Lyapunov Exponents

Lyapunov exponents $\lambda$ quantify the exponential rate at which nearby trajectories in phase space diverge or converge — the primary measure of chaos. A positive maximal exponent ${\lambda }_{\max }>0$ indicates Sensitive Dependence on Initial Conditions, while $\lambda <0$ means the system contracts toward a stable fixed point or attractor.

---

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:

$$L(x_1)=\underset{n\to \infty }{\lim }{\left(|f^{\prime }(x_1)|\cdot |f^{\prime }(x_2)|\cdots |f^{\prime }(x_n)|\right)}^{1/n}$$

The asymptotic geometric mean of derivatives — how much does a small perturbation grow per iterate, on average?

$$L<1\Rightarrow \text{contracting}L>1\Rightarrow \text{stretching (chaotic)}$$

Lyapunov Exponent

$\lambda =\ln L$, i.e. the arithmetic mean of log-derivatives:

$$\lambda (x_1)=\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{k=0}^{n-1}\ln |f^{\prime }(x_k)|$$

This is the time-average of $\ln |f^{\prime }|$ along the orbit. By the Birkhoff ergodic theorem, for an ergodic invariant measure $\mu$ this equals $\int \ln |f^{\prime }|d\mu$ for typical orbits.

---

Higher Dimensions

For maps or flows on ${\mathbb{R}}^n$ there is a full Lyapunov spectrum ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$, one per dimension. These come from the singular values of the $n$-step Jacobian $D{f}^n(x_0)$. Key facts:

• ${\sum }_i{\lambda }_i$ = asymptotic volume contraction rate
• Dissipative systems: $\sum {\lambda }_i<0$
• Hamiltonian systems: $\sum {\lambda }_i=0$ (Liouville), exponents come in $\pm$ pairs

---

Logistic Map

For $f_r(x)=rx(1-x)$, $\lambda =\ln 2\approx 0.693$ at $r=4$ (fully chaotic, conjugate to the tent map). In periodic windows $\lambda <0$, and $\lambda =0$ exactly at bifurcation points.

---

See also: Local Lyapunov Exponent, Finite-Time Lyapunov Exponent