Reading the instrument panel
Keyboard
| Space |
pause / resume |
| R · C |
reset the orbit · clear the trail |
| 1–4 |
XZ · XY · YZ · ISO views |
| ←→↑↓ |
rotate the camera |
| + · − |
zoom in · out |
| F · S |
fit the view · toggle spin |
| [ · ] |
previous · next attractor |
| G |
add / remove a ghost trajectory |
The diagnostics
| λ₁ |
The largest Lyapunov exponent, measured live by Benettin's method: two nearby orbits are advanced, their
separation logged, then renormalised. λ₁ > 0 is the definition of chaos — nearby states pull apart
exponentially, and 1/λ₁ is roughly how long a forecast survives. |
| Σλ |
The divergence of the vector field, ∇·F, averaged along the orbit — how fast a blob of initial conditions
loses volume. Negative for every dissipative system here; ≈ 0 for Nosé–Hoover, which is why that one has no
attractor. |
| D |
The Kaplan–Yorke dimension, 2 + λ₁/|λ₃|, taking λ₂ ≈ 0 and λ₃ = Σλ − λ₁. A fractional number is the
signature of a fractal. Lorenz settles near 2.06. |
| separation |
Distance between the reference orbit and the first ghost, when ghosts are on. |
All three are running estimates: they need a few thousand steps to settle, and they reset
whenever you touch a parameter.
The numbers are honest, the pixels are not
Integration is 4th-order Runge–Kutta at a fixed step. Because these systems are chaotic, the
computed orbit drifts away from the true orbit through the same initial point — no integrator avoids that. What
it does capture is the shape: the attractor, its dimension and its exponents are robust, even though the
individual trajectory drawn on screen is not the one you asked for.