Lyapunov Characteristic Number
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11-10-2021
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Lyapunov Characteristic Number
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),](https://mathworld.wolfram.com/images/equations/LyapunovCharacteristicNumber/NumberedEquation5.gif) |
(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.
0
0
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