fractals and chaos: things are complex
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Fractals and Chaos:Things are Complex
Chris Jerniganand
Estelle Diener-Stroup (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
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
Statistics
Fractal data set▪ Cannot be described by mean or variance
(Liebovich and Scheurle 2000)
Statistics and Sample Size Data distribution with increasing
amounts of new data
(Liebovich and Scheurle 2000)
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)
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)
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 cardioverter defibrillators 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 characterizesthe 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)
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)
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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