Lyapunov number and Lyapunov exponent

Question

Given two different points $x_{0}$ and $ x_{0}'$ where $x_{0}=x_{0}$ and $x'_{0}$=$x_{0}$+$\varepsilon$

where $\varepsilon_{n}$=$e^{n \lambda (x_{0})}$ is the $n^{\text{th}}$ iteration of the separation distance between the points above.

Which is the Lyapunov exponent and which is the Lyapunov number? My text doesn't explain this too well and I'm still confused. If I understood correctly, $\varepsilon_{n}$ is the Lyapunov number and ${n \lambda (x_{0})}$ is the Lyapunov exponent.

Thanks in advance.

Answer 1

Which is the Lyapunov exponent and which is the Lyapunov number? My text doesn't explains this too well and I'm still confused. If I understood correctly, $\epsilon_{n}$ is the Lyapunov number and ${n \lambda (x_{0})}$ is the Lyapunov exponent.

No, your Lyapunov exponent should be independent of $n$ and such. $λ(x_0)$ is the Lyapunov exponent (assuming that $ε$ is sufficiently small and $n$ is not too large); $e^{λ(x_0)}$ is the Lyapunov number.

Note that the Lyapunov number is only used rarely: I work in the general field and never encountered it unto today. It’s also used inconsistently; e.g., these papers define it quite differently.