Lyapunov Characteristic Number

Given a Lyapunov Characteristic Exponent $\sigma_i$, the corresponding Lyapunov characteristic number $\lambda_i$ is defined as

$$\lambda_i\equiv e^{\sigma_i}.$$ (1)

For an $n$-dimensional linear Map,

$${\bf X}_{n+1} = {{\hbox{\sf M}}}{\bf X}_n.$$ (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

$$\lim_{n\to\infty} [J(x_n,y_n)J(x_{n-1},y_{n-1})\cdots J(x_1,y_1)]^{1/n},$$ (5)

where $J(x,y)$ is the Jacobian

$$J(x,y) \equiv\left\vert\matrix{ {\partial f_1(x,y)\over\par... ...rtial x} & {\partial f_2(x,y)\over\partial y} \cr}\right\vert.$$ (6)

If $\lambda_i = 0$ for all $i$, the system is not Chaotic. If $\lambda \not= 0$ and the Map is Area-Preserving (Hamiltonian), the product of Eigenvalues is 1.

See also Adiabatic Invariant, Chaos, Lyapunov Characteristic Exponent