# GraphicalAnalysis.m

## A Brief Introduction and Tutorial

Thomas LoFaro
Department of Mathematics and Computer Science
St. Peter, Minnesota 56082
tlofaro@gustavus.edu

### Description

GraphicalAnalysis.m is a Mathematica package designed to aid students and researchers in analyzing the dynamics of functions of a single real variable.  The two main functions are PlotIterates which generates stairstep diagrams and Bifurcation which generates bifurcation diagrams.  Both of these functions provide a significant amount of flexibility and have been designed so that their syntax mimics that of most Mathematica commands.  In the following sections the use functions will be demonstrated.  To download GraphicalAnalysis.m click here.

### Iteration

The functions PlotIterates and ListIterates perform basic iteration. PlotIterates generates a stairstep diagram while ListIterates simply returns a list of iterates.

Let's consider the family of odd symmetric, cubic polynomials

(For a more complete description of the dynamics of these polynomials see T. LoFaro (1997) The Dynamics of Symmetric Bimodal Maps, International Journal of Bifurcation and Chaos, 7:2735-2744.)

Let's begin by using the PlotIterates command to plot the function and generate a stairstep diagram of the iterates.  The help cell below summarizes the syntax and options of this function.

You can also simultaneously generate stairstep diagrams for multiple initial conditions as in the following example.  Each initial condition is colored differently.

To better understand the asymptotic dynamics we can increase the total number of iterations and increase the number of hidden iterations by changing the Iterations and Transients options.

Because each of these orbits converges to a period two orbit we might want to view the iterates of f(f(x)) instead.  We can do this using the Composition option.

The ListIterates command is similar to the PlotIterates command with two significant differences; only one initial condition can be specified and the Composition option is not applicable.

We can plot this list by using the built-in ListPlot function.

### Generating Bifurcation Diagrams

This package also contains the function Bifurcation that will generate bifurcation diagrams of a family of functions.

As in the PlotIterates command, multiple initial conditions can be specified.  These intial conditions will be used for every parameter value.  The Transients and Iterations options are identical to those described above.  The Slices option is used to specify the number of equally spaced parameter values that should be used in generating the bifurcation diagram.  Below is an example using the family of functions described above.

If you find this package useful you may download it by clicking here.  Comments and/or suggestions can be emailed to tlofaro@gustavus.edu