An Introduction to Chaotic Dynamical Systems

Devaney, Robert L.

doi:10.4324/9780429502309

Cited by 392 publications

(595 citation statements)

References 3 publications

Supporting

Mentioning

541

Contrasting

Unclassified

Order By: Relevance

“…It is worth noting that Poincaré chaos admits properties similar to the ingredients of Devaney chaos [5].…”

Section: Preliminariesmentioning

confidence: 98%

“…In this section, we will show the presence of an unpredictable point in the symbolic dynamics [5,19] with a distinguishing feature.…”

Section: An Unpredictable Sequence Of the Symbolic Dynamicsmentioning

confidence: 99%

“…The first one is the homoclinic chaos [4], which is generated from the famous manuscript of Poincaré [1]. Another one is the Devaney chaos [5] whose ingredients are transitivity, Lorenz sensitivity [6] and the existence of infinitely many unstable periodic motions, which are dense in the chaotic attractor. The third one is the Li-Yorke chaos [7], which is characterized by the presence of a scrambled set in which any pair of distinct points are proximal and frequently separated.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

See 2 more Smart Citations

Poincaré chaos and unpredictable functions

Akhmet

Fen

2017

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

Highlights• Samples of unpredictable functions and sequences are constructed.• Unpredictable solutions of the logistic map and symbolic dynamics are obtained.• Non-homogeneous linear equations are verified for Poincar chaos presence.• Illustrative simulations are performed to confirm the theoretical results. Abstract The results of this study are continuation of the research of Poincaré chaos initiated in papers (Akhmet M, Fen MO. Commun Nonlinear Sci Numer Simulat 2016;40:1-5; Akhmet M, Fen MO. Turk J Math, doi:10.3906/mat-1603-51, accepted). We focus on the construction of an unpredictable function, continuous on the real axis. As auxiliary results, unpredictable orbits for the symbolic dynamics and the logistic map are obtained. By shaping the unpredictable function as well as Poisson function we have performed the first step in the development of the theory of unpredictable solutions for differential and discrete equations.The results are preliminary ones for deep analysis of chaos existence in differential and hybrid systems.Illustrative examples concerning unpredictable solutions of differential equations are provided.

show abstract

“…It is worth noting that Poincaré chaos admits properties similar to the ingredients of Devaney chaos [5].…”

Section: Preliminariesmentioning

confidence: 98%

“…In this section, we will show the presence of an unpredictable point in the symbolic dynamics [5,19] with a distinguishing feature.…”

Section: An Unpredictable Sequence Of the Symbolic Dynamicsmentioning

confidence: 99%

Section: Accepted Manuscriptmentioning

confidence: 99%

See 1 more Smart Citation

Poincaré chaos and unpredictable functions

Akhmet

Fen

2017

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

show abstract

“…f ðÁÞ is a cubic logistic map with two extrema in its range which has at most two attracting periodic orbits, whereas the logistic map has at most one [34]. As shown in Fig.…”

Section: Cl-gcm Model 21 Single Neuron Model Of Cl-gcmmentioning

confidence: 97%

A GCM neural network using cubic logistic map for information processing

Wang

Jia

2016

Neural Comput & Applic

View full text Add to dashboard Cite

A new chaotic neural network described by a modified globally coupled map (GCM) model with cubic logistic map is proposed, which is called CL-GCM model. Its rich dynamical behaviors over a wide range of parameters and the dynamics mechanism of neurons are demonstrated in detail. Furthermore, the network with delay coupling can be precisely controlled to any specifiedperiodic orbit by feedback control or modulated parameter control with variable threshold. The results of simulations and experiments suggest that the network is controlled successfully. The controlled CL-GCM model exhibits excellent associative memory performance which appears it can output unique fixed pattern or periodic patterns with specified period which contain the stored pattern closest to the initial pattern.

show abstract

“…This is also one of the most important features depicting the chaoticity of a system. This notion, the "butterfly effect", has been widely studied and is termed as sensitive dependence on initial conditions (briefly, sensitivity), introduced by Auslander and Yorke [3] and popularized by Devaney [4]. More precisely, let N = {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

On the large deviations theorem and ergodicity

Chen²

2016

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

An Introduction to Chaotic Dynamical Systems

Cited by 392 publications

References 3 publications

Poincaré chaos and unpredictable functions

Poincaré chaos and unpredictable functions

A GCM neural network using cubic logistic map for information processing

On the large deviations theorem and ergodicity

Contact Info

Product

Resources

About