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Lyapunov Characteristic Number

Given a Lyapunov Characteristic Exponent $\sigma_i$, the corresponding Lyapunov characteristic number $\lambda_i$ is defined as

\lambda_i\equiv e^{\sigma_i}.
\end{displaymath} (1)

For an $n$-dimensional linear Map,
{\bf X}_{n+1} = {{\hbox{\sf M}}}{\bf X}_n.
\end{displaymath} (2)

The Lyapunov characteristic numbers $\lambda_1$, ..., $\lambda_n$ are the Eigenvalues of the Map Matrix. For an arbitrary Map
x_{n+1} = f_1(x_n,y_n)
\end{displaymath} (3)

y_{n+1} = f_2(x_n,y_n),
\end{displaymath} (4)

the Lyapunov numbers are the Eigenvalues of the limit
\lim_{n\to\infty} [J(x_n,y_n)J(x_{n-1},y_{n-1})\cdots J(x_1,y_1)]^{1/n},
\end{displaymath} (5)

where $J(x,y)$ is the Jacobian
J(x,y) \equiv\left\vert\matrix{
{\partial f_1(x,y)\over\par...
...rtial x} & {\partial f_2(x,y)\over\partial y} \cr}\right\vert.
\end{displaymath} (6)

If $\lambda_i = 0$ for all $i$, the system is not Chaotic. If $\lambda \not= 0$ and the Map is Area-Preserving (Hamiltonian), the product of Eigenvalues is 1.

See also Adiabatic Invariant, Chaos, Lyapunov Characteristic Exponent

© 1996-9 Eric W. Weisstein