Lyapunov Exponent
The average rate at which nearby trajectories pull apart — the quantitative fingerprint of chaos.
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Lyapunov Exponent
The Lyapunov exponent \lambda measures how fast two trajectories that start infinitesimally close together separate as time passes. If their initial gap is \delta_0, then on average it grows (or shrinks) like
The sign of \lambda tells you everything about the system's predictability:
- \lambda < 0 — neighbors converge; the system forgets its initial condition and settles onto a Fixed Point or Limit Cycle.
- \lambda = 0 — neighbors drift apart only linearly; marginal, as on a torus.
- \lambda > 0 — neighbors diverge exponentially. This is Sensitive Dependence on Initial Conditions, the defining mark of Chaos.
Exponential divergence, seen
The gap between two chaotic trajectories doesn't creep — it explodes. The plot shows the same initial error of 10^{-4} growing under a positive exponent. On a log scale it would be a straight line of slope \lambda; on this linear scale it looks like nothing, then suddenly everything.
The predictability horizon
A positive exponent imposes a hard limit on forecasting. If you know the state to precision \delta_0 and can tolerate error up to \Delta, your predictions stay useful only until the Lyapunov time t_h \approx \frac{1}{\lambda}\ln\frac{\Delta}{\delta_0}. Because the dependence on your initial precision is logarithmic, buying a thousand times better measurements only buys you a little more forecast time — the reason weather prediction hits a wall a couple of weeks out.
Many trajectories, one fate
Watch a fan of nearby starts under a chaotic flow. They march together convincingly — and then, at no particular moment, they don't.