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 orbital


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 combinations. 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.


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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.