Deep Simplicity - Chapter 3

Chaos Out of Order

• Iterative Process - dependent on prior state
• Logistic Equation - can be used to illustrate period-doubling
• Example: how the population of a species changes
• x (next) = B * x (current) * (1 - a)
• B = Birth rate
• x = Population
• a = Premature death rate (prior to mating)
• If B<1:  population dies
• If B>1 & B<3:  x eventually settles at roughly 0.66 (2/3 of max population)
• If B>3:  single attractor state becomes two (period doubles to 2)
• Practically speaking, this results in 1 year of overpopulation, followed by 1 yr of underpopulation, etc - this cycle continues back and forth (2 stable states)

• Robert May - first to conduct tests of behavior of logistic equations
• B=3.4495 leads to 4 states
• B=3.56 leads to 8 states
• B=3.569 leads to 16 states
• B=3.5699 (or higher) leads to infinite states
• Mostly chaotic (infinite) beyond 3.5699, although there are small pockets of order, which eventually lead to chaos, on a smaller scale (self-similar)
• Chaos to order, chaos to order, etc.
• James Yorke & Tien Yien Li - "Period Three Implies Chaos"
• Mitchell Feigenbaum - Period-doubling is not unique to logistic equations; it is product of iterative feedback process (self-referential)
• Feigenbaum's Number - 4.669:1 is the universal ratio of period-doubling in all self-referential systems

Fractals

• Giuseppe Peano - constructed a curve (Peano Curve) that completely fills a plane (w/o ever crossing)
• Pattern is self-similar and infinitely long, but contained in finite area

• Curves (nonintersecting) are 1 dimensional and planes are 2 dimensional
• Mandelbrot -  realized that the Peano Curve is somewhere between 1 & 2 dimensions

•  George Cantor - Cantor Set
• Produced by iteration; self-similar
• Divide a line by thirds
• Divide the two outer thirds by thirds, repeat, etc.

•  Warclaw Sierpinski - Sierpinski Gasket
• Self-similar

•  "Chaos & Fractals" - Heinz-Otto Peitgen, Hartmont Jurgens, Dietmer Saupe
• We observe fractals in nature:

• Helge Von Koch - Koch Curve
• Infinitely long (although it has end points)
• Koch snowflake
• Koch island
• Richardson realized measurements of coastlines erred by up to 20%
• Due to progressively smaller scales of measurement
• Each segment can be scaled by a factor of 3 to replicate original (self-similar)

 Richardson's coastline paradox