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A frequent contributor to Acta Mathematica , the journal founded by his colleague Mittag-Leffler, Bendixson is best remembered for the Poincaré-Bendixson theorem. The theorem, which says an integral curve which does not end in a singular point has a limit cycle, was first proved by Poincaré but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.
Reference (One book/article)