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DCDS

In this paper we find necessary and sufficient conditions in order
that a planar quasi--homogeneous polynomial differential system has
a polynomial or a rational first integral. We also prove that any
planar quasi--homogeneous polynomial differential system can be
transformed into a differential system of the form $\dot{u} \, = \,
u f(v)$, $\dot{v} \, = \, g(v)$ with $f(v)$ and $g(v)$ polynomials,
and vice versa.

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

DCDS-B

We consider an autonomous differential system in $\mathbb{R}^n$
with a periodic orbit and we give a new method for computing the
characteristic multipliers associated to it. Our method works when
the periodic orbit is given by the transversal intersection of
$n-1$ codimension one hypersurfaces and is an alternative to the
use of the first order variational equations. We apply it to study
the stability of the periodic orbits in several examples,
including a periodic solution found by Steklov studying the rigid
body dynamics.

## Year of publication

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