Tuesday, Dec 4, 2012

First, we note a relation between the Mandelbrot set and the logistic map bifurcation diagram.

Next, we see that for systems with several stable behaviors, the choice of which starting condition leads to which eventual behavior can be very delicate.

In fact, we shall find the Mandelbrot set hiding in some of these situatons.

Restricting our attention to just real numbers, the Mandelbrot iteration scheme reveals an interesting relation with the familiar logistic map bifurcation diagram.

Real Newton's Method. Newton's method for finding roots. Newton's method recast as graphical iteration. A familiar shape.

Complex Newton's Method. Julia sets related to finding the roots of equations. SImilar features arise in magnetic pendula and in light reflected within a pyramid of shiny spheres.

Universality of the Mandelbrot Set. Newton's method for a family of cubic polynomials revealed more copies of the Mandelbrot set. Yet Newton's method is nothing like z → z² + c. Further investigation shows we're surrounded by Mandelbrot sets.

Restricting our attention to just real numbers, the Mandelbrot iteration scheme reveals an interesting relation with the familiar logistic map bifurcation diagram.
Real Newton's Method. Newton's method for finding roots. Newton's method recast as graphical iteration. A familiar shape.
Complex Newton's Method. Julia sets related to finding the roots of equations. SImilar features arise in magnetic pendula and in light reflected within a pyramid of shiny spheres.
Universality of the Mandelbrot Set. Newton's method for a family of cubic polynomials revealed more copies of the Mandelbrot set. Yet Newton's method is nothing like z → z² + c. Further investigation shows we're surrounded by Mandelbrot sets.