Configurations of fans and nests of limit cycles for polynomial vector fields in the plane
Consider the set of limit cycles for a differential system (*) \(\dot x=P(x,y)\), \(\dot y=Q(x,y)\), where P and Q are polynomials of degree n in x and y. Let X be the polynomial vector field associated to the system (*). Two limit cycles \(C_ 1\) and \(C_ 2\) are f-equivalent if and only if \(s(C_ 1)=s(C_ 2)\), where s(C) is the set, which is formed by the critical points of X belonging to Int C. The limit cycles \(C_ 1\) and \(C_ 2\) are n-equivalent if \(s(C_ 1)\subseteq s(C_ 2)\) and there does not exists another limit cycle \(C_ 3\neq C_ 1\) such that \(C_ 3\subset Int C_ 2\setminus Int C_ 1\) [resp. vice versa]. Using the classes of equivalence the authors introduce the concepts of nest and fan of limit cycles. These concepts are helpful in the study of configurations of limit cycles of X.
scientific article; zbMATH DE number 4152882
scientific article