Given a Lyapunov characteristic exponent 
,
the corresponding Lyapunov characteristic number 
is defined as
 |
(1)
|
For an 
-dimensional
linear map,
 |
(2)
|
The Lyapunov characteristic numbers 
, ..., 
are the eigenvalues
of the map matrix. For an arbitrary
map
 |
(3)
|
 |
(4)
|
the Lyapunov numbers are the eigenvalues of the limit
![lim_(n->infty)[J(x_n,y_n)J(x_(n-1),y_(n-1))...J(x_1,y_1)]^(1/n),](/images/equations/LyapunovCharacteristicNumber/NumberedEquation5.svg) |
(5)
|
where 
is the Jacobian
 |
(6)
|
If 
for all 
,
the system is not chaotic. If 
and the map is area-preserving
(Hamiltonian), the product of eigenvalues
is 1.
