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Arnold's Cat Map

The best known example of an Anosov Diffeomorphism. It is given by the Transformation

\left[{\matrix{x_{n+1} \cr y_{n+1}\cr}}\right] = \left[{\mat...
...\cr 1 & 2 \cr}}\right]\left[{\matrix{x_n \cr y_n \cr}}\right],
\end{displaymath} (1)

where $x_{n+1}$ and $y_{n+1}$ are computed mod 1. The Arnold cat mapping is non-Hamiltonian, nonanalytic, and mixing. However, it is Area-Preserving since the Determinant is 1. The Lyapunov Characteristic Exponents are given by
\left\vert\matrix{1-\sigma & 1 \cr 1 & 2-\sigma \cr}\right\vert = \sigma^2-3\sigma+1 = 0,
\end{displaymath} (2)

\sigma_\pm = {\textstyle{1\over 2}}(3\pm\sqrt{5}\,).
\end{displaymath} (3)

The Eigenvectors are found by plugging $\sigma_\pm$ into the Matrix Equation
\left[{\matrix{1-\sigma_\pm & 1 \cr 1 & 2-\sigma_\pm \cr}}\r...
...trix{x\cr y \cr}}\right] = \left[{\matrix{0\cr 0 \cr}}\right].
\end{displaymath} (4)

For $\sigma_+$, the solution is
y={\textstyle{1\over 2}}(1+\sqrt{5}\,)x\equiv \phi x,
\end{displaymath} (5)

where $\phi$ is the Golden Ratio, so the unstable (normalized) Eigenvector is
\boldsymbol{\xi}_+ = {\textstyle{1\over 10}}\sqrt{50-10\sqrt{5}}\,\left[{\matrix{1 \cr {1\over 2}(1+\sqrt{5}\,)\cr}}\right].
\end{displaymath} (6)

Similarly, for $\sigma_-$, the solution is
y=-{\textstyle{1\over 2}}(\sqrt{5}-1)x\equiv \phi^{-1} x,
\end{displaymath} (7)

so the stable (normalized) Eigenvector is
\boldsymbol{\xi}_- = {\textstyle{1\over 10}}\sqrt{50+10\sqrt{5}}\,\left[{\matrix{1 \cr {1\over 2}(1-\sqrt{5}\,)\cr}}\right].
\end{displaymath} (8)

See also Anosov Map

© 1996-9 Eric W. Weisstein