Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles

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 , where is the number of cycles and the number of primary cycles (a cycle is primary if there are no other cycles contained in the bounded region limited by ). 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 ). By using the same technique we also construct polynomial vector fields realizing (up to homeomorphism) any configuration of limit cycles which can be linked and knotted in . This answers a question of R. Sverdlove. [Copyright &y& Elsevier]