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

(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.