ChaosPlot
by Jason Regier

What is ChaosPlot?
	ChaosPlot is a simple program which plots the chaotic behavior of a 
damped, driven anharmonic oscillator.  This program, intended to be used for 
educational purposes, allows the user to change parameters in the differential 
equation describing the oscillator and to modify various plotting parameters.

How Does It Work?
	ChaosPlot evaluates the following differential equation:
	
	d^2x/dt^2 = F_0 sin(wt) - kx -qx^3 -b dx/dt 
                       
for various values of the parameters F0, w, k, q, and b and the initial conditions x0 
and v0.  F0 is the amplitude of the driving force and w controls its frequency.  k is 
the spring constant for the ideal spring portion of our spring potential.  q represents 
the amplitude of the anharmonic term of the spring potential which makes our 
spring vary from one which follows HookeÕs law.  b is the amplitude of the drag 
term, which depends on the velocity of the springÕs motion.
	To plot this complex equation, ChaosPlot splits the second order differential 
equation into two first order differential equations and solves them simultaneously 
using a fourth-order Runge-Kutta algorithm:

	dx/dt = v  and  dv/dt = f(x,v) = F_0 sin(wt) -kx -qx^3 -bv
  
The results of this algorithm are passed to a plotting routine which plots the points 
on either the x vs. t, v vs. t, or v vs. x axes.  The input parameters, including 
equation constants, scale of the plot, step size, and time length, can be controlled 
from the Change Plot Parameters menu item under the File menu.  To change 
between each of the 3 plots listed above, one need merely select the desired plot 
from the Plot menu.  As usual, to change the printing parameters, select Page 
SetupÉ from the File menu.  To Print the plot currently displayed on screen, 
select PrintÉ from the File menu.  The current plot parameters will be printed 
below the current plot.  To quit the program, click in the close box in the upper left 
of the Plot window or select Quit from the File menu.

Notes...
	Here are some physical interpretations of various input equation parameters:
	F0 = b = q = 0, k > 0		Simple harmonic oscillator
	F0 = k = q = 0, b > 0		Nonoscillatory damped motion
	F0 = q = 0, k and b > 0		Damped simple harmonic oscillator
	F0 = 0, k < 0, and q, b > 0	Damped anharmonic oscillator
	q = 0 and F0, w, k, b > 0		Damped, driven harmonic oscillator
	k < 0 and F0, w, q, b > 0		Damped, driven, anharmonic oscillator

(table omitted) 

	Remember, using excessively small step sizes will lead to a very accurate 
solution of the differential equation, but can take a VERY long time to execute and 
to print.
	You might notice some small wobbles in the plots produced by ChaosPlot.  
These tiny wobbles are most likely not real, but are rather a result of connecting 
successive data points.  The finite step size in the Runge-Kutta algorithm means 
that curves are not truly smooth; they are formed by connecting successive data 
points with straight lines.  I have found that a smaller step size does not necessarily 
remove the wobbles.  In fact, sometimes the wobbles may be more noticeable 
because of severe overplotting using small increments.  If you really care whether a 
wobble is truly there or is a figment of the algorithm, change the plot parameters to 
zoom in on that portion of the plot.

Legalese...
	ChaosPlot is wholly the work and a fully owner product of Jason Regier, all 
rights reserved.  In addition, regardless of whether ChaosPlot fails of its essential 
purpose, in no event will Jason Regier be liable to you for any special, 
consequential, indirect or similar damages, including any lost profits or lost data 
arising out of the use or inability to use the software.  Execution of the ChaosPlot 
program is tacit agreement to the above statements.  ChaosPlot is distributed as 
Shareware and may not be sold in any way without the express written consent of 
the author, Jason Regier.

I hope you have fun with ChaosPlot and learn a little about the fundamentals of 
chaos!  Experiment with this program and send me some interesting parameter 
settings.  If you think ChaosPlot is worth keeping, please send me $5-$10 as a 
reimbursement for the time I spent working on this project.  I can be reached at:
	Jason Regier
	Platt Campus Center
	Harvey Mudd College
	Claremont, CA  91711
or via email: Jason_Regier@hmc.edu