## Some interesting graphics:

All graphs are created using `xrk` and grabbed via `xv` unless otherwise noted.

This is the graph for where y' = 0. At y = 0, there is a discontinuity.

1. ( = 0.5 and c1 = 0.5)

The solutions aof z,w pproach a limit cycle that is a circular pattern.

The solutions of x,y approach a limit cycle that appears to be very similar to the phaseportrait of the van der Pol oscillator.

The phaseportrait of the van der Pol equation, as plotted by `xmaple`, using = -1. The model reaches a limit cycle that is a tilted rectangle.

2. ( = 0.5 and c1 = -0.5)

Various initial conditions for x and y were chosen, keeping z,w constant, creating this clockwise spiraling sink.

Various initial conditions for z and w were chosen, creating this counterclockwise spiraling sink.

3. ( = -0.5 and c1 = 0.5)

Various initial conditions for x and y were chosen, keeping z,w constant, creating this clockwise spiraling double limit cycle, which is similar to the van der Pol oscillator seen in case 1, except that there are two cycles an initial condition may fall into.

Various initial conditions for z and w were chosen, creating this counterclockwise circular limit cycle.

4. ( = -0.5 and c1 = -0.5)

Various initial conditions for x and y were chosen, keeping z,w constant, creating this clockwise spiraling bounded chaotic cycle.

Various initial conditions for z and w were chosen, creating this counterclockwise circular limit cycle.

5. ( = 0 and c1 = 0)

Here, the initial condition of x=0,y=1 was kept constant, while z and w were varied. This graph shows how the plot of x,y is dependent on the initial This graph shows how the plot of x,y is dependent on the initial conditions of z and w.

This is the plot of the initial conditions of z,w for the prior plot. Values were chosen for z,w for .5 incraments starting from -1 to 1. z and w are only dependent on each other, and create spirals similar to this one, towards the center. (Near the center, they reach a limit cycle of what appears to be a circle.)

We just thought that this looks neat... This is the periodic pattern found extremely close to the center for the x,y plot.