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| Two of the main classical problems in the Qualitative Theory of real planar polynomial vector fields are Hilbert 16th Problem and the center-focus problem.
It is well-known that the localized version of Hilbert 16th Problem is related with the center-focus problem. For instance, in case the singular point $\mathbf{s}$ is a non degenerate elliptic point, the maximum number of small amplitude limit cycles bifurcating from it, can be studied by calculating the Lyapunov quantities, which are also used to characterize the parameter for which $\mathbf{s}$ is a center. Analogously both the maximum number of large amplitude limit cycles as well as a center at infinity can be described in terms of Lyapunov quantities, that are obtained for the center at the origin after transformation $x=\frac{\cos{\theta}}{r},y=\frac{\sin{\theta}}{r}.$
Motivated by these problems we consider the cubic vector fields $\dot{x}=-y+ax^2+bxy+cy^2-y(x^2+y^2),\,\dot{x}=x+dx^2+exy+fy^2+x(x^2+y^2),$ where $a,b,c,d,e,f\in\mathbb{R}$ are such that the phase portraits all have a center simultaneously at the origin and at infinity. Recently we have obtained a complete analytic classification of the global phase portraits for it. In this talk we focus on the separatrix bifurcations that appear as a key in the complete study. |
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