Lower bounds for the local cyclicity for families of centers
This paper considers the local cyclicity of a family of centers depending on parameters. The authors extend the results of Christopher for linear and higher-order studies when the center has parameters. They prove that the number of small amplitude limit cycles bifurcating from an equilibrium of monodromic type in the class of polynomial vector fields of degrees 3 is \(M(3)\geq12\) by studying the center \(CD^{12}_{31}\). Besides, they also prove that the number of small amplitude limit cycles bifurcating from the one-parametric cubic reversible center \(CR_{17}^{12}\) and two-parameter holomorphic cubic center is 11, and that the number of small amplitude limit cycles bifurcating from the two-parameter cubic center \(CD^{11}_{10}\) is 12. For polynomial vector fields of degrees 4, they prove that the number of small amplitude limit cycles bifurcating from the center is \(M(4)\geq21\) by investigating the bifurcation diagram for a two-parameter quartic center. Moreover, they give a partial study on the bifurcation diagram for a four-parameter quartic holomorphic center.
higher-order developments and parallelization
scientific article; zbMATH DE number 7291340
scientific article