Abstract
For scalar equations
with x ε S 1 and f ε C 2 we show that the classical theorem of Poincaré and Bendixson holds: the ω-limit set of any bounded solution satisfies exactly one of the following alternatives:
-
- it consists in precisely one periodic solution, or
-
- it consists of solutions tending to equilibrium as \(t \to \pm \infty \)
This is surprising, because the system is genuinely infinite-dimensional.
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Fiedler, B., Mallet-Paret, J. A Poincaré-Bendixson theorem for scalar reaction diffusion equations. Arch. Rational Mech. Anal. 107, 325–345 (1989). https://doi.org/10.1007/BF00251553
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DOI: https://doi.org/10.1007/BF00251553