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Record 1,  Textual support,  English

Record number: 1, Textual support number: 1 CONT

The existence of limit cycles is subjected to the Bendixon criterion which states that a simply connected region of the plane cannot contain a limit cycle if the divergence... of the vector field... keeps a constant sign in that region.  1, record 1, English, - Bendixon%20criterion

Record number: 1, Textual support number: 1 OBS

The reason is nearly obvious. Suppose that such a cycle exists. The area of the domain of phase space limited by this curve should be invariant under the evolution since the curve itself is invariant, but this is impossible if the divergence keeps a constant sign since the expansion rate of phase space elements in that region is not vanishing.  1, record 1, English, - Bendixon%20criterion

Record 1,  Textual support,  French

Record number: 1, Textual support number: 1 CONT

L'existence de cycles limites doit satisfaire au critère de Bendixon qui dit qu'une région simplement connexe du plan ne peut contenir de cycle limite si la divergence [...] du champ de vecteur [...] y garde un signe constant.  1, record 1, French, - crit%C3%A8re%20de%20Bendixon

Record number: 1, Textual support number: 1 OBS

La raison en est presque évidente : supposons qu'un tel cycle existe, la surface du domaine limité par cette courbe devrait être invariante au cours de l'évolution, ce qui est impossible si la divergence, mesurant le taux d'expansion/contraction des éléments de volume dans l'espace des phases, garde un signe constant dans la région considérée.  1, record 1, French, - crit%C3%A8re%20de%20Bendixon

Record 1,  Textual support,  Spanish