I am thinking of a scenario/ examples where the stable and unstable manifold of an equilibrium of a continuous dynamical system are tangent to each other?
Any examples/ plots would be helpful?
2026-03-28 06:40:41.1774680041
Stable and unstable manifolds that are tangent to each other in a continuous dynamical system?
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A straightforward example is a planar system with a homoclinic orbit, say \begin{align} \dot{x} = y,\\ \dot{y} = x - x^2. \end{align} The unstable and stable manifolds of the origin in the right half plane coincide, and hence are tangent.