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### Browsing Articles publicats (Matemàtica) by Author "Algaba, Antonio"

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- ItemOpen AccessA New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields(Springer, 2021) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more In this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.Show more - ItemOpen AccessAnalytic integrability around a nilpotent singularity(Elsevier, 2019) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more In this work it is characterized the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions.Show more - ItemOpen AccessAnalytic integrability around a nilpotent singularity: The non-generic case(American Institute of Mathematical Sciences, 2020) Algaba, Antonio; Díaz, María; García, Cristóbal; Giné, Jaume
Show more Recently, in [9] is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. In this work we solve the remaining case completing the analytic integrability problem for such singularity.Show more - ItemOpen AccessAnalytic integrability inside a family of degenerate centers(Elsevier, 2016) Algaba, Antonio; Checa, Isabel; García, Cristóbal; Giné, Jaume
Show more In this paper we study the analytic integrability around the origin inside a family of degenerate centers or perturbations of them. For this family analytic integrability does not imply formal orbital equivalence to a Hamiltonian system. It is shown how difficult is the integrability problem even inside this simple family of degenerate centers or perturbations of them.Show more - ItemOpen AccessAnalytic integrability of some examples of degenerate planar vector fields(Springer, 2016) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more This paper is devoted to the classification of analytic integrable cases of two families of degenerate planar vector fields with a monodromic singular point at the origin. This study falls in the still open degenerate center problem. This classification can be done using the formal normal form theory and knowing a suitable normal form of any differential systems associated to each family.Show more - ItemOpen AccessAnalytically integrable system orbitally equivalent to a semi-quasihomogeneous system(Elsevier, 2023) Algaba, Antonio; García, Cristóbal; Reyes, Manuel; Giné, Jaume
Show more For perturbations of integrable non-Hamiltonian quasi-homogeneous planar vector field whose origin is a non-degenerate singular point, orbital linearization and analytic integrability are equivalent. We show a class of analytically integrable vector fields whose origin is a degenerate singular point which is orbitally equivalent to a semi-quasi-homogeneous system, that is, it is not orbital equivalent to its lowest-degree quasi-homogeneous term.Show more - ItemOpen AccessCenter conditions to find certain degenerate centers with characteristic directions(Elsevier, 2023) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more We consider the two-dimensional autonomous systems of differential equations where the origin is a monodromic degenerate singular point, i.e., with null linear part. In this work we give two heuristic procedures to obtain some center conditions (perhaps not necessary) for certain degenerate centers at the origin although they have characteristic directions.Show more - ItemOpen AccessCenter problem for generic degenerate vector fields(Elsevier, 2022) Algaba, Antonio; Díaz, María; García, Cristóbal; Giné, Jaume
Show more We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has not characteristic directions in the quasi-homogeneous leading term in certain coordinates. We apply this method to some degenerate differential systems.Show more - ItemOpen AccessGeometric criterium in the center problem(Springer Basel, 2016-10) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more In this paper we use a geometric criterium based in the classical method of the construction of Lyapunov functions to determine if a differential system has a focus or a center at a singular point. This criterium is proved to be useful for several examples studied in previous works with other more specific methods.Show more - ItemOpen AccessIntegrability of planar nilpotent differential systems through the existence of an inverse integrating factor(Elsevier, 2019) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more In this work is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. The analytic integrability problem is characterized via the existence of a formal inverse integrating factor. The relation between the analytic integrability and the existence of an algebraic inverse integrating factor is also given.Show more - ItemOpen AccessNilpotent centres via inverse integrating factors(Cambridge University Press, 2016-10) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more In this paper we are interested in the nilpotent center problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centers. More general objects are considered as the formal inverse integrating factors. However the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centers. For the family studied in this paper it is enough.Show more - ItemOpen AccessOn the Formal Integrability Problem for Planar Differential Systems(Hindawi Publishing Corporation, 2013) Giné, Jaume; Algaba, Antonio; García, Cristóbal
Show more We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems in C2, and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.Show more - ItemOpen AccessOrbital Reversibility of Planar Vector Fields(MDPI, 2021) Algaba, Antonio; García, Cristóbal; Giné, Jaume
Show more In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible.Show more - ItemOpen AccessThe center problem for Z_2-symmetric nilpotent vector fields(Elsevier, 2018) Algaba, Antonio; García, Cristóbal; Giné, Jaume; Llibre, Jaume
Show more We say that a polynomial differential system ˙x = P(x, y), ˙y = Q(x, y) having the origin as a singular point is Z2-symmetric if P(−x, −y) = −P(x, y) and Q(−x, −y) = −Q(x, y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C∞ first integral. But up to know there are no characterized these two kinks of nilpotent centers. Here we prove that the origin of any Z2-symmetric is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x, y) = y 2 + · · ·, where the dots denote terms of degree higher than two.Show more