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/>.