arXiv subjects

Jordi Villadelprat

Publications and source records attributed to Jordi Villadelprat.

12 recordsLinked to original sources

arXiv2025

The cyclicity of hyperbolic hemicycles

We consider families of planar polynomial vector fields of degree $n$ and study the cyclicity of a type of unbounded polycycle~$Γ$ called hemicycle. Compactified to the Poincaré disc,~$Γ$ consists of an affine straight line together with half of the line at infinity and has two singular points, which are hyperbolic saddles located at infinity. We prove four main results. Theorem A deals with the cyclicity of~$Γ$ when perturbed without breaking the saddle connections. For the other results we consider the case $n=2$. More concretely they are addressed to the quadratic integrable systems belonging to the class $Q_3^R$ and having two hemicycles, $Γ_u$ and $Γ_\ell$, surrounding each one a center. Theorem B gives the cyclicity of $Γ_u$ and $Γ_\ell$ when perturbed inside the whole family of quadratic systems. In Theorem C we study the number of limit cycles bifurcating simultaneously from $Γ_u$ and $Γ_\ell$ when perturbed as well inside the whole family of quadratic systems. Finally, in Theorem D we show that for three specific cases there exists a simultaneous alien limit cycle bifurcation from $Γ_u$ and $Γ_\ell$.

math.DS↗

arXiv2023

The period of the limit cycle bifurcating from a persistent polycycle

We consider smooth families of planar polynomial vector fields $\{X_μ\}_{μ\inΛ}$, where $Λ$ is an open subset of $\mathbb{R}^N$, for which there is a hyperbolic polycycle $Γ$ that is persistent (i.e., such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of $Γ$ at $μ_0$ is zero unless its graphic number $r(μ_0)$ is equal to one. It is also well known that if $r(μ_0)=1$ (and some generic conditions on the return map are verified) then the cyclicity of $Γ$ at $μ_0$ is one, i.e., exactly one limit cycle bifurcates from $Γ$. In this paper we prove that this limit cycle approaches $Γ$ exponentially fast and that its period goes to infinity as $1/|r(μ)-1|$ when $μ\toμ_0.$ Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.

math.DS↗

arXiv2022

Bifurcation analysis of the Microscopic Markov Chain Approach to contact-based epidemic spreading in networks

The dynamics of many epidemic compartmental models for infectious diseases that spread in a single host population present a second-order phase transition. This transition occurs as a function of the infectivity parameter, from the absence of infected individuals to an endemic state. Here, we study this transition, from the perspective of dynamical systems, for a discrete-time compartmental epidemic model known as Microscopic Markov Chain Approach, whose applicability for forecasting future scenarios of epidemic spreading has been proved very useful during the COVID-19 pandemic. We show that there is an endemic state which is stable and a global attractor and that its existence is a consequence of a transcritical bifurcation. This mathematical analysis grounds the results of the model in practical applications.

physics.soc-ph↗

arXiv2022

The criticality of reversible quadratic centers at the outer boundary of its period annulus

This paper deals with the period function of the reversible quadratic centers \begin{equation*} X_{\np}=-y(1-x)\partial_x+(x+Dx^2+Fy^2)\partial_y, \end{equation*} where $\np=(D,F)\in\R^2.$ Compactifying the vector field to $\Sc^2$, the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle $\out_\np$ at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of $X_\np$ that tend to $\out_{\np_0}$ in the Hausdorff sense as $\np\to\np_0.$ This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all $\np=(D,F)\in\R^2$ outside the segments $\{-1\}\times [0,1]$ and $\{0\}\times [0,2]$. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all $\np\in\R^2.$ In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to~$\np$ and where the principal part is given in a monomial scale containing a deformation of the logarithm. More precisely, Theorem~A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in~$\np$ but transcendental.

math.CA↗

arXiv2022

On the cyclicity of Kolmogorov polycycles

In this paper we study planar polynomial Kolmogorov's differential systems \[ X_μ\quad\sist{xf(x,y;μ),}{yg(x,y;μ),} \] with the parameter $μ$ varying in an open subset $Λ\subset\R^N$. Compactifying $X_μ$ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle $Γ$, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all $μ\inΛ.$ We are interested in the cyclicity of $Γ$ inside the family $\{X_μ\}_{μ\inΛ},$ i.e., the number of limit cycles that bifurcate from $Γ$ as we perturb $μ.$ In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with $N=3$ and $N=5$, and in both cases we are able to determine the cyclicity of the polycycle for all $μ\inΛ,$ including those parameters for which the return map along $Γ$ is the identity.

math.CA↗

arXiv2021

Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: Coefficient properties

We consider a $\mathscr C^\infty$ family of planar vector fields $\{X_{\hatμ}\}_{\hatμ\in\hat W}$ having a hyperbolic saddle and we study the Dulac map $D(s;\hatμ)$ and the Dulac time $T(s;\hatμ)$ from a transverse section at the stable separatrix to a transverse section at the unstable separatrix, both at arbitrary distance from the saddle. Since the hyperbolicity ratio $λ$ of the saddle plays an important role, we consider it as an independent parameter, so that $\hatμ=(λ,μ)\in \hat W=(0,+\infty)\times W$, where $W$ is an open subset of $\mathbb R^N.$ For each $\hatμ_0\in\hat W$ and $L>0$, the functions $D(s;\hatμ)$ and $T(s;\hatμ)$ have an asymptotic expansion at $s=0$ and $\hatμ\approx\hatμ_0$ with the remainder being uniformly $L$-flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so-called Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on $\hatμ$ that can be shown to be $\mathscr C^\infty$ in their respective domains and "universally" defined, meaning that their existence is stablished before fixing the flatness $L$ and the unfolded parameter $\hatμ_0.$ Each coefficient has its own domain and it is of the form $((0,+\infty)\setminus D)\times W$, where~$D$ a discrete set of rational numbers at which a resonance of the hyperbolicity ratio $λ$ occurs. In our main result we give the explicit expression of some of these coefficients and to this end a fundamental tool is the employment of a sort of incomplete Mellin transform. With regard to these coefficients we also prove that they have poles of order at most two at $D\times W$ and we give the corresponding residue, that plays an important role when compensators appear in the principal part.

math.DS↗

arXiv2021

Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centers

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^μ-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,a)$ with $\varepsilon\approx 0$ and $a$ in an open subset $A$ of $\mathbb R^α,$ and we study the Dulac time $\mathcal T(s;\varepsilon,a)$ of one of its hyperbolic sectors. We prove (Theorem A) that the derivative $\partial_s\mathcal T(s;\varepsilon,a)$ tends to $-\infty$ as $(s,\varepsilon)\to (0^+,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centers. In this regard we show (Theorem B) that no bifurcation occurs from certain semi-hyperbolic polycycles.

math.DS↗

arXiv2017

A criticality result for polycycles in a family of quadratic reversible centers

We consider the family of dehomogenized Loud's centers $X_μ=y(x-1)\partial_x+(x+Dx^2+Fy^2)\partial_y,$ where $μ=(D,F)\in\mathbb{R}^2,$ and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_μ,μ\in\mathbb{R}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set $Γ_{B}$ of codimension 1 in $\mathbb{R}^2$. In the present paper we succeed in proving that a subset of $Γ_{B}$ has criticality equal to one.

math.DS↗

arXiv2015

Unfoldings of saddle-nodes and their Dulac time

In this paper we study unfoldings of saddle-nodes and their Dulac time. By unfolding a saddle-node, saddles and nodes appear. In the first result (Theorem A) we prove uniform regularity by which orbits and their derivatives arrive at a node. Uniformity is with respect to all parameters including the unfolding parameter bringing the node to a saddle-node and a parameter belonging to a space of functions. In the second part, we apply this first result for proving a regularity result (Theorem B) on the Dulac time (time of Dulac map) of an unfolding of a saddle-node. This result is a building block in the study of bifurcations of critical periods in a neighbourhood of a polycycle. Finally, we apply Theorems A and B to the study of critical periods of the Loud family of quadratic centers and we prove that no bifurcation occurs for certain values of the parameters (Theorem C).

math.DS↗

arXiv2015

On the wave length of smooth periodic traveling waves of the Camassa-Holm equation

This paper is concerned with the wave length $λ$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$ (or "peak-to-peak amplitude"). Our main result establishes monotonicity properties of the map $a\longmapsto λ(a)$, i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of $λ(a)$, namely monotonicity and unimodality. The key point is to relate $λ(a)$ to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.

math.DS↗

arXiv2008

Bifurcation of critical periods from Pleshkan's isochrones

Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities $\mathscr C_3.$ In this paper we prove that if we perturb any of these isochrones inside $\mathscr C_3,$ then at most two critical periods bifurcate from its period annulus. Moreover we show that, for each $k=0,1,2,$ there are perturbations giving rise to exactly $k$ critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers $\mathscr C_2.$ Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in $\mathscr C_2.$ We prove that if we perturb three of them inside $\mathscr C_2,$ then at most one critical period bifurcates from its period annulus. In addition, for each $k=0,1,$ we show that there are perturbations giving rise to exactly $k$ critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper.

math.DS↗

arXiv2008

A Chebyshev criterion for Abelian integrals

We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.

math.DS↗