One of the conclusion of the Poincaré-Bendixson Theorem is that in planar dynamical systems chaotic motion could not arise. But in the Sinai Billiard trajectories in fact are chaotic...how is this possible?
2026-03-25 17:24:30.1774459470
Poincaré-Bendixson theorem and the Sinai Billiard
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Alessandro's remark is a red herring. True, Sinai billiard flows are not differentiable, but that's not the main issue.
The phase space of the billiard flow is the space of couples $(x,v)$, where $x$ belongs to the billiard and $v$ is a unit speed vector. Hence you have $3$ degrees of freedom, so the billiard flow acts on a $3$-dimensional space. Obviously Poincaré-Bendixon does not apply.
(there are further issues about the differentiability of the system and the topology of the phase space, but they seem secondary to me).