Netaji Subhas Institute of Technology Delhi University Dwarka, New Delhi Instrumentation and Control Engineering
Chaos Theory
Submitted By Submitted To:
Ayush Bhatnagar Ms. Tanushree Choudhary447/IC/13 Report Writing ICE 1 IC 221
Chaos Theory- A Brief SummaryChaos is the science of surprises, of the nonlinear and the unpredictable. Chaos explains how small changes reflect in larger systems be it as complex as electric circuits, lasers, fluids, chemical reactions or as simple as the pendulum. It also has been thought possibly to occur in the stock market. An explanation for chaos is established by the Butterfly Effect and the Lorenz Attractor. The Butterfly Effect says flapping of the wings of a butterfly here can cause a hurricane at some other place. A tiny difference in initial parameters will result in a completely different behavior of a complex system. Edward N. Lorenz uses Fractals to explain the Chaos Theory. Fractal Theory helps us in the understanding of the Lorenz Attractor, a figure which explains Chaos theory. The Chaos theory also helps us in the study of complex non linear chaotic systems such as double pendulums, which are practically used everywhere in science today. The practicality of chaos theory makes it one of the most relevant and interesting theories in the world.
(i)Introduction
The trouble with weather forecasting is that it's right too often for us to ignore it and wrong too often for us to rely on it. Patrick Young
Chaos is from the Greek word Khaos, meaning "gaping void". There are many explanations as to who or what Chaos is, but most theories state that it was the void from which all things developed into a distinctive entity, or in which they existed in a confused and amorphous shape before they were separated into genera With inquisitive scientific minds like ours, finding out causes of simple events, however random or particular they might be, causes us to delve into Chaos Theory. Chaos theory is the study of nonlinear dynamics, in which seemingly random events are actually predictable from simple deterministic equations. We also discuss The Butterfly Effect based on similar observations by Edward Lorenz. With a little knowledge of calculus, we delve into the mathematical aspect of chaos, Fractal Geometry. Using it we also discuss The Lorenz Attractor, a model to demonstrate the chaotic theory. We would like first of all to sketch some of the main steps in the historical development of the concept of chaos in dynamical systems, from the mathematical point of view.
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1. The Butterfly Effect A Groundbreaking Proposition
1.1 The History:The concept of the butterfly effect is attributed to Edward Norton Lorenz, a mathematician and meteorologist, who was one of the first proponents ofchaos theory. Edward Lorenz, a meteorology professor at MIT, entered some numbers into a computer program simulating weather patterns and then left his office to get a cup of coffee while the machine ran. When he returned, he noticed a result that would change the course of science. On this day, Lorenz was repeating a simulation hed run earlierbut he had rounded off one variable from .506127 to .506. To his surprise, that tiny alteration drastically transformed the whole pattern his program produced, over two months of simulated weather. This idea that small causes can have very large effects, what has now become the idea in chaos theory that outcomes can be very sensitively dependent on initial conditions, was popularized as the Butterfly Effect after a talk Lorenz gave in 1972 at the American Association for the Advancement of Science.
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1.2 History of the Study of Chaos
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1.3 The Theory What Happened Next:Newtonian laws of physics are completely deterministic: they assume that, at least theoretically, precise measurements are possible, and that more precise measurement of any condition will yield more precise predictions about past or future conditions. The assumption was that - in theory, at least - it was possible to make nearly perfect predictions about the behavior of any physical system if measurements could be made precise enough and that the more accurate the initial measurements were, the more precise would be the resulting predictions.The butterfly effect, also known as sensitive dependence on initial conditions, has a profound corollary: forecasting the future can be nearly impossible.Poincare discovered that in some astronomical systems (generally consisting of three or more interacting bodies), even very tiny errors in initial measurements would yield enormous unpredictability, far out of proportion with what would be expected mathematically. Two or more identical sets of initial condition measurements - which according to Newtonian physics would yield identical results - in fact, most often led to vastly different outcomes. Poincare proved mathematically that, even if the initial measurements could be made a million times more precise, that the uncertainty of prediction for outcomes did not shrink along with the inaccuracy of measurement, but remained huge. That the tiny change in his simulation mattered so much showed, by extension, that the imprecision inherent in any human measurement could become magnified into wildly incorrect forecasts.
4 Edward Lorenz Henri Poincar
1.4 Practical Applications of Chaos Theory :Lorenzs work has also led to improvements in weather forecasting, which he credited to three things: wider data collection, better modeling, and the recognition of chaos in the weather, leading to whats called ensemble forecasting. In this technique, forecasters recognize that measurements are imperfect and thus run many simulations starting from slightly different conditions; the features these scenarios share form the basis of a more reliable consensus forecast.Although Chaos Theory may appear to be an abstract concept within modern science, it has many applications within our lives.
Washing machine manufacturers have used chaotic motion to improve the performance of their machines. A tiny pump vibrates with chaotic cycles to stir the water more efficiently. The stock market is a good example as although it is somewhat random it does follow some trends Weather forecasting is chaotic, as we know from Lorenz's discoveries. The human heart also has a chaotic pattern. The time between beats doesn't remain constant but also depends on a persons activity and stress. This analysis of the heartbeat can help medical researchers in controlling irregular heartbeats. The solar system contains many chaotic patterns. The motion of gas in a vacuum is chaotic. Although the particles are flying everywhere, they follow structure and pattern. Fractals are used to produce many of the modern computer art we see in latest films.
52.2 An Example of Fractals with Fractional Dimensions- Sierpinskis Triangle
Sierpinski's Triangle demonstrates this quite well: a triangle within smaller triangles within smaller triangles within ever smaller triangles, on and on. Many shapes in nature display this same quality of self-similarity. Clouds, ferns, coastlines, mountains, etc. all possess this feature.
Sierpinskis Triangle and its fractal dimensions
72.3 The Lorenz Attractor FractalEdward Lorenz, a curious meteorologist, was looking for a way to model the action of the chaotic behavior of a gaseous system. Hence, he took a few equations from the physics field of fluid dynamics, simplified them, and got the following 3-D system Here ,, andmake up the system state,is time, and,,are the system parameters. represents the "Prandtl number," the ratio of the fluid viscosity of a substance to its thermal conductivity; however, one does not have to know the exact value of this constant; hence, Lorenz simply used 10. The variable "" represents the difference in temperature between the top and bottom of the gaseous system. The variable "" is the width to height ratio of the box which is being used to hold the gas in the gaseous system. Lorenz used 8/3 for this variable. The resultant x of the equation represents the rate of rotation of the cylinder, "y" represents the difference in temperature at opposite sides of the cylinder, and the variable "z" represents the deviation of the system from a linear, vertical graphed line representing temperature. If one were to plot the three differential equations on a three-dimensional plane, using the help of a computer of course, no geometric structure or even complex curve would appear; instead, a weaving object known as the Lorenz Attractor appears. Because the system never exactly repeats itself, the trajectory never intersects itself.
Figure 1: The Lorenz Attractor
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3. Chaos and Chaotic Systems
3.1 What are Chaotic Systems?It is a Complex systemthat showssensitivityto initialconditions, such as aneconomy, a stock market, orweather. In such systems anyuncertainty(no matter how small) in the beginning willproducerapidly escalating andcompoundingerrorsin the prediction of thesystem'sfuturebehavior. To make anaccurateprediction oflong-termbehavior of such systems, the initial conditions must be known in their entirety and to an infinite level ofaccuracy.
The typical features of a chaotic system include: Nonlinearity. If it is linear, it cannot be chaotic.
Determinism. It has deterministic (rather than probabilistic) underlying rules every future state of the system must follow.
Sensitivity to initial conditions. Small changes in its initial state can lead to radically different behavior in its final state. This butterfly effect allows the possibility that even the slight perturbation of a butterfly flapping its wings can dramatically affect whether sunny or cloudy skies will predominate days later.
Sustained irregularity in the behavior of the system: Hidden order including a large or infinite number of unstable periodic patterns (or motions). This hidden order forms the infrastructure of irregular chaotic systems---order in disorder for short.
Long-term prediction is mostly impossible due to sensitivity to initial conditions, which can be known only to a finite degree of precision.
A simple example of a chaotic system in computer science is a pseudo-random number generator. The underlying rule in this case is a simple deterministic formula. However, the resulting solutions, such as the pseudo-random numbers are very irregular and unpredictable. We also note that a small change in the initial condition (seed) can yield a significantly different sequence of random numbers. These pseudo-random number generators are chaotic but also periodic with certain periods. Such generators viewed carefully yield the hidden order characteristic of chaos.
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3.2 Some More Examples of Chaotic Systems
Chaos theory studies the behavior of dynamic systems that are highly sensitive to initial conditions. Thus, the deterministic nature of these systems does not make them predictable.Most physical system are chaotic, so it is easy to name some examples of chaotic systems. Here are a few:1) Double pendulum:It is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The double pendulum undergoes chaotic motion, and shows a sensitive dependence on initial conditions.
Figure 2 : Motion of the double compound pendulum
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2) The bifurcation diagram of the logistic map:
The bifurcation parameter r is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function.
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3.3Application of Chaotic Systems
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Chaos has already had a lasting effect on science, yet there is much still left to be discovered. Many scientists believe that twentieth century science will be known for only three theories: relativity, quantum mechanics, and chaos. Chaos shows up everywhere around the world. From the currents of the ocean and the flow of blood through fractal blood vessels to the branches of trees and the effects of turbulence, chaos has inescapably become part of modern science. As chaos changed from a little-known theory to a full science of its own, it has received widespread publicity. Chaos theory has changed the direction of science: in the eyes of the general public, physics is no longer simply the study of subatomic particles in a billion-dollar particle accelerator, but the study of chaotic systems and how they work.
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5. Bibliography
http://en.wikipedia.org/ http://www.abarim-publications.com/ http://www.wisegeek.org/ http://fractalfoundation.org/ http://www.referenceforbusiness.com/ http://www.quora.com/ http://mathworld.wolfram.com/ http://perso.ens-lyon.fr/ http://www.plouffe.fr/ http://functionspace.org/
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