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Jaume Giné

Publications and source records attributed to Jaume Giné.

At least 19 recordsLinked to original sources
arXiv2026

A new approach to the Poincaré center problem

We address the classical (degenerate or non-degenerate) center problem posed by Poincaré in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that, generically, every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincaré map is analytic. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize centers without curves of zero angular speed. Applications to nontrivial families that have resisted other methods are also provided.

math.DS↗
arXiv2026

Integrability of oscillators and transcendental invariant curves

In this work we study the integrability of a family of nonlinear oscillators. Dynamical systems from this family appear in different applications from mechanics to chemistry. We propose an approach for finding first integrals and integrating factors, which is based on the construction and classification of transcendental invariant curves whose cofactors are polynomial or rational in one of the variables. We demonstrate that this approach can be efficiently used for finding non-Liouvillian and non-Puiseux integrable dynamical systems. Its application involves finding solutions only of linear algebraic and linear ordinary differential equations. This allows one to study singularities, including essential ones, of the invariant curves in the complex plane. We illustrate this approach by proving non-Liouvillian integrability of two dynamical systems from the Painlevé--Gambier classification and non-Puiseux integrability of an oscillator from the considered family. Furthermore, we construct equivalence classes of the first two dynamical systems with respect to nonlocal transformations. We show that among these equivalence classes there are interesting examples of integrable dynamical systems.

nlin.SI↗
arXiv2026

Metrisable oscillators and (super)integrable two-dimensional metrics

We consider a family of nonlinear oscillators, which is the autonomous case of the two-dimensional projective connection. We construct several classes of these oscillators that are simultaneously integrable and metrisable. This leads to families of (super)integrable two-dimensional metrics that are parametrized by arbitrary functions. In the superintegrable case we obtain an explicit expression for the unparametrized geodesics. In the integrable case we present two families of metrics with transcendental first integrals. We introduce the concept of generalized Darboux integrability in the context of both projective equations and geodesic flows. We demonstrate that the constructed integrable metrics are generalized Darboux integrable. In addition, we establish a direct connection between relative Killing vectors and invariants of the projective vector fields that are linear in the first derivative. Finally, we compute the dimensions of the projective Lie algebra for the obtained metrics, which allows us to distinguish previously known integrable cases from new ones.

nlin.SI↗
arXiv2026

Monodromic singularities of Brunella-Miari vector fields with two edges in the Newton diagram

This work focuses on the study of monodromic singularities in planar analytic families of vector fields whose Newton diagram consists of exactly two edges. We begin by analyzing the desingularization scheme of a minimal model of polynomial vector fields, denoted by ${X}$, which includes only the monomials corresponding to the vertices of the Newton diagram. We then extend this minimal model to the so-called Brunella-Miari vector fields ${X} \subset {X}^{[1]}$, incorporating all monomials associated with points lying on the edges of the Newton diagram. As a second extension, we consider vector fields ${X}^{[1]} \subset {X}^{[2]}$ that include higher-order terms corresponding to points located above the polygonal line in the Newton diagram. The key point of our approach is to preserve the desingularization geometry at each extension step. We provide explicit desingularization procedures, which enable the computation of the linear part of the return map $Π$ in cases where the desingularized singularity is associated with a hyperbolic polycycle. Several nontrivial examples are included to illustrate the method.

math.DS↗
arXiv2024

Characterization of centers by its complex separatrices

In this work we deal with analytic families of real planar vector fields $\mathcal{X}_λ$ having a monodromic singularity at the origin for any $λ\in Λ\subset \mathbb{R}^p$ and depending analytically on the parameters $λ$. There naturally appears the so-called center-focus problem which consists in describing the partition of $Λ$ induced by the centers and the foci at the origin. We give a characterization of the centers (degenerated or not) in terms of a specific integral of the cofactor associated to a real invariant analytic curve passing through the singularity, which always exists. Several consequences and applications are also stated.

math.DS↗
arXiv2024

Principal Bautin ideal of monodromic singularities with inverse integrating factors

We analyze the structure of the Poincaré map $Π$ associated to a monodromic singularity of an analytic family of planar vector fields. We work under two assumptions. The first one is that the family possesses an inverse integrating factor that can be expanded in Laurent series centered at the singularity after a weighted polar blow-up fixed by the Newton diagram of the family. The second one is that we restrict our analysis to a subset of the monodromic parameter space that assures the non-existence of local curves with zero angular speed. The conclusions are that the asymptotic Dulac expansion of $Π$ does not contain logarithmic terms, indeed it admits a formal power series expansion with a unique independent generalized Poincaré-Lyapunov quantity, which can be computed under some explicit conditions. Moreover we also give conditions that guarantee the analyticity of $Π$, in which case we show that the Bautin ideal is principal and therefore the cyclicity of the singularity with respect to perturbation within the family is zero.

math.DS↗
arXiv2024

On the geometric and analytical properties of the anharmonic oscillator

Here we consider the anharmonic oscillator that is a dynamical system given by $y_{xx}+δy^{n}=0$. We demonstrate that to this equation corresponds a new example of a superintegrable two-dimensional metric with a linear and a transcendental first integrals. Moreover, we show that for particular values of $n$ the transcendental first integral degenerates into a polynomial one, which provides an example of a superintegrable metric with additional polynomial first integral of an arbitrary even degree. We also discuss a general procedure of how to construct a superintegrable metric with one linear first integral from an autonomous nonlinear oscillator that is cubic with respect to the first derivative. We classify all cubic oscillators that can be used in this construction. Furthermore, we study the Liénard equations that are equivalent to the anharmonic oscillator with respect to the point transformations. We show that there are nontrivial examples of the Liénard equations that belong to this equivalence class, like the generalized Duffing oscillator or the generalized Duffing--Van der Pol oscillator.

math.DS↗
arXiv2022

Generalized interacting Barrow Holographic Dark Energy: cosmological predictions and thermodynamic considerations

We construct a generalized interacting model of Barrow Holographic Dark Energy (BHDE) with infrared cutoff being given by the Hubble horizon. We analyze the cosmological evolution of a flat Friedmann-Lemaître-Robertson-Walker Universe filled by pressureless dark matter, BHDE and radiation fluid. The interaction between the dark sectors of the cosmos is assumed of non-gravitational origin and satisfying the second law of thermodynamics and Le Chatelier-Braun principle. We study the behavior of various model parameters, such as the BHDE density parameter, the equation of state parameter, the deceleration parameter, the jerk parameter and the square of sound speed. We show that our model satisfactorily retraces the thermal history of the Universe and is consistent with current observations for certain values of parameters, providing an eligible candidate to describe dark energy. We finally explore the thermodynamics of our framework, with special focus on the validity of the generalized second law.

gr-qc↗
arXiv2021

Gravitational effects on the Heisenberg Uncertainty Principle: a geometric approach

The Heisenberg Uncertainty Principle (HUP) limits the accuracy in the simultaneous measurements of the position and momentum variables of any quantum system. This is known to be true in the context of non-relativistic quantum mechanics. Based on a semiclassical geometric approach, here we propose an effective generalization of this principle, which is well-suited to be extended to general relativity scenarios as well. We apply our formalism to Schwarzschild and de Sitter space-time, showing that the ensuing uncertainty relations can be mapped into well-known deformations of the HUP. We also infer the form of the perturbed metric that mimics the emergence of a discrete spacetime structure at Planck scale, consistently with the predictions of the Generalized Uncertainty Principle. Finally, we discuss our results in connection with other approaches recently appeared in the literature.

gr-qc↗
arXiv2017

A counterexample to the composition condition conjecture for polynomial Abel differential equations

The Polynomial Abel differential equations are considered a model problem for the classical Poincaré center--focus problem for planar polynomial systems of ordinary differential equations. Last decades several works pointed out that all the centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.

math.CA↗
arXiv2014

Essential perturbations of polynomial vector fields with a period annulus

In this paper we first give the explicit definition of essential perturbation. Secondly, given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which we explicitly know its Poincaré--Liapunov constants, we give the structure of its $k$-th Melnikov function. This result generalizes the result obtained by Chicone and Jacobs for perturbations of degree at most two of any center of a quadratic polynomial system. Moreover we study the essential perturbations for all the centers of the differential systems \[ \dot{x} \, = \, -y + P_{\rm d}(x,y), \quad \dot{y} \, = \, x + Q_{\rm d}(x,y), \] where $P_{\rm d}$ and $Q_{\rm d}$ are homogeneous polynomials of degree ${\rm d}$, for ${\rm d}=2$ and ${\rm d}=3$.

math.DS↗
arXiv2012

Towards a quantum universe

In this short review we study the state of the art of the great problems in cosmology and their interrelationships. The reconciliation of these problems passes undoubtedly through the idea of a quantum universe.

physics.gen-ph↗
arXiv2011

The Pioneer anomaly and the holographic scenario

In this paper we discuss the recently obtained relation between the Verlinde's holographic model and the first phenomenological Modified Newtonian dynamics. This gives also a promising possible explanation to the Pioneer anomaly.

physics.gen-ph↗
arXiv2009

Weierstrass integrability of differential equations

The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems.

math.DS↗
arXiv2006

On the origin of the anomalous precession of Mercury's perihelion

Action at distance in Newtonian physics is replaced by finite propagation speeds in classical post--Newtonian physics. As a result, the differential equations of motion in Newtonian physics are replaced by functional differential equations, where the delay associated with the finite propagation speed is taken into account. Newtonian equations of motion, with post--Newtonian corrections, are often used to approximate the functional differential equations. In ``On the origin of quantum mechanics'', preprint, physics/0505181, May 2005, a simple atomic model based on a functional differential equation which reproduces the quantized Bohr atomic model was presented. The unique assumption was that the electrodynamic interaction has a finite propagation speed. In ``On the origin of the gravitational quantization: The Titius--Bode Law'', preprint, physics/0507072, Jul 2005, a simple gravitational model based on a functional differential equation which gives a gravitational quantification and an explanation of the modified Titius--Bode law is described. Firstly, in this work, we recall the calculations made by Einstein to arrive at the explanation of the anomalous precession of Mercury's perihelion. Secondly, we recover an ancient work of Gerber in 1898 as a precursor of the retarded theories. In this paper Gerber gave an explanation of the anomalous precession of the Mercury's perihelion in terms of a velocity--dependent potential. In this paper an explanation of the anomalous precession of Mercury's perihelion is given in terms of a simple retarded potential, which, at first order, coincides with Gerber's potential, and which agrees with the author's previous works.

physics.gen-ph↗
arXiv2006

On the origin of the deflection of light

Action at distance in Newtonian physics is replaced by finite propagation speeds in classical post--Newtonian physics. As a result, the differential equations of motion in Newtonian physics are replaced by functional differential equations, where the delay associated with the finite propagation speed is taken into account. Newtonian equations of motion, with post--Newtonian corrections, are often used to approximate the functional differential equations. In ``On the origin of quantum mechanics'', preprint, physics/0505181, May 2005, a simple atomic model based on a functional differential equation which reproduces the quantized Bohr atomic model was presented. The unique assumption was that the electrodynamic interaction has a finite propagation speed. In ``On the origin of the gravitational quantization: The Titius--Bode Law'', preprint, physics/0507072, Jul 2005, a simple gravitational model based on a functional differential equation which gives a gravitational quantification and an explanation of the modified Titius--Bode law is described. In ``On the origin of the anomalous precession of Mercury's perihelion'', preprint, physics/0510086, Oct 2005, an explanation of the anomalous precession of Mercury's perihelion is given in terms of a simple retarded potential, which, at first order, coincides with Gerber's potential of 1898, and which agrees with the author's previous works. In this paper, it is shown how the simple retarded potential presented in physics/0510086 also gives the correct value of the gravitational deflection of fast particles of General Relativity.

physics.gen-ph↗
arXiv2006

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré--Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré--Liapunov method.

math.DS↗
arXiv2005

On the origin of quantum mechanics

Action at distance in Newtonian physics is replaced by finite propagation speeds in classical post--Newtonian physics. As a result, the differential equations of motion in Newtonian physics are replaced by functional differential equations, where the delay associated with the finite propagation speed is taken into account. Newtonian equations of motion, with post--Newtonian corrections, are often used to approximate the functional differential equations. Are the finite propagation speeds the origin of the quantum mechanics? In this work a simple atomic model based on a functional differential equation which reproduces the quantized Bohr atomic model is presented. As straightforward application of the result the fine structure of the hydrogen atom is approached.

physics.gen-ph↗