Effective construction of Poincar\'e--Bendixson regions |
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Authors: | Armengol Gasull Hector Giacomini and Maite Grau |
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Institution: | Departament de Matem\`{a}tiques, Universitat Aut\`{o}noma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain,Laboratoire de Math\''ematiques et Physique Th\''eorique. C.N.R.S. UMR 7350., Facult\''e des Sciences et Techniques. Universit\''e de Tours, Parc de Grandmont 37200 Tours, France and Departament de Matem\`atica, Universitat de Lleida, Avda. Jaume II, 69; 25001 Lleida, Catalonia, Spain |
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Abstract: | This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\''e--Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Li\''{e}nard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system. |
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Keywords: | Transversal curve Poincar\''e--Bendixson region limit cycle bifurcation planar differential system |
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