A trajectory connecting two different saddle equilibria — the skeleton organizing the global flow structure
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System: ẋ = y, ẏ = x - x³ + 0.1·y(1-x²)
Saddles: At (±1, 0) — both unstable along x, stable along y (index -1).
Heteroclinic connection: The unstable manifold of S₁=(−1,0) connects to the stable manifold of S₂=(+1,0) along the central homoclinic-like curve.
Significance: Heteroclinic connections organize "transition states" — in chemistry, the heteroclinic orbit over a saddle in the potential energy surface IS the reaction pathway. In mechanics, they bound regions of libration vs. rotation.
Double-well potential: V(x) = -x²/2 + x⁴/4. The heteroclinic orbits are the separatrices of the double-well.