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Recall that when we installed Duffing's equations as a time-dependent vector field,
we defined
92#92
as an auxiliary function and claimed that we could
use it to study the time-
89#89
stroboscopic map.
In theory, there is nothing wrong with
this, however in practice we will encounter numerical errors in the evaluation of
transcendental functions such as
92#92
for large values of t.
Since we are often interested in generating
Poincaré maps for extremely long times, and since the function
93#93also appears in the definition of our vector field, the user may want to extend phase space
by introducing the variable 94#94.
Then we can rewrite Duffing's
equations in the form
95#95
where
96#96,
and S1 is the circle of length
89#89.
That is, 94#94
takes values in
97#97.
The problem with this formulation is that DsTool cannot handle periodic variables
whose length depends on a parameter! To overcome this difficulty, we change coordinates
via the transformation
98#98.
Thus we could study Duffing's equations
on extended phase space in the form
99#99
where
100#100,
and S1 is now the circle of length 82#82.
The advantage to an extended phase space such as we have for Equation
is that it is trivial to plot Poincaré sections for this set of equations because we
can make 101#101
a periodic variable. This allows us to request that DsTool only plot
points when
102#102
for some 103#103.
In contrast, DsTool never treats time as a periodic variable,
so we needed to define the auxiliary function
92#92
in order to be able to generate a
Poincaré map for Duffing's equations.
Next: Deleting Dynamical Systems
Up: A Vector Field Example:
Previous: A Vector Field Example:
John Lapeyre
1998-09-04