Introduction to anti-control of discrete chaos: theory and applications

2006 ◽  
Vol 364 (1846) ◽  
pp. 2433-2447 ◽  
Author(s):  
Guanrong Chen ◽  
Yuming Shi
Keyword(s):  
Dynamical System ◽  
Feedback Control ◽  
Chaos Theory ◽  
Control Of Chaos ◽  
Main Interest ◽  
Chaotic Dynamical System ◽  
Infinite Dimensional ◽  
Control Techniques ◽  
Finite Dimensional ◽  
Discrete Chaos

In this paper, the notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system. The main interest in this paper is to employ the classical feedback control techniques. Only the discrete case is discussed in detail, including both finite-dimensional and infinite-dimensional settings.

FEEDBACK CONTROL OF THE LIU CHAOTIC DYNAMICAL SYSTEM

2010 ◽  
Vol 24 (03) ◽  
pp. 397-404 ◽  
Author(s):  
XINGYUAN WANG ◽  
XINGUANG LI
Keyword(s):  
Dynamical System ◽  
Feedback Control ◽  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical System ◽  
Feedback Method ◽  
Liu System

Classical feedback method is used to control chaos in the Liu dynamical system. Based on the Routh–Hurwitz criteria, the conditions of the asymptotic stability of the steady states of the controlled Liu system are discussed, and they are also proved theoretically. Numerical simulations show that the method can suppress chaos to both unstable equilibrium points and unstable periodic orbits (limit cycles) successfully.

Center Manifold of Fractional Dynamical System

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Dynamical System ◽  
Center Manifold ◽  
High Dimensional ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Reduction Methods ◽  
Fractional Dynamical System ◽  
Lower Dimensional

Dimension reduction of dynamical system is a significant issue for technical applications, as regards both finite dimensional system and infinite dimensional systems emerging from either science or engineering. Center manifold method is one of the main reduction methods for ordinary differential systems (ODSs). Does there exists a similar method for fractional ODSs (FODSs)? In other words, does there exists a method for reducing the high-dimensional FODS into a lower-dimensional FODS? In this study, we establish a local fractional center manifold for a finite dimensional FODS. Several examples are given to illustrate the theoretical analysis.

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS GOVERNED BY CONTINUOUS MAPS

2005 ◽  
Vol 15 (02) ◽  
pp. 547-555 ◽  
Author(s):  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Relevant Literature ◽  
Discrete Dynamical Systems ◽  
One Dimensional ◽  
Control Techniques ◽  
Controlled Systems ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Continuous Maps

This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

2020 ◽  
Vol 31 (12) ◽  
pp. 2050172
Author(s):  
Henryk Fukś ◽  
Yucen Jin
Keyword(s):  
Dynamical System ◽  
Local Structure ◽  
Structure Theory ◽  
Dimensional System ◽  
Perfect Agreement ◽  
Infinite Dimensional ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Dimensional Dynamical System

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely dimensional system. While it is well known that this approximation works surprisingly well for some CA, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.

CHAOTIFICATION OF NONAUTONOMOUS DISCRETE DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (11) ◽  
pp. 3359-3371 ◽  
Author(s):  
QIULING HUANG ◽  
YUMING SHI ◽  
LIJUAN ZHANG
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Banach Spaces ◽  
Discrete Dynamical Systems ◽  
Strong Sense ◽  
Control Techniques ◽  
Controlled Systems ◽  
Finite Dimensional ◽  
Time Invariant

This paper focuses on the chaotification of nonautonomous discrete dynamical systems in finite-dimensional and general Banach spaces by feedback control techniques. Several chaotification schemes with general controllers and sawtooth function are established, respectively, where the controllers are time-invariant. The controlled systems are proved to be chaotic in the strong sense of Li–Yorke.

scholarly journals Adaptive Feedback Control for Chaos Control and Synchronization for New Chaotic Dynamical System

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
M. M. El-Dessoky ◽  
M. T. Yassen
Keyword(s):  
Dynamical System ◽  
Adaptive Control ◽  
Feedback Control ◽  
Chaos Control ◽  
Control Technique ◽  
Unknown Parameters ◽  
Chaotic Dynamical System ◽  
Adaptive Feedback ◽  
Adaptive Feedback Control ◽  
Control And Synchronization

This paper investigates the problem of chaos control and synchronization for new chaotic dynamical system and proposes a simple adaptive feedback control method for chaos control and synchronization under a reasonable assumption. In comparison with previous methods, the present control technique is simple both in the form of the controller and its application. Based on Lyapunov's stability theory, adaptive control law is derived such that the trajectory of the new system with unknown parameters is globally stabilized to the origin. In addition, an adaptive control approach is proposed to make the states of two identical systems with unknown parameters asymptotically synchronized. Numerical simulations are shown to verify the analytical results.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

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