Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields
In a previous work the authors of this article demonstrated that there are six classes of Hamiltonian planar polynomial vector fields (identified with systems of polynomial differential equations on the plane) that have the form \(X = L + C\), where \(L\) and \(C\) are homogeneous polynomials of degrees 1 and 3, respectively, and where the eigenvalues of \(L\) are nonzero pure imaginary numbers, so that \(X\) has a ``linear center'' at the origin. That is, for any such vector field there exist a linear change of coordinates and a time rescaling that expresses it in one of six forms. They also found, up to topological equivalence and time reversal, all possible phase portraits on the Poincaré sphere of such systems. In this work they give an exhaustive description of conditions on the parameters in each class that determine the phase portrait of members of the class. For example, one class corresponds to the systems of differential equations of the form \[ \dot x = a x + b y, \] \[ \dot y = -((a^2+1)/b) x - a y + x^3. \] All such systems with \(b < 0\) have one particular phase portrait, all those with \(b > 0\) have another.
scientific article; zbMATH DE number 6371762
scientific article