REAL LIFE

1996 ◽  
Vol 06 (11) ◽  
pp. 2077-2086 ◽  
Author(s):  
GARY MAR ◽  
PAUL ST. DENIS
Keyword(s):  
Chaotic Systems ◽  
Initial Conditions ◽  
Real Life ◽  
Game Of Life ◽  
Life And Death ◽  
Sensitive Dependence ◽  
Conway’S Game Of Life ◽  
Special Case

In Conway’s Game of Life every cell is either fully alive (has the value of 1) or completely dead (has the value 0). In Real Life this restriction to bivalence is lifted to countenance “real-valued” degrees of life and death. Real Life contains Conway’s Game of Life as a special case; however, Real Life, in contrast to Conway’s Game of Life, exhibits sensitive dependence on initial conditions which is characteristic of chaotic systems.

CLARIFYING CHAOS: EXAMPLES AND COUNTEREXAMPLES

1996 ◽  
Vol 06 (02) ◽  
pp. 219-249 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Closed Form Solutions ◽  
Power Spectral ◽  
Sensitive Dependence ◽  
Universal Definition ◽  
Definition Of ◽  
One Dimensional Maps ◽  
Continuous Power

Over the past fifteen years there have been various attempts to define chaos. In an effort to find a universally acceptable definition we began constructing new examples of chaotic systems in the hope that the salient features of chaos could be captured. Our efforts to date have failed and the examples we have constructed seem to suggest that no such definition exists. However, these examples have proved to be valuable in spite of our inability to hone a universal definition of chaos from them. Consequently, we present this list of examples and their significance. Some interesting conclusions that we can draw from them are: It is possible to construct simple closed form solutions of chaotic one-dimensional maps; sensitive dependence on initial conditions, the most widely used definition of chaos, has many counterexamples; there are invertible chaotic dynamical systems defined by simple differential equations that do not have horseshoes; three important properties that are thought to characterize chaos, continuous power spectral density, exponentially sensitive dependence on initial conditions, and exponential loss of information (Chaitin’s concept of algorithmic complexity), are independent. Chaos seems to be tied to our notion of rates of divergence of orbits or degradation of information such as is found in systems with positive Lyapunov exponents. The reliance on rates seems to open the door to a pandora’s box of rates, both higher and lower than exponential. The intuitive notion of pseudo-randomness, a practical feature of chaos, is present in examples that do not have positive Lyapunov exponents. And in general, nonlinear polynomial rates of degradation of information are also quite “unpredictable”. We conclude that it appears that for any given definition of chaos, there may always be some “clearly” chaotic systems which do not fall under that definition, thus making chaos a cousin to Gödel’s undecidability.

Life turns fifty

2021 ◽  
Vol 49 (3) ◽  
pp. 10-10
Author(s):  
SIGCAS Team
Keyword(s):  
Initial Conditions ◽  
50Th Anniversary ◽  
Game Of Life ◽  
Single Person ◽  
Computer Scientists ◽  
Person Game ◽  
Conway’S Game Of Life

Mention to computer scientists, gliders, glider guns, birth and death rules and they smile remembering their efforts to study societal life. October marked the 50th anniversary of the publication of John Conway's game of Life in Martin Garner's Mathematical Games column [1], For the lay person with no knowledge of Life, it's difficulty to imagine how popular a single person game with only a single move (i.e. setting the initial conditions) could be.

A Comparative Study to Quantify Sensitive Dependence in Numerical Models for a Developing Low in the Southern Plains

2010 ◽  
Vol 44-45 (2010-2011) ◽  
pp. 29-40
Author(s):  
Amy E. Schnetzler ◽  
Justin M. Glisan ◽  
H. Athar ◽  
Patrick S. Market ◽  
Anthony R. Lupo
Keyword(s):  
Chaotic Systems ◽  
Initial Conditions ◽  
Numerical Models ◽  
Wrf Model ◽  
Convective Precipitation ◽  
Southern Plains ◽  
Southeast United States ◽  
Sensitive Dependence ◽  
Trajectory Divergence

Abstract Studies have shown that numerical models display the characteristics of chaotic systems, and that the solutions can be sensitive to the initial conditions, the model used, or the parameterizations used. Using the Kain-Fritsch, Grell, and modified Kuo convective parameterizations in the MASS and the WRF model, the results from a case study show that 48-h forecasts were not identical. Lyapunov exponents were calculated by plotting forecast trajectories in a phase diagram and estimating the rate of trajectory divergence for two time periods outside the study of the main cyclone. These calculations did show divergence at a rate which was consistent with differences in model height in 48-h forecasts from other studies. Additionally, the integrated enstrophy can be used to estimate the Lyapunov value. Finally, a qualitative analysis comparing various model runs (pseudo-ensemble) was performed to determine if there were regions or areas where consistent differences in the runs existed between the indexes used for forecasting convective precipitation. Results demonstrated that the region of the southeast United States associated with the developing cyclone showed the most significant differences in these indexes and for heights and temperatures. The differences in the model forecasts between convective parameterizations (intramodel forecasts) in this case were not as great as the model-to-model forecast differences (intermodel forecasts).

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

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

IS SENSITIVE DEPENDENCE ON INITIAL CONDITIONS NATURE’S SENSORY DEVICE?

1992 ◽  
Vol 02 (01) ◽  
pp. 193-199 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA ◽  
BECKY POPP
Keyword(s):  
Initial Conditions ◽  
Sensory Perception ◽  
Sensory Features ◽  
Sensitive Dependence ◽  
Nonlinear Devices

In this letter we illustrate three methods of using nonlinear devices as sensors. We show that the sensory features of these devices is a result of sensitive dependence on parameters which we show is equivalent to sensitive dependence on initial conditions. As a result, we conjecture that sensitive dependence on initial conditions is nature’s sensory device in cases where remarkable feats of sensory perception are seen.

CELLULAR NEURAL NETWORKS CAN MIMIC SMALL-WORLD NETWORKS

1999 ◽  
Vol 09 (10) ◽  
pp. 2105-2126 ◽  
Author(s):  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Neural Networks ◽  
Coupling Strength ◽  
Initial Conditions ◽  
Computing Time ◽  
Small World ◽  
Cellular Neural Networks ◽  
Clustering Coefficient ◽  
High Survival ◽  
Game Of Life ◽  
Hole Detection

Small-world phenomenon can occur in coupled dynamical systems which are highly clustered at a local level and yet strongly coupled at the global level. We show that cellular neural networks (CNN's) can exhibit "small-world phenomenon". We generalize the "characteristic path length" from previous works on "small-world phenomenon" into a "characteristic coupling strength" for measuring the average coupling strength of the outputs of CNN's. We also provide a simplified algorithm for calculating the "characteristic coupling strength" with a reasonable amount of computing time. We define a "clustering coefficient" and show how it can be calculated by a horizontal "hole detection" CNN, followed by a vertical "hole detection" CNN. Evolutions of the game-of-life CNN with different initial conditions are used to illustrate the emergence of a "small-world phenomenon". Our results show that the well-known game-of-life CNN is not a small-world network. However, generalized CNN life games whose individuals have strong mobility and high survival rate can exhibit small-world phenomenon in a robust way. Our simulations confirm the conjecture that a population with a strong mobility is more likely to qualify as a small world. CNN games whose individuals have weak mobility can also exhibit a small-world phenomenon under a proper choice of initial conditions. However, the resulting small worlds depend strongly on the initial conditions, and are therefore not robust.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

scholarly journals Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters

Algorithms ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 85
Author(s):  
Andreas Rauh ◽  
Julia Kersten
Keyword(s):  
Linear Systems ◽  
Initial Conditions ◽  
A Priori ◽  
Real Life ◽  
Reachability Analysis ◽  
Interval Methods ◽  
Space Representation ◽  
Uncertain Parameters ◽  
Time Varying ◽  
Similarity Transformations

Continuous-time linear systems with uncertain parameters are widely used for modeling real-life processes. The uncertain parameters, contained in the system and input matrices, can be constant or time-varying. In the latter case, they may represent state dependencies of these matrices. Assuming bounded uncertainties, interval methods become applicable for a verified reachability analysis, for feasibility analysis of feedback controllers, or for the design of robust set-valued state estimators. The evaluation of these system models becomes computationally efficient after a transformation into a cooperative state-space representation, where the dynamics satisfy certain monotonicity properties with respect to the initial conditions. To obtain such representations, similarity transformations are required which are not trivial to find for sufficiently wide a-priori bounds of the uncertain parameters. This paper deals with the derivation and algorithmic comparison of two different transformation techniques for which their applicability to processes with constant and time-varying parameters has to be distinguished. An interval-based reachability analysis of the states of a simple electric step-down converter concludes this paper.

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.

Sign in / Sign up

Export Citation Format

Share Document