How can tiny starting differences change a model future?

What will you explore?

Lorenz, 1963

Exact rules do not remove initial uncertainty.

How does it work?

Watch a 90-second cinematic explanation, or explore a story that pauses so you can change its central mechanism. Inspect the source evidence, compare outcomes and resume the narration. This free Lesson Worlds adaptation needs no account or download; younger learners may want help from a parent or teacher.

Inside the cinematic story

1. Same rules. Different futures.

   A computed path through model state space

2. Three numbers. A twisting world.

   Model variables, not geographic coordinates

3. Change only the beginning

   Identical versus slightly different initial states

   Take control: Do both computations start from exactly the same state? Identical start · Tiny perturbation

4. Change the regime

   Compare r = 10 and r = 28

   Take control: Does this numerical trajectory approach a steady state? r = 28 · r = 10

5. What the paper established

   Numerical evidence and explicit boundaries

6. Follow the rules. Respect uncertainty.

   Explore the equations and their limits

Discuss the evidence

Before starting, explain what you think this claim means: Exact rules do not remove initial uncertainty. After exploring, choose one stage above and describe what its model helps you see. Separate what you observed in the adaptation from what the original researchers measured.

Read the original publication for its methods, evidence and limitations. Ask which part of the explanation the interactive model leaves out, and what additional evidence you would need to test the claim yourself.

New numerical integration of Lorenz equations 25–27: x prime=10(y-x), y prime=x(r-z)-y, z prime=xy-(8/3)z. Fixed-step RK4 dt=.005; samples every .02; 65 dimensionless model-time units. Initial state (1,1,1), compared with (1.0001,1,1); r=28 or illustrative steady-regime r=10. The paper used initial (0,1,0), a double-approximation method and dt=.01. Not a reproduction of Tables 1–2 or Figures 1–5. Spatial axes and trajectories show model states, not geography, air parcels or measured weather. Designed observatory, finite trace, colors and compressed/nonuniform screen-time mapping are illustrative. No weather forecast horizon, physical apparatus, empirical data, proof of aperiodicity, Lyapunov exponent or calibrated error growth is claimed.

Original research

Deterministic Nonperiodic Flow

Edward N. Lorenz · Journal of the Atmospheric Sciences 20(2), 130–141 · 1963

Educational adaptation by Lesson Worlds. Not affiliated with the author, MIT or AMS. The original publication may have its own access terms.