Non-algebraic invariant curves for polynomial planar vector fields
Isaac A. García Jaume Giné
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.
keywords: Nonlinear differential equations Liouvillian and non-Liouvillian first integral. non-algebraic invariant curves
Center conditions for generalized polynomial kukles systems
Jaume Giné

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 Pi(x) are polynomials of degree n, P0(0) = 0 and P0′(0) < 0. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems when P0 is of degree 2 and Pi for i = 1; 2; 3 are of degree 3 without constant terms. We also establish a conjecture about the center conditions for such systems.

keywords: Center problem analytic integrability polynomial Generalized Kukles systems Gröbner bases decomposition in prime ideals
Liénard and Riccati differential equations related via Lie Algebras
Isaac A. García Jaume Giné Jaume Llibre
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.
keywords: Lie algebra integrability. Liénard system Riccati differential equation
Center problem for systems with two monomial nonlinearities
Armengol Gasull Jaume Giné Joan Torregrosa
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.
keywords: Poincaré--Lyapunov constants Nondegenerate center Darboux center Holomorphic center Reversible center Persistent center.
Linearization of smooth planar vector fields around singular points via commuting flows
Isaac A. García Jaume Giné Susanna Maza
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.
keywords: Lie symmetries linearization problem. Planar differential equations
Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems
Jaume Giné Maite Grau Jaume Llibre
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.
keywords: polynomial first integral rational first integral. Quasi--homogeneous polynomial differential equations integrability problem
Essential perturbations of polynomial vector fields with a period annulus
Adriana Buică Jaume Giné Maite Grau
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
Analytic integrability for some degenerate planar systems
Antonio Algaba Cristóbal García Jaume Giné
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.
keywords: integrability problem degenerate center problem. Nonlinear differential systems

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