Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles |
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Authors: | D. Peralta-Salas |
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Affiliation: | Departamento de Fisica Teorica II, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain |
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Abstract: | In a recent paper of Llibre and Rodríguez (J. Differential Equations 198 (2004) 374-380) it is proved that every configuration of cycles in the plane is realizable (up to homeomorphism) by a polynomial vector field of degree at most 2(n+r)-1, where n is the number of cycles and r the number of primary cycles (a cycle C is primary if there are no other cycles contained in the bounded region limited by C). In this letter we prove the same theorem by using an easier construction but with a greater polynomial bound (the vector field we construct has degree at most 4n-1). By using the same technique we also construct R3 polynomial vector fields realizing (up to homeomorphism) any configuration of limit cycles which can be linked and knotted in R3. This answers a question of R. Sverdlove. |
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