Governor’s School for the Sciences Mathematics Day 4

The truth knocks on the door and you say, "Go away, I'm looking for the truth," and so it goes away. -- Robert Pirsig, from “Zen and the Art of Motorcycle Maintenance”

MOTD: Benoit Mandelbrot (Poland) Created interest in Fractal Geometry from ‘length of the coast of England’ question Mandelbrot set

Nonlinear Dynamics Dynamical System: x(n+1) = f(x(n)) Sequence: x(0), x(1), x(2), … A sequence can (1) tend towards a value (finite or infinite), (2) oscillate between several values, or (3) none of the above (chaos) Even if bounded, the orbit can be quite interesting

Periodic Points; Cycles x* is a periodic point of order k if f k (x*) = x* x* generates a k-cycle, an orbit O + (x*) = {x*, f(x*), …, f k-1 (x*)} k-cycles can be attracting or repelling; analyze the DE y(n+1) = f k (y(n))

Example y = f(x) = x 2 -1y = f 2 (x) = (x 2 -1) 2 -1

Example (cont.) f(x) = x 2 -1 has 2 equilibrium points f 2 (x) has 4 equilibrium points, 2 are from f(x), other 2 form 2-cycle: {0,-1}, i.e. f(0) = -1, f(-1) = 0 Since 0 and –1 are stable equilibria for f 2, the cycle {0,-1} is an attracting cycle, i.e. if start near 0 (or –1) then the iterates tend towards {0, -1}.

Bifurcation Diagram Suppose f(x) depends on parameter As varies, the equilibria points change and their stability status changes Stable cycles come and go also Graph of stable objects (pts, cycles) vs. is a Bifurcation Diagram

Example: f(x) = x 2 -

Complex DE Uses complex arithmetic: (a+bi)(c+di) = ac-bd + (ad+bd)i Plot value a+bi as point (a,b) Orbits are paths in plane Bounded means: ‘stays near the origin’ Main example: Quadratic map z(n+1) = z(n) 2 + c

Julia Set Restrict to bounded vs. unbounded Fix c in quadratic map Filled Julia set J(c) is the set of all z(0) such that the orbit of z(0) is bounded (True Julia set is the boundary of the Filled Julia set)

Mandelbrot Set Set of all c such that the Julia set J(c) is connected Equivalent to set of all c such that the orbit of 0 under the map z 2 +c is bounded Both Mandelbrot and Julia sets are fractals meaning they have non-integer dimension, they are also self-similar meaning certain parts look like the whole thing

Today’s Lab Julia and Mandelbrot Sets (Lab 4) If you have time, go back and finish parts of Labs 1-3 you haven’t done No Homework!