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Commercial Tom Kreitinger MAT 305 – 9/28/2009

Can an insect create a tornado ?. Commercial Tom Kreitinger MAT 305 – 9/28/2009. Edward Norton Lorenz & Chaos theory. Commercial Overview: Part I - Who was Edward Norton Lorenz and what did he do? Part II - What is Chaos theory and what are some examples?. Part I.

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Commercial Tom Kreitinger MAT 305 – 9/28/2009

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  1. Can an insect create a tornado? Commercial Tom KreitingerMAT 305 – 9/28/2009

  2. Edward Norton Lorenz& Chaos theory Commercial Overview: • Part I - Who was Edward Norton Lorenz and what did he do? • Part II - What is Chaos theory and what are some examples?

  3. Part I • Who was Edward Norton Lorenz and what did he do?

  4. Edward Norton Lorenz1917- 2008 • American Mathematician & Meteorologist • 1948 - Postdoctoral work at MIT Department of Meteorology • 1955 - Director of statistical weather forecasting (pioneered by his department)‏

  5. 1961 Computer Weather Models • Lorenz studied the relationships between temperature, pressure and wind speed using computer models. • One day he takes a short cut and restarts a program half way through its run by manually entering initial values.

  6. 1961 Computer Weather Models • Enters a value .506 for .506127 (from a printout)‏ • The differences was only .000127 and was thought to be inconsequential. • He gets a cup of coffee while the program runs

  7. Tiny initial change major impact • Lorenz finds the tiny difference of .000127 (.506 and .506127) generates a weather pattern nowhere near the original. • The end result: Two completely different weather results!

  8. Lorenz Conclusions • Lorenz realized that a minute differences in the initial conditions - like a puff of wind could prove catastrophic to weather models. • Lorenz: “It implies that two states differing by imperceptible amounts may eventually evolve into two considerably different states, If, then, there is any error whatever in observing the present state – and in any real system such errors seem inevitable – an acceptable predication of the instantaneous state in the distant future may well be impossible.”

  9. Butterfly Effect • 1972 – Lorenz presents a paper at a conference in Washington, entitled “Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?” • Lorenz coins the phrase ”Butterfly Effect” (Lorenz Attractor)‏

  10. Lorenz – main idea • The “butterfly effect” shows how slightly different initial conditions can produce an actual result very different from previous. • Plinko – Try as you might dropping a puck produces a different result

  11. What is Chaos theory and what are some examples? Part II

  12. Some Terms: System • Any entity that changes with time is called a ”system” • Time prevents everything from happening at once. • All systems have variables (e.g. temperature, pressure and wind speed)‏ Another example: population of fish

  13. System Feedback • Feedback is a characteristic of any system in which the output or result affects the input of the system • Example: Feedback from a public address amplifier system Robert May fish population: x(next)=rx(1-x)‏

  14. Periodic System • A variable in a periodic system exactly repeats its past behavior after the passage of a fixed interval of time • Example: A swinging pendulum is periodic and deterministic

  15. Aperiodic System • Aperiodic behavior occurs when no variable affecting the state of the system undergoes a completely regular repetition of values • Example: Water as it goes down a drain

  16. Unstable Aperiodic System • Unstable aperiodic behavior is highly complex. It never repeats itself and continues to show the effects of any small change to the system. This makes exact predications impossible and appear random

  17. Double Pendulum • A Double Pendulum is a pendulum with an extra pivot joint added

  18. Unstable Aperiodic Example • The motion of a double pendulum is seemingly random • A characteristic of chaos is that a deterministic system may appear to generate random behavior.

  19. Chaos (definitions)‏ • ”The ability of simple models, without inbuilt random features, to generate highly irregular behavior” • ”Apparently random recurrent behavior in a simple deterministic (clock-work-like) system” • ”The qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamic systems”

  20. Summary • Edward Norton Lorenz's “butterfly effect” illustrates how slight changes in the initial conditions of a system can greatly affect outcome. • Chaos is the occurrence of aperiodic, apparently random events in a deterministic system

  21. Sources • Book: “Introducing Chaos” - Ziauddin Sardar and Iwona Abrams ISBN 1-84046-078-4 • Book: “50 mathematical ideas you really need to know” - Tony Crilly ISBN 1-84724-147-6 • Double Pendulum - http://en.wikipedia.org/wiki/Double_pendulum • http://en.wikipedia.org/wiki/File:Double-compound-pendulum.gif • Chaos Theory - http://en.wikipedia.org/wiki/Chaos_theory • Edward Lorenz - http://en.wikipedia.org/wiki/Edward_Norton_Lorenz • Attractor - http://en.wikipedia.org/wiki/Attractor • Butterfly Effect - http://en.wikipedia.org/wiki/Butterfly_effect

  22. The End Can an insect create a tornado? Thank you Tom Kreitinger

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