Governor’s School for the Sciences

MathematicsDay 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

• 1924- (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 fk(x*) = x*

• x* generates a k-cycle, an orbit O+(x*) = {x*, f(x*), …, fk-1(x*)}

• k-cycles can be attracting or repelling; analyze the DE y(n+1) = fk(y(n))

Example

y = f(x) = x2-1

y = f2(x) = (x2-1)2-1

Example (cont.)

• f(x) = x2-1 has 2 equilibrium points• f2(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 f2, 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) = x2 -

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 z2+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!