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.

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

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.

Reverse Butterfly Effect: Saving Alternate Histories of Virtual Worlds

2016 ◽  
Author(s):  
Zi Ye
Keyword(s):  
Popular Culture ◽  
Virtual Worlds ◽  
Chaos Theory ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Storage Model ◽  
Simulation Gaming ◽  
Non Linear Systems ◽  
Butterfly Effect ◽  
Random Nature

Chaos theory is a recent field of study which has become extremely influential in science and in popular culture. Chaos theory deals with complex, non‐linear systems which are extremely sensitive to their initial conditions (commonly known as the butterfly effect), and whose behaviour quickly become unpredictable over short periods of time. Despite their seemingly random nature, chaotic systems are fully deterministic. This means that the same initial conditions will always yield the same future states. When I looked at the butterfly effect backwards, and applied it to computer simulations, the result was a way to store many alternate histories of virtual worlds in a very small amount of data. This time storage model may have applications in scientific simulation, gaming, and cryptography, and provides a different look at chaos theory.

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.

The Features of Chaos

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Average Rate ◽  
Quantitative Measure ◽  
Fractal Dimensions ◽  
Recurrence Plot ◽  
Sensitivity To Initial Conditions

In this chapter we introduce the features of Chaotic systems. We describe “sensitivity to initial conditions” and its quantitative measure, the Lyapunov exponent, which reflect the average rate of divergence (if any) between two neighboring trajectories. We describe the dynamic “strangeness” of the system. Which has its counterpart in the “strangeness” of the attractor's geometry and concerns with the texture woven by the system in phase space. Fractal dimensions are measures of such strange geometries and they are here described. The concept of recurrence is introduced and the recurrence plot is described, and code provided to generate it. The correlation dimension is addressed and the R code to compute is listed and detailed. Poincare map is introduced and applied to the study of the damped, driven pendulum.

scholarly journals Design of Initial Value Control for Modified Lorenz-Stenflo System

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Yuan-Long Wu ◽  
Cheng-Hsiung Yang ◽  
Chang-Hsi Wu
Keyword(s):  
Circuit Design ◽  
Chaotic Systems ◽  
Analog Circuit ◽  
Initial Conditions ◽  
Control Circuit ◽  
Initial Value ◽  
Sensitivity To Initial Conditions

For the sake of complexity, unpredictability, and exceeding sensitivity to initial conditions in the chaotic systems, there were many studies for information encryption of chaotic systems in recent years. Enhancing the security in information encryption of chaotic systems, an initial value control circuit for chaotic systems is proposed in this paper. By way of changing the initial value, we can change the behavior of chaotic systems and also change the key of information encryption. An analog circuit is implemented to verify the initial value control circuit design.

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 Spontaneous Synchronization in Two Mutually Coupled Memristor-Based Chua’s Circuits: Numerical Investigations

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Eleonora Bilotta ◽  
Francesco Chiaravalloti ◽  
Pietro Pantano
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Synchronization Control ◽  
Time Varying ◽  
Cubic Nonlinearity ◽  
Numerical Investigations ◽  
Self Organized ◽  
Coupling Resistance ◽  
Coupled Circuits

Chaotic dynamics of numerous memristor-based circuits is widely reported in literature. Recently, some works have appeared which study the problem of synchronization control of these systems in a master-slave configuration. In the present paper, the spontaneous dynamic behavior of two chaotic memristor-based Chua’s circuits, mutually interacting through a coupling resistance, was studied via computer simulations in order to study possible self-organized synchronization phenomena. The used memristor is a flux controlled memristor with a cubic nonlinearity, and it can be regarded as a time-varying memductance. The memristor, in effect, retains memory of its past dynamic and any difference in the initial conditions of the two circuits results in different values of the corresponding memductances. In this sense, due to the memory effect of the memristor, even if coupled circuits have the same parameters they do not constitute two completely identical chaotic oscillators. As is known, for nonidentical chaotic systems, in addition to complete synchronizations (CS) other weaker forms of synchronization which provide correlations between the signals of the two systems can also occur. Depending on initial conditions and coupling strength, both chaotic and nonchaotic synchronization are observed for the system considered in this work.

Whale optimization based synchronization and control of two identical fractional order financial chaotic systems

2021 ◽  
pp. 1-14
Author(s):  
Sangeeta Gupta ◽  
Pragya Varshney ◽  
Smriti Srivastava
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Optimization Techniques ◽  
Complete Synchronization ◽  
Whale Optimization ◽  
Parameter Perturbations ◽  
And Control ◽  
Method Analysis ◽  
Sensitivity To Initial Conditions

This paper proposes a scheme to synchronize fractional order chaotic systems employing fractional PID controller. The parameters of FOPID are tuned using Swarm based optimization techniques, viz.: Whale optimization algorithm and Particle swarm optimization techniques. To assert the complete synchronization, master-slave method has been implemented. Chaotic systems are highly dependent upon initial conditions and parameter perturbations. Therefore, taking these properties into consideration, synchronization of two identical fractional order financial chaotic systems is performed with distinct initial conditions. To show the efficacy of the proposed method, analysis is performed for orders between 0 to 1, and also for sensitivity to initial conditions.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

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