General Chaos Theory Notes
Deterministic chaos:
Three or more bodies in phase space have unpredictable interaction but are deterministic.
Peano curve (1890): 1 dimensional curve that tries to fill a 2 dimensional plane.
Cantor set (1883): 1 dimensional line that tries to fragment to zero dimensional points.
Siepinski Gasket (1916): Random plot based on simple rules creates self-similar pyramid with dimension between 1 and 2 and infinite complexity.
Koch curve (1904): Coastline like plot, has approx. same fractal dimension as a real coastline.
Feigenbaum (mid 1970s), period doubling route to chaos is a general feature of feedback, the ratio in size between one step and the next is universal, the Feigenbaum number, 4.669…
Lev Landau (1940s): turbulence is a cascade of periodic cycles.
Mandelbrot (1975): objects like Peano curve can be described as having a fractional dimension which lies between whole number dimensions, fractals.
Link between chaos and fractals:
Period doubling plot of, say, logistic map: Feigenbaum point, between order and chaos (i.e. where interval between steps is negligible), branches of bifurcation form Cantor set.
A fractal dimension can be measured by taking a piece of a fractal and scaling it to match a larger piece, then measuring the ratio of this scaling factor to the original length.
Action of DNA is a recipe rather than a blueprint.
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