Adams-Bashforth Basins for z^3-1, step size h=1.99

        Magifractificator

This page contains movies that accompany the paper Approximations of Continuous Newton's Method: An Extension of Cayley's Problem by Jon Jacobsen, Owen Lewis, and Brad Tennis. The movies show fractal basins of attraction for several numerical approximations of the continuous Newton flow. For example, using an Euler approximation of step size delta yields the "damped Newton's method" with damping coefficient delta. When delta=1 this is the classical Newton's method. The first movie below (Newton's Method) shows what happens to the basins of attraction for the cubic polynomial p(z)=z3 - 1 as delta ranges from 0.4 to 1.995. The other movies show what happens when different numerical approximations of the continuous Newton flow are made. Enjoy! At the bottom of this page is a link where you can download a Mac program that will allow you to generate your own fractals and movies for arbitrary 2 dimensional vector fields.

Newton's Method Heun's Method
Runge-Kutta Order 2 Method RK Order 2 Quadrant II Zoom
Adams-Bashforth Refined Newton
Singular Set A for PDE with u - u3 Singular Set B for PDE with u - u3

The images and movies were generated using Magifractificator. This application was written by Brad Tennis and will run on the Macintosh OS X operating system. User's interested in the program can download it here:

  • Magifractificator (dmg)
  • The mathematics behind the ideas presented here are discussed in the reference Approximations of Continuous Newton's Method: An Extension of Cayley's Problem, Electron. J. Diff. Eqns, Conference 15 (2007), pp. 163-173.
  • Other good references are: