Given a Lyapunov Characteristic Exponent
, the corresponding Lyapunov characteristic number
is defined as
![\begin{displaymath}
\lambda_i\equiv e^{\sigma_i}.
\end{displaymath}](l2_1258.gif) |
(1) |
For an
-dimensional linear Map,
![\begin{displaymath}
{\bf X}_{n+1} = {{\hbox{\sf M}}}{\bf X}_n.
\end{displaymath}](l2_1259.gif) |
(2) |
The Lyapunov characteristic numbers
, ...,
are the Eigenvalues of the Map
Matrix. For an arbitrary Map
![\begin{displaymath}
x_{n+1} = f_1(x_n,y_n)
\end{displaymath}](l2_1262.gif) |
(3) |
![\begin{displaymath}
y_{n+1} = f_2(x_n,y_n),
\end{displaymath}](l2_1263.gif) |
(4) |
the Lyapunov numbers are the Eigenvalues of the limit
![\begin{displaymath}
\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}](l2_1264.gif) |
(5) |
where
is the Jacobian
![\begin{displaymath}
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}](l2_1266.gif) |
(6) |
If
for all
, the system is not Chaotic. If
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
1999-05-25