Monday, May 11, 2009

57) The Iterated Logistic Map and The Mandelbrot Set: An Intro To Mathematical Dynamical Systems

An Integer Dynamical System With A Curious Array Of Orbits

fn+1= (fn+fn-1)/2 if even odd else (fn+fn-1)/4 if even even or odd odd

58) Iterates Of The Unimodal Map: Intro To Concepts In Mathematical Dynamical Systems

Xnew=mXold(1-Xold) and Mandelbrot set stability, sensitive dependence on initial conditions, repellors, periodic orbits, chaos, phase space, bifurcations

Pick a number, say 3. double it, 6. double it again 12, well you see where that goes.

pick a fraction like 1/2, double it, 1, double it again, 2, double it again, 4

what if we square numbers: 2, 4, 16, 256 grows wildly

how about start with 1/2? 1/4, 1/16, 1/256... that one keeps shrinking forever, but at least it's not unbounded. in fact it approaches a particular number: 0.

what if i decide to multiply a number by -2? start with 3, we get -6, 18, -54, 162, that one bounces back and forth wildly

what if i decide to multiply by -1/2? start with 2, -1, -1/2, 1/4, -1/8, 1/16... that one swings back and forth but the swings are smaller and smaller and that one zeros in on 0.

what if i decide to multiply by -1? 2, -2, 2, -2.. huh, this one keeps oscillating back and forth between two values.

god it would take days to write this! can i make an anthology instead? or at least pilfer a chapter from Devaney and rewrite it to emphasize my own points?

find a system with period four i.e. multiply i

then look at exp(2pi/3) for period 3.

now follow mx(1-x), look at fixed, periodic, attractor, repellor, bifurcation,
structural stability
sensitive dependence on initial condition
wandering orbit
infinite many orbits
whole bifurcation cascade
cantor dust

then look at z^2+c
the space of bifurcations on c
Mandelbrot set.

what were the concepts from Liu: singularity theory? structural stability, genericity? ...

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