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Fractals and Chaos: Things are Complex

Chris Jernigan and Estelle Diener -Stroup. Fractals and Chaos: Things are Complex. (Can and Moore 2010). Spatial Chaos. Chaos in space Infinitely detailed line or object..... Can you find the exact area of the shaded region?. ( Baranger 2010). Examples of Fractals.

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Fractals and Chaos: Things are Complex

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  1. Chris Jernigan and Estelle Diener-Stroup Fractals and Chaos:Things are Complex (Can and Moore 2010)

  2. Spatial Chaos • Chaos in space • Infinitely detailed line or object..... • Can you find the exact area of the shaded region? (Baranger 2010)

  3. Examples of Fractals • http://upload.wikimedia.org/wikipedia/commons/6/6d/Animated_fractal_mountain.gif • http://upload.wikimedia.org/wikipedia/commons/f/fd/Von_Koch_curve.gif • http://upload.wikimedia.org/wikipedia/en/7/74/Animated_construction_of_Sierpinski_Triangle.gif

  4. Statistics • Fractal data set • Cannot be described by mean or variance (Liebovich and Scheurle 2000)

  5. Statistics and Sample Size • Data distribution with increasing amounts of new data (Liebovich and Scheurle 2000)

  6. Games of Chance • Normal Coin Toss • Tails Win nothing, Head Win $1 • (1/2)*1 + (1/2)*0 = $0.5 • On average you should win $0.5, so could fairly gamble $1 (Liebovich and Scheurle 2000)

  7. Games of Chance • St. Petersburg Coin Toss Game • Flip a coin until it lands on heads • Lands: heads = $2; tails, heads = $4; tails, tails, heads = $8 • (1/2)*2+ (1/4)*4+(1/8)*8.... = 1+1+1......= ∞ • Half the time you win at least $2 so could fairly wager $4, however casino will correctly argue that the mean winnings per game is infinite and therefore should put up more than all the money in the universe to play the game (Liebovich and Scheurle 2000)

  8. Probability Density Function • The probability that any measurement has a value between x and x+d(x).... The PDF of the times between episodes of the onset of rapid heart rate measured in patients with implanted cardioverterdefibrillators from the work of Liebovitch et al. [1]. Most often the time between episodes is brief. Less often the time is longer. Infrequently it is very long. There is no single average time that characterizes the times between these events. The PDF has a power law form that is a straight line on a plot of log[PDF(t)] versus log(t) (Liebovich and Scheurle 2000)

  9. So what does that Mean????? Anyone?... Anyone? • “Even when events occur at random, they are often bunched together and the bunches have bunches which have bunches......” • “One purpose of studying chaos though fractals is to predict patterns in dynamical systems that on the surface seem unpredictable” (Liebovich and Scheurle 2000) (Presley 2010)

  10. Refrences • Baranger, M. 2010. Chaos, Complexity, and Entropy: A physics talk for non-physicists. MIT. <http://necsi.org/projects/baranger/cce.pdf>. • Can, T. And Moore, W. Fractals and Chaos in the Driven Pendulum: A Review and Numerical Study of a Strange Attractor. 2010 • Liebovich, L.S. and Scheurle D. 2000. Two Lessons from Fractals and Chaos: Changes in the way we see the world. Complexity 5(4). John Wiley & Sons, Inc. 2000. • Presley, R.E. 2010. Fractals in Nature. <http://people.bathac.ulc/rjp30/>.

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