Control Lorenz System with Vibration Estimation with Averaging Method

2010 ◽  
Vol 43 ◽  
pp. 36-39
Author(s):  
Chun Zhou
Keyword(s):  
Chaotic Systems ◽  
Inverted Pendulum ◽  
Lorenz System ◽  
Stable Equilibrium ◽  
Averaging Method ◽  
Control Method ◽  
Initial Conditions ◽  
Vibrational Control ◽  
Extreme Sensitivity ◽  
Sensitivity To Initial Conditions

The vibrational control theory stems from the well-known of stabilization of the upper unstable equilibrium position of the inverted pendulum having suspension point vibration along the vertical line with amplitude as small as desired and a frequency reason high. Chaotic phenomena have been found in many nonlinear systems including continuous time and discrete time. The chaotic systems are characterized by their extreme sensitivity to initial conditions, nonperiodic and boundary. The trajectories start even from close initial states will diverge from each other at an exponential rate as time goes. The vibrational control method was applied to Lorenz system. The effect of the control can be estimated with the APAZ method. It was showed that vibrational control brought the controlled Lorenz system to stable equilibrium with appropriate parameters. Numerical simulation demonstrated validity of the proposed method.

scholarly journals Nonlinear control of chaotic systems:A switching manifold approach

2000 ◽  
Vol 4 (4) ◽  
pp. 257-267 ◽  
Author(s):  
Jin-Qing Fang ◽  
Yiguang Hong ◽  
Huashu Qin ◽  
Guanrong Chen
Keyword(s):  
Nonlinear Control ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Control Method ◽  
Initial Conditions ◽  
Design Strategy ◽  
Control Law ◽  
The Lorenz System ◽  
Bang Bang Control ◽  
Nonlinear Control Law

In this paper, a switching manifold approach is developed for nonlinear feed-back control of chaotic systems. The design strategy is straightforward, and the nonlinear control law is the simple bang–bang control. Yet, this control method is very effective; for instance, several desired equilibria can be stabilized by using one control law with different initial conditions. Its effectiveness is verified by both theoretical analysis and numerical simulations. The Lorenz system simulation is shown for the purpose of illustration.

A New Image Encryption Scheme Using Dual Chaotic Map Synchronization

2020 ◽  
Vol 18 (1) ◽  
Keyword(s):  
Image Encryption ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Chaotic Map ◽  
Security Applications ◽  
The Common ◽  
And Diffusion ◽  
Confusion And Diffusion ◽  
Sensitivity To Initial Conditions ◽  
Diffusion Properties

Chaotic systems behavior attracts many researchers in the field of image encryption. The major advantage of using chaos as the basis for developing a crypto-system is due to its sensitivity to initial conditions and parameter tunning as well as the random-like behavior which resembles the main ingredients of a good cipher namely the confusion and diffusion properties. In this article, we present a new scheme based on the synchronization of dual chaotic systems namely Lorenz and Chen chaotic systems and prove that those chaotic maps can be completely synchronized with other under suitable conditions and specific parameters that make a new addition to the chaotic based encryption systems. This addition provides a master-slave configuration that is utilized to construct the proposed dual synchronized chaos-based cipher scheme. The common security analyses are performed to validate the effectiveness of the proposed scheme. Based on all experiments and analyses, we can conclude that this scheme is secure, efficient, robust, reliable, and can be directly applied successfully for many practical security applications in insecure network channels such as the Internet

Comparison of Times of Synchronization of Chaotic Burke-Shaw Attractor with Active Control and Integer and Optimum Fractional-Order Pecaro Carroll Method

2021 ◽  
Author(s):  
Ali Durdu ◽  
Yılmaz Uyaroğlu
Keyword(s):  
Active Control ◽  
Fractional Order ◽  
Chaotic Attractor ◽  
Secure Communication ◽  
Chaotic Systems ◽  
Control Method ◽  
Initial Conditions ◽  
Data Communication ◽  
Experimental Results ◽  
Active Control Method

Abstract Many studies have been introduced in the literature showing that two identical chaotic systems can be synchronized with different initial conditions. Secure data communication applications have also been made using synchronization methods. In the study, synchronization times of two popular synchronization methods are compared, which is an important issue for communication. Among the synchronization methods, active control, integer, and fractional-order Pecaro Carroll (P-C) method was used to synchronize the Burke-Shaw chaotic attractor. The experimental results showed that the P-C method with optimum fractional-order is synchronized in 2.35 times shorter time than the active control method. This shows that the P-C method using fractional-order creates less delay in synchronization and is more convenient to use in secure communication applications.

scholarly journals Sources of uncertainty in deterministic dynamics: an informal overview

2011 ◽  
Vol 369 (1956) ◽  
pp. 4705-4729 ◽  
Author(s):  
Ian Stewart
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Sources Of Uncertainty ◽  
Deterministic Systems ◽  
Coin Tossing ◽  
Random Switching ◽  
Butterfly Effect ◽  
Basin Boundaries ◽  
Sensitivity To Initial Conditions

The discovery of chaotic dynamics implies that deterministic systems may not be predictable in any meaningful sense. The best-known source of unpredictability is sensitivity to initial conditions (popularly known as the butterfly effect), in which small errors or disturbances grow exponentially. However, there are many other sources of uncertainty in nonlinear dynamics. We provide an informal overview of some of these, with an emphasis on the underlying geometry in phase space. The main topics are the butterfly effect, uncertainty in initial conditions in non-chaotic systems, such as coin tossing, heteroclinic connections leading to apparently random switching between states, topological complexity of basin boundaries, bifurcations (popularly known as tipping points) and collisions of chaotic attractors. We briefly discuss possible ways to detect, exploit or mitigate these effects. The paper is intended for non-specialists.

SYNCHRONIZATION STABILITY ANALYSIS OF THE CHAOTIC RÖSSLER SYSTEM

1996 ◽  
Vol 06 (11) ◽  
pp. 2153-2161 ◽  
Author(s):  
QINGXIAN XIE ◽  
GUANRONG CHEN
Keyword(s):  
Initial Conditions ◽  
Sufficient Conditions ◽  
General Situation ◽  
Wide Range ◽  
Rossler System ◽  
Extreme Sensitivity ◽  
System Synchronization ◽  
Rössler System ◽  
Type Of Stability ◽  
Sensitivity To Initial Conditions

In this paper we show, both analytically and experimentally, that the Rössler system synchronization is either asymptotically stable or orbitally stable within a wide range of the system key parameters. In the meantime, we provide some simple sufficient conditions for synchronization stabilities of the Rössler system in a general situation. Our computer simulation shows that the type of stability of the synchronization is very sensitive to the initial values of the two (drive and response) Rössler systems, especially for higher-periodic synchronizing trajectories, which is believed to be a fundamental characteristic of chaotic synchronization that preserves the extreme sensitivity to initial conditions of chaotic systems.

Generalized Predictive Control of Discrete-Time Chaotic Systems

1998 ◽  
Vol 08 (07) ◽  
pp. 1591-1597 ◽  
Author(s):  
Kwang-Sung Park ◽  
Jin-Bae Park ◽  
Yoon-Ho Choi ◽  
Tae-Sung Yoon ◽  
Guanrong Chen
Keyword(s):  
Predictive Control ◽  
Discrete Time ◽  
Control Method ◽  
Initial Conditions ◽  
Generalized Predictive Control ◽  
System Sensitivity ◽  
Initial Sensitivity ◽  
Armax Model ◽  
Time Systems ◽  
Sensitivity To Initial Conditions

A generalized predictive control method based on an ARMAX model is suggested for chaos control in discrete-time systems. Both control performance and system sensitivity to initial conditions of this approach are compared with the conventional model-referenced adaptive control via numerical simulations. Simulation results show that this controller yields faster settling time, more accurate target tracking, and less initial sensitivity.

GENERALIZED (LAG, ANTICIPATED AND COMPLETE) PROJECTIVE SYNCHRONIZATION IN TWO NONIDENTICAL CHAOTIC SYSTEMS WITH UNKNOWN PARAMETERS

2012 ◽  
Vol 26 (16) ◽  
pp. 1250121
Author(s):  
XINGYUAN WANG ◽  
LULU WANG ◽  
DA LIN
Keyword(s):  
Adaptive Control ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Control Method ◽  
Projective Synchronization ◽  
Unknown Parameters ◽  
Chen System ◽  
Control Approach ◽  
Response System ◽  
Hyperchaotic Lorenz System

In this paper, a generalized (lag, anticipated and complete) projective synchronization for a general class of chaotic systems is defined. A systematic, powerful and concrete scheme is developed to investigate the generalized (lag, anticipated and complete) projective synchronization between the drive system and response system based on the adaptive control method and feedback control approach. The hyperchaotic Chen system and hyperchaotic Lorenz system are chosen to illustrate the proposed scheme. Numerical simulations are provided to show the effectiveness of the proposed schemes. In addition, the scheme can also be extended to research generalized (lag, anticipated and complete) projective synchronization between nonidentical discrete-time chaotic systems.

DIVERSE STRUCTURE SYNCHRONIZATION OF FRACTIONAL ORDER HYPER-CHAOTIC SYSTEMS

2013 ◽  
Vol 27 (11) ◽  
pp. 1350034 ◽  
Author(s):  
XING-YUAN WANG ◽  
GUO-BIN ZHAO ◽  
YU-HONG YANG
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Control Method ◽  
Rossler System ◽  
Rössler System ◽  
Predictor Corrector ◽  
Fractional Orders ◽  
Active Control Method

This paper studied the dynamic behavior of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system, then numerical analysis of the different fractional orders hyper-chaotic systems are carried out under the predictor–corrector method. We proved the two systems are in hyper-chaos when the maximum and the second largest Lyapunov exponential are calculated. Also the smallest orders of the systems are proved when they are in hyper-chaos. The diverse structure synchronization of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system is realized using active control method. Numerical simulations indicated that the scheme was always effective and efficient.

scholarly journals Active Sliding Mode Control Antisynchronization of Chaotic Systems with Uncertainties and External Disturbances

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Wafaa Jawaada ◽  
M. S. M. Noorani ◽  
M. Mossa Al-sawalha
Keyword(s):  
Sliding Mode Control ◽  
Chaotic Systems ◽  
Control Method ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
External Disturbances ◽  
Mode Control ◽  
Active Sliding Mode Control

The antisynchronization behavior of chaotic systems with parametric uncertainties and external disturbances is explored by using robust active sliding mode control method. The sufficient conditions for achieving robust antisynchronization of two identical chaotic systems with different initial conditions and two different chaotic systems with terms of uncertainties and external disturbances are derived based on the Lyapunov stability theory. Analysis and numerical simulations are shown for validation purposes.

scholarly journals The Formation and Early Evolution of the Greensburg, Kansas, Tornadic Supercell on 4 May 2007

2009 ◽  
Vol 24 (4) ◽  
pp. 899-920 ◽  
Author(s):  
Howard B. Bluestein
Keyword(s):  
Doppler Radar ◽  
Initial Conditions ◽  
Radar Data ◽  
Early Evolution ◽  
Convective Storm ◽  
Extreme Sensitivity ◽  
The Right ◽  
Subsequent Splitting ◽  
Operational Data ◽  
Sensitivity To Initial Conditions

Abstract During the evening of 4 May 2007, a large, powerful tornado devastated Greensburg, Kansas. The synoptic and mesoscale environments of the parent supercell that spawned this and other tornadoes are described from operational data. The formation and early evolution of this long-track supercell, within the context of its larger-scale environment, are documented on the basis of Weather Surveillance Radar-1988 Doppler (WSR-88D) data and mobile Doppler radar data. The storm produced tornadoes cyclically for about 30 min before producing a large, long-lived tornado. It is shown that in order to have forecasted the severe weather locations and times accurately, it would have been necessary to have predicted 1) the localized formation of an isolated convective storm near/east of a dryline, 2) the subsequent splitting and resplitting of the storm several times, 3) the growth of a new storm along the right-rear flank of an existing storm, and 4) the transition from the cyclic production of small tornadoes to the production of one, large, long-track tornado. It is therefore suggested that both extreme sensitivity to initial conditions associated with storm formation and the uncertainty of storm behavior made it unusually difficult to forecast this event accurately.

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