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A Torus Diffeomorphism

Despite the richness of the Lorenz system, there are aspects of dstool that we did not explore in the previous example. This example is designed to explore some of the remaining elementary features of dstool.

This example is a diffeomorphism of the two dimensional torus ${\cal T}^2 = S^1 \times S^1$ onto itself. The map is given by a non-linear perturbation of translations of the torus:

\begin{displaymath}
{\bf f}({\bf x})= {\bf x} + {\bf\Omega}
+ {\bf g}({\bf x})
\end{displaymath}

where ${\bf x} = (x, y) \in {\cal T}^2$, ${\bf\Omega} = (\omega_x, \omega_y) \in {\bf R}^2$, and

\begin{displaymath}
{\bf g}({\bf x}) = \left(\frac{-ab}{2 \pi}\sin 2 \pi y,
\frac{-a}{2 \pi b}\sin 2 \pi x \right).
\end{displaymath}

Because the map is from the torus to itself, the phase space in this example is the unit square with opposite edges identified.





root
1998-11-02