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| A way to study the limit cycles which bifurcate from the periodic orbits of a center of a planar polynomial differential system is to perturbate the system with the center in a certain parametric family of systems. In this context, it appears the notion of essential perturbation used for the first time by Iliev (1998) in a paper on quadratic systems. We will give its explicit definition.
Given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which we explicitly know its Poincar\'e--Liapunov constants, we find its essential perturbations. As a consequence we give the structure of its $k$-th Melnikov function.
This result generalizes the result obtained by Chicone and Jacobs (1991) for perturbations of degree at most two of any center of a quadratic polynomial system. Moreover we study the essential perturbations for all the centers of the differential systems
\[ \dot{x} \, = \, -y + P_{\rm d}(x,y), \quad \dot{y} \, = \, x + Q_{\rm d}(x,y), \]
where $P_{\rm d}$ and $Q_{\rm d}$ are homogeneous polynomials of degree ${\rm d}$, for ${\rm d}=2$ and ${\rm d}=3$. |
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