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Mathematics
Differential Equations
Dynamic Systems: Poincaré-Bendixson Theorem finite # of equilibria
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[QUOTE="S.G. Janssens, post: 6581180, member: 571630"] What (iii) is trying to capture, are homoclinic and heteroclinic orbits. Have you seen those before? The lecture notes or book probably discuss those. In the homoclinic case, the unstable and stable manifolds of a single equilibrium coincide. In the heteroclinic case, the unstable manifold of one equilibrium coincides with the stable manifold of another equilibrium. At first, this is easiest to see geometrically, without attempting to write down a vector field explicitly. (Note that the Poincaré-Bendixson theorem is valid for vector fields on the plane. Its classification is not exhaustive in higher dimensions.) [/QUOTE]
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Dynamic Systems: Poincaré-Bendixson Theorem finite # of equilibria
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