Lyapunov Exponents

Lyapunov exponents $(\lambda )$ quantify the exponential rate of separation of infinitesimally close trajectories in a dynamical system’s phase space, serving as a primary measure of chaos. A positive maximum Lyapunov exponent $({\lambda }_{max}>0)$ indicates sensitive dependence on initial conditions, known as the “butterfly effect,” while negative values signify convergence towards a stable fixed point or attractor.

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Lyapunov Number

Let $f$ be a continuous map on $\mathbb{R}$. The Lyapunov number $L(x_1)$ of the orbit $\{x_1,x_2,\dots \}$ is defined as:

$$L(x_1)=\underset{n\to \infty }{\lim }(|f^{{}^{\prime }}(x_1)|\cdot |f^{{}^{\prime }}(x_2)|\dots |f^{{}^{\prime }}(x_N)|{)}^{\frac{1}{n}}$$

$h(x_1)$ is defined as:

$$h(x_1)=\underset{n\to \infty }{\lim }\frac{1}{n}\left(\ln |f^{{}^{\prime }}(x_1)|+\ln |f^{{}^{\prime }}(x_2)|+\cdots +\ln |f^{{}^{\prime }}{(}_n)|\right)$$