Sensitive Dependence

The shorthand is sensitive dependence on initial conditions. If two states start a distance $\delta_0$ apart, a chaotic system often grows that separation like:

$$\delta(t) \approx \delta_0 e^{\lambda t}, \qquad \lambda > 0$$

Copy LaTeXwith $$ delimitersCopy mathexpression only

That $\lambda$ is a Lyapunov exponent. Positive means the system stretches nearby states apart. The interesting part is that the Lorenz attractor is also bounded. Trajectories do not fly to infinity; they stay trapped in the butterfly-shaped region. So the system is both constrained and unpredictable. It has structure, but not long-term forecastability.

Prediction horizon

Weather is the usual example because Lorenz found this behaviour while modelling convection. The lesson generalises. A chaotic system can have accurate short-term forecasts and useless long-term forecasts without changing its equations. The limiting factor is the precision of the state estimate. If each unit of time roughly multiplies uncertainty, then eventually the uncertainty is the size of the whole attractor.

How to use this lab

Set Trajectories to 2 and watch the pair. They start nearly identical, then split. Raise the speed if you want to see the divergence quickly, or lower the trail to make the current path easier to read. Now move $\rho$. The attractor changes character because $\rho$ controls how strongly the convective loop is driven. Below some values the motion is tame; around the classic $\rho = 28$ case, the butterfly appears.

There is a nice conundrum here. The attractor makes the system easier to describe statistically. You can talk about where trajectories tend to spend time. But it makes individual futures harder to promise. That is a good pattern to remember: when point predictions fail, distributions may still tell the truth.