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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 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 up previous contents
Next: Deleting Dynamical Systems Up: A Vector Field Example: Previous: A Vector Field Example:
John Lapeyre
1998-09-04