THE ORIGIN OF A CONTINUOUS TWO-DIMENSIONAL "CHAOTIC" DYNAMICS

2005 ◽  
Vol 15 (09) ◽  
pp. 3023-3029
Author(s):  
JOSE ALVAREZ-RAMIREZ ◽  
JOAQUIN DELGADO-FERNANDEZ ◽  
GILBERTO ESPINOSA-PAREDES
Keyword(s):  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Dimensional System ◽  
Two Dimensional ◽  
Time System ◽  
Paradoxical Situation ◽  
Numerical Studies ◽  
Regularization Techniques ◽  
Sensitivity To Initial Conditions ◽  
Continuous Time System

Ten years ago, Dixon et al. [1993] studied the behavior of a continuous-time system displaying erratic, apparently chaotic, dynamics. This is a paradoxical case since the system is two-dimensional, which is seemingly a violation of the Poincare–Bendixon theorem. Using numerical studies, Dixon et al. explained such a behavior from the presence of an attracting singularity, which induces arbitrarily large sensitivity to initial conditions. The aim of this letter is to use singularity regularization techniques to study the dynamics around the system singularity. The results obtained in this way explain the paradoxical situation of having continuous "chaotic" dynamics in a two-dimensional system.

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.

scholarly journals Market Share Delegation in a Bertrand Duopoly: Synchronisation and Multistability

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Luciano Fanti ◽  
Luca Gori ◽  
Cristiana Mammana ◽  
Elisabetta Michetti
Keyword(s):  
Market Share ◽  
Discrete Time ◽  
Price Competition ◽  
Initial Conditions ◽  
Two Dimensional ◽  
Discrete Time System ◽  
Multiple Attractors ◽  
Time System ◽  
Market Share Delegation

This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their substitutability. This means that when products are complements, players may coordinate their behaviour even if initial conditions are different. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very different dynamic patterns.

Effects of hypercapnia and hypocapnia on ventilatory variability and the chaotic dynamics of ventilatory flow in humans

2007 ◽  
Vol 292 (5) ◽  
pp. R1985-R1993 ◽  
Author(s):  
Marie-Noëlle Fiamma ◽  
Christian Straus ◽  
Sylvain Thibault ◽  
Marc Wysocki ◽  
Pierre Baconnier ◽  
...  
Keyword(s):  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Lung Ventilation ◽  
Largest Lyapunov Exponent ◽  
Ventilatory Control ◽  
Method Noise ◽  
Noise Limit ◽  
Breathing Variability ◽  
Sensitivity To Initial Conditions

In humans, lung ventilation exhibits breath-to-breath variability and dynamics that are nonlinear, complex, sensitive to initial conditions, unpredictable in the long-term, and chaotic. Hypercapnia, as produced by the inhalation of a CO2-enriched gas mixture, stimulates ventilation. Hypocapnia, as produced by mechanical hyperventilation, depresses ventilation in animals and in humans during sleep, but it does not induce apnea in awake humans. This emphasizes the suprapontine influences on ventilatory control. How cortical and subcortical commands interfere thus depend on the prevailing CO2 levels. However, CO2 also influences the variability and complexity of ventilation. This study was designed to describe how this occurs and to test the hypothesis that CO2 chemoreceptors are important determinants of ventilatory dynamics. Spontaneous ventilatory flow was recorded in eight healthy subjects. Breath-by-breath variability was studied through the coefficient of variation of several ventilatory variables. Chaos was assessed with the noise titration method (noise limit) and characterized with numerical indexes [largest Lyapunov exponent (LLE), sensitivity to initial conditions; Kolmogorov-Sinai entropy (KSE), unpredictability; and correlation dimension (CD), irregularity]. In all subjects, under all conditions, a positive noise limit confirmed chaos. Hypercapnia reduced breathing variability, increased LLE ( P = 0.0338 vs. normocapnia; P = 0.0018 vs. hypocapnia), increased KSE, and slightly reduced CD. Hypocapnia increased variability, decreased LLE and KSE, and reduced CD. These results suggest that chemoreceptors exert a strong influence on ventilatory variability and complexity. However, complexity persists in the quasi-absence of automatic drive. Ventilatory variability and complexity could be determined by the interaction between the respiratory central pattern generator and suprapontine structures.

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

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