Abstract
In this paper we first characterize all quasi--homogeneous but non--homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.
Highlights
The problem on the classification of polynomial differential systems having a center has been intensively studied
Bautin [5] completed the classification of the center–focus problem for quadratic differential systems
Malkin [19] and Vulpe and Sibirskii [27] classified the centers of cubic polynomial systems formed by linear plus homogeneous nonlinearities of degree three
Summary
The problem on the classification of polynomial differential systems having a center has been intensively studied. The first is to complete the classification of all planar quintic quasi–homogeneous but non–homogeneous polynomial differential systems. Every planar real quintic quasi–homogeneous but non–homogeneous coprime polynomial differential system (1) can be written as one of the following 15 systems. Our second result completes the characterization of planar quintic quasi– homogeneous but non–homogeneous polynomial vector fields which have a center at the origin. The quintic quasi–homogeneous but non–homogeneous coprime polynomial differential system (1) having a center at the origin, together with possible rescalings of variables, must be of the form (2). The center is not isochronous and the period of the periodic orbits is a monotonic function We remark that this last theorem solves the center–focus problem for quintic quasi–homogeneous but non–homogeneous polynomial differential systems. We prove Theorem 3 by using the Poincare compactification
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