Nonlinear Dynamics and Chaos                                                                          Daylight Program

 The graph shows the phase portrait for a pendulum with no damping.  The differential equation is d^2(theta)/dt^2 + sin(theta) = 0.  The vertical axis represents the angular velocity:  d(theta)/dt  and the horizontal axis represents theta.  The inside loop corresponds to the pendulum bob rocking back and forth in a periodic motion.  The largest closed loop corresponds to the pendulum bob being raised to the top (theta = pi) and then given a slight displacement to the right, the pendulum bob swings back and forth, but very slowly near theta = pi or theta = -pi.  The curve at the top represents the situation where the pendulum bob is given a large positive velocity.  Thereafter the bob will swing through the bottom point but with enough velocity so that it goes over the top and repeats this motion indefinitely.   This is one example of an application that is studied in the Nonlinear Dynamics and Chaos course (Math 3020).
 

Links to websites for mathematics, research, and other favorites:

1)  Example of application in nonlinear dynamics:  Insect Outbreak Example:  Spruce Budworm
2)  General undergraduate mathematics website:  http://archives.math.utk.edu
3)  Evans Harrell Homepage (at Georgia Tech)  http://www.math.gatech.edu/~harrell
4)  American Mathematical Society:  http://www.ams.org
5)  American Physical Society:   http://www.aps.org
6)  Mathematics Physics:    http://www.ma.utexas.edu/mp_arc/
7)  Weather:   Current U.S. Weather
8)   Swiss Tourism.
9)  Mark Harden's Archive:  First Impressionist Exhibition 1874.

COMPLEX ANALYSIS EXAMPLE:  The graph shown illustrates constant temperature curves which are
determined via a function of complex variables.   The steady-state temperature is zero on the x axis and 100
on the union of two rays which meet at the point (0,pi) at an angle of 90 degrees.
 
 

The other curves illustrate the lines of constant temperature for temperatures of 25 (green),  50 (yellow)
and 75 degrees (purple)..