Homepage of Benjamin Wright

Math 595:

Project Proposal

I am interested in working on a project related to dynamical systems (e.g., the logistic map). Specifically, I would like to create an HTML5/Javascript-based webpage with real-time interactive animation of functional iteration.

In particular, I aim to implement a method of extending the process of iterating a function n times to fractional/real values of n. That is, instead of looking at the sequence of discrete iterations x, f(x), f(f(x)), f(f(f(x))), ... , f^n(x), ..., I will consider a smooth deformation of the line y = x such that, when evaluated at the times t=1, t=2, t=3, ..., the deformation becomes the functions f(x), f(f(x)), f(f(f(x)))..., respectively.

To this end, I may follow one of the methods outlined on the Wikipedia article for functional iSteration: https://en.wikipedia.org/wiki/Iterated_function#Fractional_iterates_and_flows.2C_and_negative_iterates

The goal will be to produce an animation of this deformation for various functions f(x), including the logistic maps f(x) = rx(1-x). If feasible, I'm hoping to be able to produce a visualization of this deformation for complex-valued functions as well as real-valued functions.