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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 $\sin \omega t$ as an auxiliary function and claimed that we could use it to study the time-$2 \pi / \omega$ 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 $\sin \omega t$ for large values of $t$. Since we are often interested in generating Poincaré maps for extremely long times, and since the function $\cos \omega t$ also appears in the definition of our vector field, the user may want to extend phase space by introducing the variable $\theta$. Then we can rewrite Duffing's equations in the form
\begin{displaymath}\begin{array}{rcl}
\dot{u} &=& v, \\
\dot{v} &=& u - u^3 -...
...\gamma \cos \omega \theta, \\
\dot{\theta} &=& 1,
\end{array}\end{displaymath} (4.6)

where $(u,v,\theta) \in {\mbox{\Bbb R}} ^2 \times S^1$, and $S^1$ is the circle of length $2 \pi / \omega$. That is, $\theta$ takes values in $[0, 2 \pi/ \omega)$. 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 $\psi = \omega \theta$. Thus we could study Duffing's equations on extended phase space in the form
\begin{displaymath}
\begin{array}{rcl}
\dot{u} &=& v, \\
\dot{v} &=& u - u^3 ...
...a v - \gamma \cos \psi, \\
\dot{\psi} &=& \omega,
\end{array}\end{displaymath} (4.7)

where $(u,v,\psi) \in {\mbox{\Bbb R}} ^2 \times S^1$, and $S^1$ is now the circle of length $2 \pi$.

The advantage to an extended phase space such as we have for Equation 4.7 is that it is trivial to plot Poincaré sections for this set of equations because we can make $\psi$ a periodic variable. This allows us to request that DsTool only plot points when $\theta=\theta_0$ for some $\theta_0$. In contrast, DsTool never treats time as a periodic variable, so we needed to define the auxiliary function $\sin \omega t$ 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
2008-05-14