In the rotating (synodic) frame of the Sun-Earth system, a test mass obeys the circular restricted three-body problem (CR3BP), normalized so the Sun-Earth distance is 1 and mass ratio μ = mEarth/(mSun+mEarth) ≈ 3.003×10⁻⁶:
ẍ − 2ẏ = Ω_x, ÿ + 2ẋ = Ω_y, z̈ = Ω_z
Ω = (1−μ)/r₁ + μ/r₂ + (x²+y²)/2
L1/L2 are the points on the Sun-Earth line where Ω's gradient vanishes. Linearizing the equations of motion there gives a Hessian with Ωxx>0, Ωyy<0, Ωzz<0 — a saddle × center × center structure. Solving the characteristic equation
λ⁴ + (4 − Ω_xx − Ω_yy)λ² + Ω_xx·Ω_yy = 0
yields one real pair ±λ (the saddle — this is the instability every L1/L2 mission fights) plus two oscillatory pairs ±iωxy (in-plane) and ±iωz (out-of-plane, ωz=√(−Ωzz)). Combining the two center modes gives the quasi-periodic Lissajous orbit real missions like JWST, Gaia and SOHO actually fly. A true halo orbit is the special nonlinear case where amplitude-dependent frequency corrections lock ωz to ωxy, closing the path into one repeating 3D loop — the "Mode" button reproduces that resonance directly for illustration.
The saddle eigenvalue ±λ also defines the orbit's stable and unstable invariant manifolds: displace a state along the +λ eigendirection and the nonlinear dynamics carry it away exponentially (∝eλt) — no correction, and a spacecraft drifts off within weeks, which is exactly why real L1/L2 halo missions need station-keeping burns every few weeks. Displace along the −λ eigendirection instead and the trajectory decays onto the orbit — the same manifold mission designers ride for near-zero-fuel arrivals (as with the Genesis and ISEE-3 missions). "Launch tube" seeds 24 points around the nominal orbit with this local perturbation and integrates all of them through the full nonlinear CR3BP equations (RK4), so the fan/funnel shape you see is the real manifold geometry, not a cartoon.
- Ax / Az — independent linear-theory amplitudes of the in-plane and out-of-plane oscillations.
- Mode — toggles the vertical frequency between its natural value (Lissajous, does not close) and the resonant halo value.
- ε — size of the manifold-launch displacement along the local saddle eigendirection.