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Chaos Theory By Nadha CHAOS THEORY What is Chaos Theory? . It is a field of study within applied mathematics . It studies the behavior of dynamical systems that are highly sensitive to initial conditions . It deals with nonlinear systems . It is commonly referred to as the Butterfly Effect What is a nonlinear system? . In mathematics, a nonlinear system is a system which is not linear . It is a system which does not satisfy the superposition principle, or whose output is not directly proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. the SUPERPOSITION PRINCIPLE states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). So a nonlinear system does not satisfy this principal. There are 3 criteria that a chaotic system satisfies 1) sensitive dependence on initial conditions 2) topological mixing 3) periodic orbits are dense 1) Sensitive dependence on initial conditions . This is popularly known as the "butterfly effect” . It got its name because of the title of a paper given by Edward Lorenz titled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? . The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. A consequence of sensitivity to initial conditions is that if we start with only a finite amount of information about the system, then beyond a certain time the system will no longer be predictable. This is most familiar in the case of weather, which is generally predictable only about a week ahead. 2) Topological Mixing . Topological mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition. The mixing of colored dyes or fluids is an example of a chaotic system. 3) Dense Periodicity . Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. This means that there is no true periodicity . There is only quasi- periodicity . Quasiperiodic orbits: Periodic solutions with at least two incommensurable frequencies (i.e., the ratio of the frequencies is an irrational number). The Butterfly Effect . Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather. It didn't predict the weather itself. However this computer program did theoretically predict what the weather might be. The Butterfly Effect Figure 1: Lorenz's experiment: the difference between the starting values of these curves is only .000127. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141) The Butterfly Effect . One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run. When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up completely different from the original. The Butterfly Effect • This effect came to be known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings. • The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, the atmosphere actually does diverge from what it would have been. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141) So… to sum up so far . Chaos theory is the study of nonlinear dynamics, where seemingly random events are actually predictable from simple deterministic equations. Deterministic Chaos . Most prominent effect of nonlinear dynamics . A key element of deterministic chaos is the sensitive dependence of the trajectory on the initial conditions . http://www.elmer.unibas.ch/pendulum/chaos .htm Deterministic Chaos . A horizontally driven pendulum . Both initial conditions differ in one arcsec only. But 10 seconds later they behave very differently. Deterministic Chaos . At the beginning the distance of the trajectories increases on average exponentially (see the inset which shows the distance in phase space). The rate of divergence is measured by the largest Lyapunov exponent. The Lyapunov exponent . The Lyapunov exponent characterises the extent of the sensitivity to initial conditions. Quantitatively, two trajectories in phase space with initial separation diverge . where λ is the Lyapunov exponent. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov Exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic. Stability and Instability . Local instability versus global stability: In order to have amplification of small errors and noise, the behavior must be locally unstable: over short times nearby states move away from each other. But for the system to consistently produce stable behavior, over long times the set of behaviors must fall back into itself. The tension of these two properties leads to very elegantly structured chaotic attractors. http://www.exploratorium.edu/complexity/Com pLexicon/chaos.html Some examples of chaotic systems . Double Pendulum . Rayleigh-Bénard convection Double Pendulum . This is considered to be a dynamical system . It is a pendulum with another pendulum attached to its end . The motion of this system is controlled by a set of coupled ordinary differential equations . This motion can be considered chaotic Double Pendulum Video: To demonstrate the motion of the double pendulum within its constraints Motion of the double compound pendulum (from numerical integration of the equations of motion) Double Pendulum Double Pendulum long exposure, tracked with LED light at its end. http://en.wikipedia.org/ wiki/File:DPLE.jpg Rayleigh-Bénard convection . An example of a self- organizing nonlinear system . A type of natural convection that occurs in a plane of fluid heated from below . The fluid develops a regular pattern of convection cells known as Benard cells . It is well studied because of its accessibility both experimentally and analytically Rayleigh-Bénard convection . The experimental set-up uses a layer of liquid, e.g. water, between two parallel planes. The height of the layer is small compared to the horizontal dimension. We can consider different cases . 1) The temperature of the bottom plane is the same as the top plane. The liquid will then tend towards an equilibrium, where its temperature is the same as its surroundings. At equilibrium, the liquid is perfectly uniform: to an observer it would appear the same from any position Rayleigh-Bénard convection . 2) If the temperature of the bottom plane is increased slightly yielding a flow of thermal energy conducted through the liquid. The system will begin to have a structure of thermal conductivity Rayleigh-Bénard convection . The temperature, and the density and pressure with it, will vary linearly between the bottom and top plane. A uniform linear gradient of temperature will be established. Once conduction is established, the microscopic random movement spontaneously becomes ordered on a macroscopic level, forming Bénard convection cells, with a characteristic correlation length. Convection cells in a gravity field Simulation of Rayleigh-Bénard convection in 3D Rayleigh-Bénard convection . the deterministic law at the microscopic level produces a non-deterministic arrangement of the cells: if the experiment is repeated, a particular position in the experiment will be in a clockwise cell in some cases, and a counter-clockwise cell in others. Microscopic perturbations of the initial conditions are enough to produce a (non- deterministic) macroscopic effect. This inability to predict long-range conditions and sensitivity to initial-conditions are characteristics of chaotic systems (i.e., the butterfly effect). If the temperature of the bottom plane was to be further increased, the structure would become more complex in space and time; the turbulent flow would become chaotic. THE END.
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