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

[摘要] In a recent paper of Llibre and Rodriguez (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 R-3 polynomial vector fields realizing (up to homeomorphisin) any configuration of limit cycles which can be linked and knotted in R-3. This answers a question of R. Sverdlove. (c) 2005 Elsevier Inc. All rights reserved.

[发布日期] 2005-10-01