Some interesting graphics:

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


Picture: [3D plot]
This is the graph for where y' = 0. At y = 0, there is a discontinuity.


Steady state stability

  1. (lambda = 0.5 and c1 = 0.5)

    Picture: [counterclockwise spirals
approaching a limit cycle (circle)]
    The solutions aof z,w pproach a limit cycle that is a circular pattern.

    Picture: [clockwise loops approaching a
limit cycle (circle)]
    The solutions of x,y approach a limit cycle that appears to be very similar to the phaseportrait of the van der Pol oscillator.

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


  2. (lambda = 0.5 and c1 = -0.5)

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

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


  3. (lambda = -0.5 and c1 = 0.5)

    Picture: [clockwise limit cycle (bowtie)]
    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.

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


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

    Picture: [clockwise limit cycle (bowtie), chaotic pattern]
    Various initial conditions for x and y were chosen, keeping z,w constant, creating this clockwise spiraling bounded chaotic cycle.

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


  5. (lambda = 0 and c1 = 0)

    Picture: [loops spiraling from one point
towards origin]
    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.

    Picture: [counterclockwise spiral
towards center]
    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.)

    Picture:  [neat circular pattern, path
around torus]
    We just thought that this looks neat... This is the periodic pattern found extremely close to the center for the x,y plot.


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Michelle Vadeboncoeur <mrvahs@wpi.edu>
Forest Lee-Elkin <yusuf@wpi.edu>