Monodromy and Dulac's problem for piecewise analytical planar vector fields
The piecewise analytical vector fields \(X\) defined by a pair \((X_+, X_-)\), where \(X_+\) and \(X_-\) are analytical vector fields on regions of the plane separated by an analytical curve \(\sum\). The curve \(\sum\) is called switching curve which separates the plane in two regions \(\sum_+\) and \(\sum_-\) having defined on each region the analytical vector fields \(X_+\) and \(X_-\), respectively. In this paper, the authors study the monodromic singular point in \(\sum\) (called \(\sum\)-monodromic singular point) and Dulac's problem of piecewise analytical vector fields \(X\). First, they characterize all \(\sum\)-monodromic singular points of piecewise analytical vector fields \(X\). Then they consider Dulac's problem restricted to piecewise analytical planar vector fields and prove that there is a neighborhood of the \(\sum\)-monodromic singular point free of limit cycles. Their results are interesting and enrich the dynamics of piecewise analytical vector fields.
piecewise analytical planar vector fields
scientific article; zbMATH DE number 7771501
scientific article