Is the purple region “proved” chaotic?

No — it is a finite-time separation proxy, not a rigorous Lyapunov exponent. It highlights sensitive dependence on initial conditions.

Why zero initial velocity?

It is a simple, repeatable sweep of initial positions. Different velocity choices would change the map.

Restricted 3-Body (map)

In the circular restricted three-body problem, a massless particle moves in the rotating frame of two primaries in circular orbit. The effective potential includes gravitational and centrifugal terms; Coriolis forces appear in the equations of motion. This page integrates many initial conditions with zero corotating velocity to classify short-horizon outcomes: collision with a primary, escape to large radius, or remaining bound. A shadow trajectory with a tiny initial offset estimates sensitivity to initial conditions — a qualitative stand-in for chaotic regions with fractal-like basin boundaries.

Who it's for: Follows the Lagrange points page; for advanced students before full Poincaré sections.

Key terms

restricted three-body problem
corotating frame
Jacobi-like dynamics
chaos
basin boundaries

How it works

The **circular restricted three-body problem** tracks a **massless** particle in the **corotating** frame of two **massive** bodies in a **circular** orbit. Here each grid cell starts the particle with **zero velocity** in that frame; forward integration reveals whether the orbit **hits** a primary (**collision**), **escapes** a large radius, or remains **bound** for a fixed horizon. A **nearby** duplicate initial condition measures **exponential-like** separation (**chaotic** sensitivity) — a **pedagogical** stand-in for **Lyapunov** structure. **Boundaries** between outcomes can be **fractally** complicated.

Key equations

U = −(1−μ)/r₁ − μ/r₂ − ½r² · ẍ = −∂U/∂x + 2ẏ

Frequently asked questions

Is the purple region “proved” chaotic?: No — it is a finite-time separation proxy, not a rigorous Lyapunov exponent. It highlights sensitive dependence on initial conditions.
Why zero initial velocity?: It is a simple, repeatable sweep of initial positions. Different velocity choices would change the map.