xrk
and grabbed via
xv
unless otherwise noted.
This is the graph for where y' = 0. At y = 0, there is a
discontinuity.
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.
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.
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.
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.
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.
Michelle Vadeboncoeur <mrvahs@wpi.edu>