Deterministic does not mean indefinitely predictable
Exact equations can still produce limited trajectory predictability.
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Lorenz and the
Butterfly Effect
How a tiny difference in the present can place a limit on long-range prediction.
Begin the Story ↓Scene 01
A scientist restarts a calculation
During an early numerical experiment, Edward Lorenz restarted a calculation using rounded values printed from an earlier run. The equations were unchanged. The computer was unchanged. Only a few apparently insignificant digits were missing.
0.506127and0.506Phase space · x–z projection
The same trajectories over time · x(t)
Scene 02
At first, the two futures agree
For a while, the difference is almost invisible. A small error does not immediately produce a dramatically different result.
Phase space · x–z projection
The same trajectories over time · x(t)
Drag time to reveal the experiment.
Scene 03
Then the futures divide
The small initial difference grows. Eventually, the two calculations describe completely different trajectories, even though both follow exactly the same deterministic equations.
Sensitive dependence on initial conditionsThe same trajectories over time · x(t)
Scene 04
A smaller error delays the separation, but does not remove it
Nearby trajectories separate approximately exponentially during the early growth stage: a small difference is repeatedly amplified by the nonlinear dynamics. On the logarithmic vertical axis at right, exponential growth appears as a nearly straight rising segment.
More accurate initial information can extend the useful forecast, but it cannot preserve one exact trajectory forever.
This behavior is a property of chaotic dynamical systems such as the Lorenz system, not a claim that all systems behave this way.
Trajectory separation · logarithmic vertical scale
Initial difference1e-3Time until forecasts separate18.5
The same trajectories over time · x(t)
Scene 05
Information about the present has a finite lifetime
δ(t) ≈ δ(0)eλt
- δ(0)
- the initial uncertainty
- δ(t)
- the uncertainty after time t
- λ
- the average rate at which nearby trajectories separate
Idealized exponential error growth
The same trajectories over time · x(t)
Scene 06
Chaos is not the same as randomness
What does the experiment show?
x(t)same state + same equationssame future
Scene 07
Instead of predicting one future, predict a set of possible futures
An ensemble does not eliminate uncertainty. It represents how uncertainty evolves. The question becomes: “What outcomes remain plausible, and with what probability?”
Atmosphere and ocean prediction use ensembles to follow uncertainty through nonlinear models.
Phase space · x–z projection
The same forecast members over time · x(t)
The butterfly effect is a limit on information, not a claim that every tiny cause creates a disaster
Better observations extend prediction
Reducing initial uncertainty can delay forecast divergence.
Uncertainty must be evolved, not ignored
Ensemble prediction describes a range of possible futures.
The future is constrained by the equations, but our knowledge of that future is constrained by information.
Why Prediction Has Limits (Even with Perfect Models)
A direct video on predictability, model structure, and the limits of a single trajectory.
Watch Video →Explore the Lorenz System
Change parameters and investigate nearby trajectories.
Open the Exploration →The Planet We Never See Directly
How sparse observations and mathematical models are combined to reconstruct an incomplete Earth.
Begin the next story →