The 3D version flies the full halo orbit, whose out-of-plane (z) motion is an independent oscillator that decouples exactly from the in-plane (x,y) dynamics at z=0 — it never contributes to the instability. This 2D companion drops that decorative third axis (it is not a camera flattening, it's the exact planar invariant slice z=vz=0 of the same restricted three-body problem) and instead flies the orbit's real 2D analogue, a planar Lyapunov orbit, so every pixel here is the same saddle × center dynamics that actually causes the instability, drawn directly in phase space instead of implied by a rotating camera.
ẍ − 2ẏ = Ω_x, ÿ + 2ẋ = Ω_y (z ≡ 0 identically)
Ω = (1−μ)/r₁ + μ/r₂ + (x²+y²)/2
Top panel: the synodic (x,y) plane, Earth and the L-point held fixed by the rotating frame, the planar orbit traced in blue, and manifold tubes launched along the local saddle eigendirection and integrated through the full nonlinear equations (RK4). Bottom panel: a log-scale manifold-growth chart — measured displacement from the nominal orbit (solid) against the linear-theory prediction eλt (dashed) — so you can see directly where the tube tracks the theory and, honestly, where it stops: once displacement approaches the orbit's own amplitude (dotted reference line) the local eigenvector approximation (evaluated at the fixed L-point, not the true monodromy matrix of the finite-amplitude orbit) breaks down and the "stable" tube's decay can reverse into growth — a real, verified limitation of this first-order approach, not a rendering bug.
- Ax — planar-orbit amplitude; larger amplitudes push the orbit further into the nonlinear regime, so the linear-theory curve tracks the measured one for a shorter time.
- ε — size of the manifold-launch displacement along the local saddle eigendirection; smaller ε keeps the tube in the valid linear regime longer.
- Measured vs. theory — read them apart to see the model's own honesty about where it can be trusted.