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Here is an amusing differential equation that is relevant to the
upcoming elections and/or primaries. Suppose that there are three
candidates, *x*,*y* and *z*. Suppose that in an *x*,*y* race, *x* wins,
in a *y*,*z* race, *y* wins, and in a *z*,*x* race, *z* wins. What happens
in a three-way race? Here are the differential equations:

We choose *a*<1,*b*>1 to mimic the pairwise behavior. The *XPPAUT *file is
called `voter.ode` and is:
x'=x*(1-x-a*y-b*z)
y'=y*(1-y-b*x-a*z)
z'=z*(1-z-a*x-b*y)
par a=.5,b=2
init x=.28,y=.281,z=.279
@ total=1000,njmp=5
@ xhi=1000,ylo=-.1,yhi=1.1
done

Run this program. Add the two curves *y*(*t*) and *z*(*t*) as
follows. Click on
(tap
)
and fill in the
dialog box as:
Repeat this for the variable *z* filling the dialog box as

You should see a red and yellow curve on the graph. Note how at first
one wins the competition, then the next, and so on but at each stage,
the amount of time that any given variable wins gets longer and
longer. In fact, the solution is period in the logarithm of time. This
means that in each cycle, each one stays active exponentially longer
than it did the previous cycle! This is called the voter paradox.

You can sort of see what is going on by plotting in
three-dimensions. Delete all the added curves by clicking
(tap
). Click on
(tap
). Fill in the first three
entries of the dialog box as:

You will see nothing particularly meaningful on the screen. Click on
(tap
)
and the graphics
window will choose a view that optimally fits the graph. (This is
**always** the way I do three-d graphs, letting *XPPAUT *figure out the
best way to show it.) You should see a triangular surface with a
spiral coming out the center. The three corners of the triangle
represent the fixed points (1,0,0), (0,1,0), and (0,0,1). There are
no stable fixed points, nor a stable limit cycle for this model.

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*G. Bard Ermentrout*

*1999-10-26*