TOPICS
Search

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

 J(x,y)=|(partialf_1(x,y))/(partialx) (partialf_1(x,y))/(partialy); (partialf_2(x,y))/(partialx) (partialf_2(x,y))/(partialy)|.
(6)

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 Exponent

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Lyapunov Characteristic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LyapunovCharacteristicNumber.html

Subject classifications