## 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.

DCDS

In this work we extend techniques based on computational algebra for bounding the cyclicity of nondegenerate centers to
nilpotent centers in a natural class of polynomial systems, those of the form $\dot x = y + P_{2m + 1}(x,y)$,
$\dot y = Q_{2m + 1}(x,y)$, where $P_{2m+1}$ and $Q_{2m+1}$ are homogeneous polynomials of degree $2m + 1$ in $x$ and $y$.
We use the method to obtain an upper bound (which is sharp in this case) on the cyclicity of all centers in the cubic family
and all centers in a broad subclass in the quintic family.

DCDS

In this work we extend well-known techniques for solving the Poincaré-Lyapunov nondegenerate analytic center problem in the plane to the 3-dimensional center problem at the zero-Hopf singularity. Thus we characterize the existence of a neighborhood of the singularity completely foliated by periodic orbits (including continua of equilibria) via an analytic Poincaré return map. The vanishing of the first terms in a Taylor expansion of the associated displacement map provides us with the necessary 3-dimensional center conditions in the parameter space of the family whereas the sufficiency is obtained through symmetry-integrability methods. Finally we use the proposed method to classify the 3-dimensional centers of some quadratic polynomial differential families possessing a zero-Hopf singularity.

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

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.

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