Saddle Connection

System: ẋ = x + μ, ẏ = -y + x²
(two saddles near (±1, 1))

Stable manifold: (green) — trajectories flowing INTO the saddle as t→+∞.

Unstable manifold: (red) — trajectories flowing OUT of the saddle as t→-∞.

At μ=0: The unstable manifold of saddle S₁ exactly coincides with the stable manifold of S₂ — a saddle connection (also called a heteroclinic orbit).

Breaking it: For μ≠0, the manifolds separate. The Melnikov function measures the signed distance — a zero of Melnikov = a saddle connection.