This model has three main trajectories based on the values of lambda and c1.
The motion of the model seems to either decay to the origin, stablize into
an orbit, or enter into a chaotic state where it seems to stay within a defined
Usint XRK software we were able to see some strange behavior of the
system. While there are many combinations of parameters and initial
conditions that give that weird behavior, we only looked at a few
Here is an example of such behavior. If x and y initial conditions are
set to a positive number (smaller than, say 5), and z is a small positive
number, and w is a negative number around -3.0, varying c1 and lambda can
get the system from more or less stable oscillations to chaotic behavior
which the program cannot handle. Because the graph grows very fast right
before the program crashes, it is extremely hard to stop it just in time
and be able to show a graph of the system. Unfortunately, we were unable
to do that very well. The strangest behavior was obtained when the parameters were
c1 approximately equals to -2, lambda equals to 2. The x-y graph does not
even complete one oscillation, it crashes just a few seconds after
starting. When we made a graph of lambda vs. c1, it shows that for a
negative c1 and a positive lambda we have a source. Therefore, varying
those parameters and initial conditions will give us chaotic behavior.
Project Home Page
Conclusions and Credits
We would like to extend a special thanks to the XRK program, "Introductory Differential
Equations: From Linearity to Chaos" (Kostelich and Armbruster), and Professor William
Farr (Worcester Polytechnic Institute) for his support and guidance during this endeavor.