## Journals

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### Open Access Journals

CPAA

We deal with nonlinear periodic differential systems depending on a small parameter. The
unperturbed system has an invariant manifold of periodic solutions. We provide sufficient
conditions in order that some of the periodic orbits of this invariant manifold persist after the
perturbation. These conditions are not difficult to check, as we show in some applications. The
key tool for proving the main result is the Lyapunov--Schmidt reduction method applied to the
Poincaré--Andronov mapping.

CPAA

Chicone--Jacobs and Iliev found the essential perturbations of quadratic systems
when considering the problem of finding the cyclicity of a period
annulus. Given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which the expressions of its Poincaré--Liapunov constants are known, we give the structure of its $k$-th Melnikov function. This allows to find the essential perturbations in concrete cases. We study here in detail the essential perturbations
for all the centers of the differential systems
\begin{eqnarray}
\dot{x} = -y + P_{\rm d}(x,y), \quad \dot{y} = x +
Q_{d}(x,y),
\end{eqnarray}
where $P_d$ and $Q_d$ are
homogeneous polynomials of degree $d$,
for $ d=2$ and $ d=3$.

keywords:
bifurcation.
,
non-degenerated center
,
Melnikov functions
,
essential perturbation
,
Cyclicity

## Year of publication

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