Visualising the invisible: performing chaos theory

Leonardo ◽  
2020 ◽  
pp. 1-8
Author(s):  
Emma Weitkamp
Keyword(s):  
Complex Systems ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Starting Point ◽  
Weather And Climate ◽  
Butterfly Effect ◽  
Long Term Behavior ◽  
The Invisible

Edward Lorenz, the pioneering figure in the field of chaos theory coined the phrase “butterfly effect” and posed the famous question “Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?” In posing the question, Lorenz sought to highlight the intrinsic difficulty of predicting the long term behavior of complex systems that are sensitive to initial conditions, like, for example, the weather and climate; these systems are often referred to as chaotic. Taking Lorenz' butterfly as a starting point, Chaos Cabaret sought to explore the nuances of chaos theory through performance and music.

scholarly journals Complexity In Psychological Self-Ratings: Implications for research and practice

2020 ◽  
Author(s):  
Merlijn Olthof ◽  
Fred Hasselman ◽  
Anna Lichtwarck-Aschoff
Keyword(s):  
Complex Systems ◽  
Systems Approach ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Regime Shifts ◽  
Fundamental Aspect ◽  
Time Varying ◽  
Term Prediction ◽  
Sensitive Dependence

Background: Psychopathology research is changing focus from group-based ‘disease models’ to a personalized approach inspired by complex systems theories. This approach, which has already produced novel and valuable insights into the complex nature of psychopathology, often relies on repeated self-ratings of individual patients. So far it has been unknown whether such self-ratings, the presumed observables of the individual patient as a complex system, actually display complex dynamics. We examine this basic assumption of a complex systems approach to psychopathology by testing repeated self-ratings for three markers of complexity: memory, the presence of (time-varying) short- and long-range temporal correlations, regime shifts, transitions between different dynamic regimes, and, sensitive dependence on initial conditions, also known as the ‘butterfly effect’, the divergence of initially similar trajectories.Methods: We analysed repeated self-ratings (1476 time points) from a single patient for the three markers of complexity using Bartels rank test, (partial) autocorrelation functions, time-varying autoregression, a non-stationarity test, change point analysis and the Sugihara-May algorithm.Results: Self-ratings concerning psychological states (e.g., the item ‘I feel down’) exhibited all complexity markers: time-varying short- and long-term memory, multiple regime shifts and sensitive dependence on initial conditions. Unexpectedly, self-ratings concerning physical sensations (e.g., the item ‘I am hungry’) exhibited less complex dynamics and their behaviour was more similar to random variables. Conclusions: Psychological self-ratings display complex dynamics. The presence of complexity in repeated self-ratings means that we have to acknowledge that (1) repeated self-ratings yield a complex pattern of data and not a set of (nearly) independent data points, (2) humans are ‘moving targets’ whose self-ratings display non-stationary change processes including regime shifts, and (3) long-term prediction of individual trajectories may be fundamentally impossible. These findings point to a limitation of popular statistical time series models whose assumptions are violated by the presence of these complexity markers. We conclude that a complex systems approach to mental health should appreciate complexity as a fundamental aspect of psychopathology research by adopting the models and methods of complexity science. Promising first steps in this direction, such as research on real-time process-monitoring, short-term prediction, and just-in-time interventions, are discussed.

Investigation of the Dynamic Behavior of Railway Ballast Using Molecular Dynamics Simulations

2001 ◽  
Author(s):  
Holger Kruse ◽  
Karl Popp
Keyword(s):  
Molecular Dynamics ◽  
Initial Conditions ◽  
Contact Forces ◽  
Contact Network ◽  
Railway Ballast ◽  
Particle Layer ◽  
Dynamics Simulations ◽  
Long Term Behavior ◽  
Overlap Area

Abstract The molecular dynamics method (MD method) is a powerful tool for the investigation of granular materials like the railway ballast. The characteristics of this method are explained in detail. In contrast to a continuum description, each single stone of the ballast is taken into account. Since the ballast settlement is strongly influenced by the shape of the stones, in the two-dimensional model polygonal particles are used. These particles are surrounded by fixed boundary walls. At the top of the ballast particle layer, a single sleeper is positioned which is loaded by forces occurring at the real track. The contact forces are calculated from the overlap area of the particle geometries. The paper includes information about the sensitivity of the model behavior on initial conditions and contact law parameters. Furthermore, the contact network, the quasi-static stiffness of the ballast layer and its long-term behavior are addressed. Particular emphasis is put on the description of current difficulties and challenges in applying the MD method.

Long-term behavior of positive solutions of an exponentially self-regulating system of difference equations

2017 ◽  
Vol 10 (03) ◽  
pp. 1750045 ◽  
Author(s):  
N. Psarros ◽  
G. Papaschinopoulos ◽  
K. B. Papadopoulos
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
System Of Difference Equations ◽  
Long Term Behavior

In this paper, we study the asymptotic behavior of the positive solutions of a system of the following difference equations: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants and the initial conditions [Formula: see text] and [Formula: see text] are positive numbers.

scholarly journals Further Analysis of Lorenz’s Maximum Simplification Equations

2006 ◽  
Vol 63 (11) ◽  
pp. 2673-2699 ◽  
Author(s):  
S. Lakshmivarahan ◽  
Michael E. Baldwin ◽  
Tao Zheng
Keyword(s):  
Initial Conditions ◽  
Complete Picture ◽  
Long Term Behavior

Abstract The goal of this paper is to provide a complete picture of the long-term behavior of Lorenz’s maximum simplification equations along with the corresponding meteorological interpretation for all initial conditions and all values of the parameter.

Chaos-Based Cryptography for Voice Secure Wireless Communication

2014 ◽  
pp. 97-132 ◽  
Author(s):  
Sattar B. Sadkhan Al Maliky ◽  
Rana Saad
Keyword(s):  
Wireless Communication ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Security Research ◽  
Speech Encryption ◽  
Modern Cryptography ◽  
Long Term Prediction ◽  
The Voice ◽  
Sensitivity To Initial Conditions

Chaos theory was originally developed by mathematicians and physicists. The theory deals with the behaviors of nonlinear dynamic systems. Chaos theory has desirable features, such as deterministic, nonlinear, irregular, long-term prediction, and sensitivity to initial conditions. Therefore, and based on chaos theory features, the security research community adopts chaos theory in modern cryptography. However, there are challenges of using chaos theory with cryptography, and this chapter highlights some of those challenges. The voice information is very important compared with the information of image and text. This chapter reviews most of the encryption techniques that adopt chaos-based cryptography, and illustrates the uses of chaos-based voice encryption techniques in wireless communication as well. This chapter summarizes the traditional and modern techniques of voice/speech encryption and demonstrates the feasibility of adopting chaos-based cryptography in wireless communications.

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.

Advising Undecided Students: Lessons From Chaos Theory

1999 ◽  
Vol 19 (1) ◽  
pp. 45-49 ◽  
Author(s):  
Amy Beck
Keyword(s):  
Complex Systems ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Emergent Behavior ◽  
Strange Attractors ◽  
Undecided Students

Chaos theory can be used as a metaphor for advising undecided students. Concepts from chaos theory viewed in this context include dependence on initial conditions, strange attractors, emergent behavior in complex systems, and fractals. Looking at advising in a new light often gives advisors new ways of responding to traditional problems. The lessons advisors can take from chaos theory may simply be “get back to the basics,” or they may open advisor and student minds to avenues for change.

scholarly journals Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Michael A. Radin ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Boundedness Of Solutions ◽  
Real Numbers ◽  
Unbounded Solutions ◽  
The Difference ◽  
Long Term Behavior

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: , where the initial conditions and are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.

scholarly journals Chaos-Based Cryptography for Voice Secure Wireless Communication

Economics ◽  
2015 ◽  
pp. 460-493
Author(s):  
Sattar B. Sadkhan Al Maliky ◽  
Rana Saad
Keyword(s):  
Wireless Communication ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Security Research ◽  
Speech Encryption ◽  
Modern Cryptography ◽  
Long Term Prediction ◽  
The Voice ◽  
Sensitivity To Initial Conditions

Chaos theory was originally developed by mathematicians and physicists. The theory deals with the behaviors of nonlinear dynamic systems. Chaos theory has desirable features, such as deterministic, nonlinear, irregular, long-term prediction, and sensitivity to initial conditions. Therefore, and based on chaos theory features, the security research community adopts chaos theory in modern cryptography. However, there are challenges of using chaos theory with cryptography, and this chapter highlights some of those challenges. The voice information is very important compared with the information of image and text. This chapter reviews most of the encryption techniques that adopt chaos-based cryptography, and illustrates the uses of chaos-based voice encryption techniques in wireless communication as well. This chapter summarizes the traditional and modern techniques of voice/speech encryption and demonstrates the feasibility of adopting chaos-based cryptography in wireless communications.

scholarly journals Modelling the second wave of COVID-19 infections in France and Italy via a Stochastic SEIR model

2021 ◽  
Author(s):  
Davide Faranda ◽  
Tommaso Alberti
Keyword(s):  
Initial Conditions ◽  
Asymptotic Estimates ◽  
Seir Model ◽  
Actual Knowledge ◽  
The World ◽  
Log Normal ◽  
Long Term Behavior ◽  
Second Wave ◽  
Log Normal Distribution

<p>COVID-19 has forced quarantine measures in several countries across the world. These measures have proven to be effective in significantly reducing the prevalence of the virus. To date, no effective treatment or vaccine is available. In the effort of preserving both public health as well as the economical and social textures, France and Italy governments have partially released lockdown measures. Here we extrapolate the long-term behavior of the epidemics in both countries using a Susceptible-Exposed-Infected-Recovered (SEIR) model where parameters are stochastically perturbed with a log-normal distribution to handle the uncertainty in the estimates of COVID-19 prevalence and to simulate the presence of super-spreaders. Our results suggest that uncertainties in both parameters and initial conditions rapidly propagate in the model and can result in different outcomes of the epidemics leading or not to a second wave of infections. Furthermore, the presence of super-spreaders adds instability to the dynamics, making the control of the epidemics more difficult. Using actual knowledge, asymptotic estimates of COVID-19 prevalence can fluctuate of order of ten millions units in both countries.</p>

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