Iterate

Copyright 2015 by Alden Walker. See my software page for other programs. Thanks to Sarah Koch for the idea. I found useful WebGL examples here and here.

Iterate numerically finds periodic orbits of a rational function near a specified point. This works best on the Julia set, where there are many periodic orbits. Left mouse drag and scroll to navigate the plot. Right click either selects the point near which to search for a periodic orbit or sets the initial point of an orbit. The function syntax is straightforward; the default {0:4,1:-4,2:1}/{2:1} is the function (4-4z+z^2)/z^2. Plotting the Julia set does not make sense for functions whose Julia set is the entire plane, as with the default. Click "go" for a simple example.

Navigation

Click and drag the left mouse to move the plot. The mouse wheel zooms. There are also navigation buttons on the right. Recenter centers the plot on the selecting search start point.

Right mouse

The right mouse either selects a search start point (default), or selects a point from which an orbit is started. Change the behavior of the right mouse or the orbit length on the right.

Function input

To use the rational function (a1*z^p1 + ... + ak*z^pk)/(b1*z^q1 + ... + bl*z^ql), set the function input to {p1:a1, ..., pk:ak}/{q1:b1, ..., ql:bl}. If the denominator is omitted, it is assumed to be the constant function 1. Complex coefficients are allowed. The function is updated (and replotted, if the Julia set plot is selected) as soon as the input box loses focus.

Julia set plotting

If the Julia set of the function is not the entire plane and does not contain infinity, it can be useful to plot it. It is plotted by checking which points go to infinity under iteration. In particular, the plot shows the filled Julia set. The monochrome and color options determine whether the exterior of the filled Julia set is colored according to how quickly the point escapes to infinity.

The Julia set is plotted using WebGL, which may not be supported in some mobile and old browsers. It may also be slow, depending on your graphics hardware.

Periodic orbit searching

After selecting a search start point, radius, and maximum orbit length, click the "go" button to initiate the search. The program iterates the chosen point until it comes back close to itself and then uses Newton's method to try to find an exactly periodic orbit. The output area shows the search progress and, if the search is successful, the found periodic orbit. If the Julia set isn't everything, the start point should lie on the Julia set to have any real hope of finding an orbit.

Technical comments

An iterate of the initial point is defined to be "close" to itself if the derivative of the iterated map at the initial point is large enough that, if the iterated map were linear, we could close up the orbit by moving the initial point less than the specified radius. The function is not linear, so this doesn't always work.

The Julia set overlay is created by having one 2d canvas directly on top of a webgl canvas. The webgl canvas plots the Julia set, and the 2d canvas plots everything else. The shader code source in this website contains a placeholder where the function should be computed. Every time the function is updated, the program reloads the source and replaces the placeholder with dynamically generated code for the given function.

There are ways to rigorously certify the existence of a periodic orbit near a numerical solution (see the Kantorovich theorem and Lemma 11.1.1 in here), but this program does not do this.

Functions whose Julia set is the entire Riemann sphere

As long as the maximum orbit length is large enough, the program should be able to find an orbit close to any point.

• {0:4,1:-4,2:1}/{2:1}

Functions with interesting Julia set

• {0:-1,2:1}
• {0:i,2:1}
• {0:1,3:0.18}/{4:0.25, 2:1}
• {0:1,2:-1,5:1}/{0:1-0.35i, 2:1}
• {0:-0.4+0.6i,2:1}