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A Few Remarks on the Definition of Duffing's Equations

Recall that when we installed Duffing's equations as a time-dependent vector field, we defined 48#48 as an auxiliary function and claimed that we could use it to study the time-45#45 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 48#48 for large values of t. Since we are often interested in generating Poincaré maps for extremely long times, and since the function 49#49 also appears in the definition of our vector field, the user may want to extend phase space by introducing the variable 50#50. Then we can rewrite Duffing's equations in the form
51#51
where 52#52, and S1 is the circle of length 45#45. That is, 50#50 takes values in 53#53. 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 54#54. Thus we could study Duffing's equations on extended phase space in the form
55#55
where 56#56, and S1 is now the circle of length 38#38.

The advantage to an extended phase space such as we have for Equation 3.7 is that it is trivial to plot Poincaré sections for this set of equations because we can make 57#57 a periodic variable. This allows us to request that dstool only plot points when 58#58 for some 59#59. In contrast, dstool never treats time as a periodic variable, so we needed to define the auxiliary function 48#48 in order to be able to generate a Poincaré map for Duffing's equations.


next up previous contents
Next: Deleting Dynamical Systems Up: A Vector Field Example Previous: A Vector Field Example   Contents
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1998-11-02