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Deterministic Chaos

Estiklal Basem. Deterministic Chaos . Objective Dictionary Meaning of Chaos What is the chaos theory? Areas utilizing chaos theory History Butterfly Effect Features of Chaotic Systems Examples of chaotic system Edward Lorenz work Conclusion References.

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Deterministic Chaos

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  1. EstiklalBasem Deterministic Chaos

  2. Objective • Dictionary Meaning of Chaos • What is the chaos theory? • Areas utilizing chaos theory • History • Butterfly Effect • Features of Chaotic Systems • Examples of chaotic system • Edward Lorenz work • Conclusion • References

  3. Dictionary Meaning of Chaos . • “a state of things in which chance is supreme; especially : the confused unorganized state of primordial matter before the creation of distinct forms” • “The inherent unpredictability in the behavior of a complex natural system (as the atmosphere, boiling water, or the beating heart)”

  4. What is the chaos theory ? • It is the branch of mathematics that deals with complex systems whose behavior is highly sensitive to slight changes in conditions, so that small alterations can give rise to strikingly great consequences. • Traditional notion of chaos – unorganized, disorderly, random etc. • But Chaos Theory has nothing do with the traditional notion .On the contrary, it actually tells you that not all that ‘chaos’ you see is due to chance, or random or caused by unknown factors.

  5. Chaos theory is a mathematical field of study which states that non-linear dynamical systems that are seemingly random are actually deterministic from much simpler equations • As chaos theory was developed by inputs of various mathematicians and scientists, it found applications in a large number of scientific fields. • Everything in the universe is in some way or the other under control of Chaos or product of Chaos. Every motion, behavior or tendency can be explained by Chaos Theory

  6. Areas utilizing chaos theory • mathematics • meteorology • physics • engineering • economics • andbiology

  7. This phenomenon has been observed in diverse areas as: • Fluid dynamics • The motion of planets • Economic cycles • General relativity • The behavior of water boiling on a stove, • Migratory patterns of birds • The spread of vegetation across a continent.

  8. History • An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. • In 1898 Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature.

  9. The pioneer of the theory was Edward Lorenz . Chaos theory is most commonly attributed to the work of Edward Lorenz. His 1963 paper, Deterministic Non periodic Flow, is credited for laying the foundation for Chaos Theory. Lorenz was a meteorologist who developed a mathematical model used to model the way the air moves in the atmosphere. He discovered by chance that when he entered a starting value at three decimal 0.506 instead of entering the full 0.506127. It caused vast differences in the outcome of the model.

  10. In this way he discovered the principle of Sensitive Dependence on Initial Conditions(SDIC), which is now viewed as a key component in any chaotic system. This idea was then immortalized when Lorenz gave a talk at the 139th meeting of the American Association for the Advancement of Science in 1972 entitled “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas”. With this speech the idea of the Butterfly Effect was born and has been used when talking about chaos theory ever since.

  11. Butterfly Effect • Is the sensitive dependency on initial conditions in which a small change at one place in a deterministic nonlinear system can result in large differences in a later state. • Is derived from the theoretical example of a hurricane's formation being contingent on whether or not a distant butterfly had flapped its wings several weeks earlier.

  12. That was a joke this has nothing to do with weather

  13. Features of Chaotic Systems • 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.

  14. 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 • Long-term prediction is mostly impossible due to sensitivity to initial conditions, which can be known only to a finite degree of precision.

  15. Chaos is qualitative in that it seeks to know the general character of a systems long-term behavior, rather than seeking numerical predictions about a future state. • Chaotic systems are unstable since they tend not to resist any outside disturbances but instead react in significant ways. • These systems are deterministic because they are made up of few, simple differential equations, and make no references to implicit chance mechanisms.

  16. These systems are described using differential equations specifying the rates of change for each variable . • A dynamic system is a simplified model for the time-varying behavior of an actual system. • A deterministic system is a system in which no randomness is involved in the development of future states of the system. • Chaotic whenever its evolution sensitively depends on the initial conditions. This property implies that two trajectories emerging from two different close-by initial conditions separate exponentially in the course of time.

  17. Examples of chaotic system 1) One of the most famous is the three-body problem. Newton's theory of gravitation provides a simple solution to the problem of two mutually attracting bodies, for example the sun and one of its planets. However, as soon as a third body comes into play, for example another planet, the problem becomes mathematically unsolvable. In practice, astronomers work with approximations, where the attraction to the most important body, in this case the sun, is taken as the basis, while the effect of a third body is brought in as a perturbation.

  18. Predictions based on this approximation are in practice very reliable. The reason this works is because the gravitation exerted by the planets is tiny compared to the gravitation exerted by the sun. However, nobody can prove that they are absolutely reliable. It is very well possible that the solar system is unstable, and that the gravitational attractions between the different planets may lead one of the planets to suddenly escape into outer space.

  19. 2) A double pendulum is a classic chaotic system A double pendulum is simply two pendulums attached end to end. * the motion of a double pendulum is highly dependent on the starting position. * Due to erratic behavior different starting position lead to very different paths. * Sensitivity on initial conditions means that starting the pendulum in a slightly different position would cause drastically different behavior. * The double pendulum is also an example of dynamical system.

  20. Take a dynamical system as a point in the space. As time passes the point moves around. • For the double pendulum this point would be the position of the pendulum tip. • Sometimes, a dynamical system moves to and stays near a certain point such a point is called Attractor. • Attractors can also be sets of point for example a circle could be an attractor.

  21. blue image is one trajectory of the Lorenz system with (σ, ρ, β) = (10, 28, 8/3) started from the initial point (0, 0, 1). The yellow image is for the same parameters but a different initial condition, (0, 0, 1+ε) where ε = 10-5.

  22. The double pendulum at small angle.

  23. The double pendulum at large angle.

  24. Edward Lorenz work * Developed a set of equations to model a simplified weather system. * Basically the equations model the flow of fluid (Particularly air)from a hot area to a cold area. This is called (Convective Fluid Flow).

  25. The model is a system of three ordinary differential equations now known as the Lorenz equations:

  26. The solution to this set of differential equation is a dynamical system. • We can plot the solution with varying initial conditions. • This plots is sometimes called the Lorenz Butterfly due to its shape. • The plots seems to spiral a round two distinct points one on the left and one on the right

  27. These two points are attractors, but of different sort than normal they are known as “Strange Attractors”. • Basically this means that we cannot predict when a solution will jump from one attractor to another  .

  28. A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3

  29. Conclusion Finally dynamical systems can be chaotic if these conditions emerge on it determinism,nonlinearity, instabilities, and sensitivity on initial conditions. Chaotic systems can only be solved numerically. Only with the availability of modern computers has it become possible to study these phenomena . The related areas of nonlinear mechanics and Chaos are very modern and are hot topics of current research.

  30. References • Chaotic Dynamics: An Introduction Based on Classical Mechanics, Tamás Tél‏،Márton Gruiz‏,Cambridge University PressNewYork,2006,P3 • http://www.physicsplanet.com/articles/chaos-theory-simplified • Deterministic Chaos: An Introduction By Heinz Georg Schuster, Wolfram Just,John Wiley & Sons,2005,P89 • Physics and Theoretical Computer Science: From Numbers and Languages , Jean-Pierre Gazeau, Jaroslav Nešetřil, Branislav Rovan, IOS Press,2007,P71

  31. Chaos, Complexity, Curriculum and Culture: A Conversation, William E. Doll, M. Jayne Fleener, John St. Julien ,Peter Lang, 2005,P154 • http://www.merriam-webster.com/dictionary/chaos • http://en.wikipedia.org/wiki/Chaos_theory • Chaos Theory, a suggestion toward deterministic reality,Jason Turonie,Faculty Sponsor: David Miller, Department of Philosophy • http://pespmc1.vub.ac.be/CHAOS.html

  32. Any Question

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