Abstract:

In the qualitative theory of real planar polynomial differential systems two of the main problems are the determination of limit cycles and the center-focus problem, i.e. to distinguish when a singular point is either a focus or a center. If an analytic system has a center, then after an affine change of variables and a rescaling of the time variable, it can be written in one of the following three forms:

i) $x'=-y+P(x,y), y'=x+Q(x,y),$ called a non-degenerate center,

ii) $x'=y+P(x,y), y'=Q(x,y),$ called a nilpotent center,

iii) $x'=P(x,y), y'=Q(x,y),$ called a degenerate center,

where $P(x,y)$ and $Q(x,y)$ are real analytic functions without constant nor linear terms, defined in a neighborhood of the origin.

The centers for cubic polynomial differential systems of the form linear with homogeneous nonlinearities of degree 3 were characterized by Malkin [1], and by Vulpe and Sibiirski [2].

In this talk we will provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian non-degenerate centers of linear plus cubic homogeneous planar polynomial vector fields up to topological equivalence.

This is a joint work with Jaume Llibre and Cláudia Valls.

References:

[1] K.E. Malkin, “Criteria for the center for a certain differential equation”, Vols. Mat. Sb. Vyp. 2 (1964), 87-91 (in Russian).

[2] N.I. Vulpe and K.S. Sibirskii, “Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities”, Dokl. Akad. Nauk. SSSR 301 (1988), 1297-1301 (in Russian); translation in: Soviet Math. Dokl. 38 (1989), 198-201.