Lyapunov Characteristic Number
Given a Lyapunov characteristic exponent sigma_i, the corresponding Lyapunov characteristic number lambda_i is defined as
| lambda_i=e^(sigma_i). |
(1)
|
For an n-dimensional linear map,
| X_(n+1)=MX_n. |
(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) |
(3)
|
| y_(n+1)=f_2(x_n,y_n), |
(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), |
(5)
|
where J(x,y) is the Jacobian
If lambda_i=0 for all i, the system is not chaotic. If lambda!=0 and the map is area-preserving (Hamiltonian), the product of eigenvalues is 1.
See also
Adiabatic Invariant, Chaos, Lyapunov Characteristic ExponentExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Lyapunov Characteristic Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LyapunovCharacteristicNumber.html