## Journals

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### Open Access Journals

DCDS

In this paper we give, as far as we know,
the first method to detect non-algebraic invariant curves for
polynomial planar vector fields. This approach is based on the
existence of a generalized cofactor for such curves. As an
application of this algorithmic method we give some Lotka-Volterra
systems with non-algebraic invariant curves.

CPAA

Abstract. In this paper we study the center problem for certain generalized Kukles systems $\dot{x}= y, \qquad \dot{y}= P_0(x)+ P_1(x)y+P_2(x) y^2+ P_3(x) y^3, $\end{document} where *P _{i}*(

*x*) are polynomials of degree n,

*P*

_{0}(0) = 0 and

*P*

_{0}′(0) < 0. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems when

*P*

_{0}is of degree 2 and

*P*for

_{i}*i*= 1; 2; 3 are of degree 3 without constant terms. We also establish a conjecture about the center conditions for such systems.

DCDS-B

In this work we study the Liénard systems that can be transformed into Riccati differential equations, using changes of variables more general than the ones used by the classical Lie theory.

CPAA

We study the center problem for planar systems with a linear center at the origin that in complex
coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several
centers inside this family. To the best of our knowledge this list includes a new class of Darboux
centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the
given list is exhaustive. We get several partial results that seem to indicate that this is the case.
In particular, we solve the question for several general families with arbitrary high degree and for
all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest
known order for weak-foci of planar polynomial systems of some given degrees.

CPAA

In this paper we propose a constructive procedure to get the
change of variables that linearizes a smooth planar vector field
on $\mathbb C^2$ around an elementary singular point (i.e., a
singular point with associated eigenvalues $\lambda, \mu \in \mathbb C$
satisfying $\mu$≠$0$) or a nilpotent singular point from a
given commutator. Moreover, it is proved that the near--identity
change of variables that linearizes the vector field $\mathcal X
= (x+\cdots) \partial_x + (y+\cdots) \partial_y$ is unique and linearizes simultaneously all the
centralizers of $\mathcal X$. The method is used in order to
obtain the linearization of some extracted examples of
the existent literature.

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

CPAA

In the present paper we characterize the analytic integrability around the origin
of a family of degenerate differential systems. Moreover, we study the analytic integrability of some degenerate systems through the orbital reversibility and from the existence of a Lie's symmetry for these systems. The results obtained for this family are similar to the results obtained when we characterize the analytic integrability of
non-degenerate and nilpotent systems. The obtained results can be applied to compute the analytic integrable systems of any particular family of degenerate systems studied.

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