunov Functions ◽  
The Stability ◽  
Zero Equilibrium

Abstract The algorithm for estimating the stability domain of zero equilibrium to the system of nonlinear differential equations with a quadratic part and a fractional part is proposed in the article. The second Lyapunov method with quadratic Lyapunov functions is used as a method for studying such systems.

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