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Chaos

Chaos. Spring 2006 Tyler Hardman Brittany Pendleton Shi-Hau Tang. Outline. Motivation Historical Background Theories & Examples of Chaos Sensitivity of Initial Conditions Feedback and Mapping Non-linearity The PASCO Pendulum: a damped & driven SHO Lorenz’s Waterwheel Conclusions

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Chaos

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  1. Chaos Spring 2006 Tyler Hardman Brittany Pendleton Shi-Hau Tang

  2. Outline • Motivation • Historical Background • Theories & Examples of Chaos • Sensitivity of Initial Conditions • Feedback and Mapping • Non-linearity • The PASCO Pendulum: a damped & driven SHO • Lorenz’s Waterwheel • Conclusions • What chaotic behaviors were shown in the experiments? • Room for Improvement • What aspects could be investigated further?

  3. Motivation Chaos theory offers ordered models for seemingly disorderly systems, such as: • Weather patterns • Turbulent Flow • Population dynamics • Stock Market Behavior • Traffic Flow

  4. Pre-Lorenz History • The qualitative idea of small changes sometimes having large effects has been present since ancient times • Henry Poincaré recognizes this chaos in a three-body problem of celestial mechanics in 1890 • Poincaré conjectures that small changes could commonly result in large differences in meteorology Modern version of Lorenz’s model

  5. Theoretical Roots • Edward Lorenz stumbled upon chaotic behavior while studying weather prediction in 1960 • In 1962 Lorenz ran a computer simulation of three differential equations describing turbulent flow • The solutions are chaotic and can only be solved numerically • Strange Attractors are areas of equilibrium • Motion never reaches a steady state because it is chaotically bounded by attractors

  6. First noted by Edward Lorenz, 1961 Changing initial value by very small amount produces drastically different results Sensitivity of Initial Conditions The Strange Attractor of the turbulent flow equations. Each color represents varying ICs by 10^-5 seconds. Motion seems chaotic, but when you piece it together there is order.

  7. The Butterfly Effect The effect Lorenz accidentally discovered in 1960 A change as small as air displaced by the wing of a butterfly can dramatically change resulting weather conditions Long-term weather predictions are impossible The Butterfly is used as the insect of choice due to the strange attractors’ butterfly-like appearance.

  8. Logistic Equations Feedback and Mapping Feedback is when inputs of a system depend on previous outputs Mapping uses the iterative relationship of systems involving feedback to graphically solve chaotic systems Bifurcation occurs when the number of equilibrium states jumps from one to two

  9. Fractals & Population Growth • Fractals illustrate self-similarity, or lack of scale, due to feedback • The Mandelbrot Setcomplex iterative relationship • Population Dynamics • Evolution of chaotic behavior modeled with mapping

  10. Non-Linearity • Most physical relationships are not linear and aperiodic • Usually these equations are approximated to be linear • -Ohm’s Law, Newton’s Law of Gravitation, Friction Nonlinear diffraction patterns of alkali metal vapors.

  11. The Damped & Driven SHO This motion is determined by the nonlinear equation x = oscillating variable (θ) r = damping coefficient F0 = driving force strength ω= driving angular frequency t = dimensionless time Motion is periodic for some values of F0, but chaotic for others Driven here with F0 Damped here with r

  12. The PASCO Pendulum Weight attached to rotating disc Springs attached to either side of disc in pulley fashion One spring is driven by sinusoidal force Sensors take angular position, angular velocity and driving frequency data

  13. Mapping the Potential • Let the weight rotate all the way around once, without driving force • Take angular position vs. angular velocity data for the run • Potential energy is defined by the equation Two “wells” represent equilibrium points. In the lexicon of chaos theory, these are “strange attractors”.

  14. Phase Space Plots Each point represents angular position and velocity at a given time This particular plot shows a bigger potential well on the left

  15. Magnetic Damping • The magnetic damping of our pendulum has the most visible affect on Poincaré plots. • Phase space plots in 3D • The width of the phase path and damping are inversely related. Larger Damping Constant Smaller Damping Constant

  16. Non-Chaotic Oscillations Angular position is sinusoidal with respect to time. Frequency is same as driving frequency. Results in highly localized Poincaré plots. Blue data is chaotic Pink data is non-chaotic

  17. Chaotic Oscillations Angular position is disorderly and unpredictable Results in intricately patterned Poincaré plots Order seems to be present, but is much more complex than that found in the non-Chaotic case

  18. The Lorenz Water Wheel Follows the turbulent flow system of equations Water fills cups at a steady rate Holes in bottom of cups empty at steady rate As certain cups fill, others empty Wheel exhibits both clockwise and counterclockwise motion Sensitive to initial conditions, aperiodic, two attractors Two Attractors

  19. Conclusions • Which chaotic behaviors were successfully demonstrated? • Strange attractors were demonstrated well by the PASCO pendulum • What modifications could improve the present setup? • Increase the amplitude of the driving force • Increase tension in the springs as much as possible • Add supports for structural stability • Questions that future projects might answer • Quantitatively, how does the magnetic damping affect the Poincare plots? • What do the Poincare plots look like for different potential configurations? • How sensitive is the PASCO pendulum to initial conditions?

  20. Sources • http://en.wikipedia.org/wiki/Lorenz_attractor • http://www.imho.com/grae/chaos/chaos.html • http://www.adver-net.com/mmonarch.jpg • http://datalib.library.ualberta.ca/tornado/new1b.jpg • http://ib.cnea.gov.ar/~thelerg/rueda_loca.php • http://www.sparknotes.com/physics/oscillations/applicationsofharmonicmotion/section2.rhtml • http://www.uni-muenster.de/Physik.AP/Lange/Welcome-e.html • http://mkeadle.org/?m=200502 • http://home.att.net/~fractalia/newton/newton.htm Gleick, James. Chaos: Making a New Science. Penguin Books: New York, 1987. Lorenz, Edward. The Essence of Chaos. University of Washington Press: Olympia, 2003. Sardar, Ziauddin and Iwona Abrams. Introducing Chaos. Totem Brooks: New York, 1999.

  21. http://www.pha.jhu.edu/~ldb/seminar/images/bifurcation.gif http://amath.colorado.edu/faculty/jdm/gifs/RT3Bsection.gif http://www.gap-system.org/~history/Mathematicians/Poincare.html

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