Analysis:


We used the Maple software to investigate this model. Here is our worksheet that we used to explore this model.
[worksheet]


ANALYSIS

Using Maple software we found that the only steady state of the system is x = 0, y = 0, z = 0, w = 0.

Plugging these values into the Jacobian matrix obtained through Maple as well, we get eigenvalues for the given Jacobian.

The steady states depend on the values of lambda and c1, and therefore, varying the values of those two parameters allowes us to change the stability of the points. Using Maple we can draw phaseportraits for different values of lambda.

Lambda = -.1


Lambda = 0


Lambda = 2

The stability of the fixed point (0,0,0,0) depends on the values of lambda and c1. Since the stability of a fixed point is determined based on the signs of the real parts of eigenvalues, we are only interested in signs for:

- from the first eigenvalue: lambda
- from the second eigenvalue: lambda
- from the third eigenvalue; -1/2*c1
- from the fourth eigenvalue: -1/2*c1

Therefore, for an unstable source: lambda > 0, c1 < 0 for a stable sink: lambda < 0, c1 > 0; for an unstable saddle: both lambda and c1 have the same sign.
If all real parts of all eigenvalues are non-zero, the fixed point is hyperbolic. Since our eigenvalues depend on the values of lambda and c1, we can determine if the point is hyperbolic or not from the values. If either lambda or c1 are equal to zero, we cannot determine the stability of the point, and the point is non-hyperbolic, which indicates that a bifurcation occurs. There is also a possibility that it is a Hopf bifurcation.

When lambda and c1 are both equal to zero, we have a possible Hopf bifurcation. Looking at phase portrats below, it is clear that, while it may seem that we have a transcritical bifurcation at (0,0,0,0), it is actually a Hopf bifurcation, since a stable fixed point at the origin is replaced by a limit cycle (periodic orbit) as lambda increases.

Exotic Behavior:

(-.1,1)
This model does have some chaotic tendancies when the values of lambda and c1 take certain values. In this case, (-.1,1).

This system also has a chaotic trajectory for these values.

Here, we can see a chaotic trajectory that has a higher cyclic rate than the forcing terms. This photo is the trajectory after one complete cycle of the forcing terms. With these values of c1 and Lambda, the system quickly goes to infinity shortly after this.



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