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David Rojas

Publications and source records attributed to David Rojas.

14 recordsLinked to original sources
arXiv2024

Monotonous period function for equivariant differential equations with homogeneous nonlinearities

We prove that the period function of the center at the origin of the $\mathbb{Z}_k$-equivariant differential equation $\dot{z}=iz+a(z\overline{z})^nz^{k+1}, a\ne0,$ is monotonous decreasing for all $n$ and $k$ positive integers, solving a conjecture about them. We show this result as corollary of proving that the period function of the center at the origin of a sub-family of the reversible quadratic centers is monotonous decreasing as well.

math.DS↗
arXiv2024

Relativistic effects in the dynamics of a particle in a Coulomb field

We prove that Bertrand's property cannot occur in a special-relativistic scenario using the properties of the period function of planar centres. We also explore some integrability properties of the relativistic Coulomb problem and the asymptotic behavior of collision solutions.

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arXiv2023

Characterization of the tree cycles with minimum positive entropy for any period

Consider, for any integer $n\ge3$, the set $\text{Pos}_n$ of all $n$-periodic tree patterns with positive topological entropy and the set $\text{Irr}_n\subset\text{Pos}_n$ of all $n$-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\text{Pos}_n$, $\text{Irr}_n$ and $\text{Pos}_n\setminus\text{Irr}_n$. Let $\lambda_n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty)$. We explicitly construct an irreducible $n$-periodic tree pattern $\mathcal{Q}_n$ whose entropy is $\log(\lambda_n)$. We prove that this entropy is minimum in $\text{Pos}_n$. Since the pattern $\mathcal{Q}_n$ is irreducible, $\mathcal{Q}_n$ also minimizes the entropy in the family $\text{Irr}_n$. We also prove that the minimum positive entropy in the set $\text{Pos}_n\setminus\text{Irr}_n$ (which is nonempty only for composite integers $n\ge6$) is $\log(\lambda_{n/p})/p$, where $p$ is the least prime factor of $n$.

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arXiv2022

Saddle-node bifurcation of limit cycles in an epidemic model with two levels of awareness

In this paper we study the appearance of bifurcations of limit cycles in an epidemic model with two types of aware individuals. All the transition rates are constant except for the alerting decay rate of the most aware individuals and the rate of creation of the less aware individuals, which depend on the disease prevalence in a non-linear way. For the ODE model, the numerical computation of the limit cycles and the study of their stability are made by means of the Poincar\'e map. Moreover, sufficient conditions for the existence of an endemic equilibrium are also obtained. These conditions involve a rather natural relationship between the transmissibility of the disease and that of awareness. Finally, stochastic simulations of the model under a very low rate of imported cases are used to confirm the scenarios of bistability (endemic equilibrium and limit cycle) observed in the solutions of the ODE model.

q-bio.PE↗
arXiv2020

Robustness of behaviourally-induced oscillations in epidemic models under a low rate of imported cases

This paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ODE model defined on contact networks. The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a non-oscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters.

q-bio.PE↗
arXiv2019

Resonance of bounded isochronous oscillators

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the phenomenon of resonance may appear. We give a sufficient condition on the perturbation in order that resonance occurs when the period annulus of the isochronous oscillator is bounded. In this context, resonance means that all solutions escape from the period annulus.

math.DS↗
arXiv2019

Asymptotic development of an integral operator and boundedness of the criticality of potential centers

We study the asymptotic development at infinity of an integral operator. We use this development to give sufficient conditions in order to upper bound the number of critical periodic orbits that bifurcate from the outer boundary of the period function of planar potential centers. We apply the main results to two different families: the power-like potential family $\ddot x=x^p-x^q$, $p,q\in\mathbb{R}$, $p>q$; and the family of dehomogenized Loud's centers.

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arXiv2018

Periodic oscillators, isochronous centers and resonance

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context, resonance means that all solutions are unbounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators.

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arXiv2018

On the upper bound of the criticality of potential systems at the outer boundary using the Roussarie-Ecalle compensator

This paper is concerned with the study of the criticality of families of planar centers. More precisely, we study sufficient conditions to bound the number of critical periodic orbits that bifurcate from the outer boundary of the period annulus of potential centers. In the recent years, the new approach of embedding the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary has shown to be fruitful in this issue. In this work, we tackle with a remaining case that was not taken into account in the previous studies in which the Roussarie-Ecalle compensator plays an essential role. The theoretical results we develop are applied to study the bifurcation diagram of the period function of two different families of centers: the power-like family $\ddot x=x^p-x^q$, $p,q\in\mathbb{R}$ with $p>q$; and the family of dehomogenized Loud's centers.

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arXiv2017

The monotonicity of the apsidal angle using the theory of potential oscillators

In a central force system the angle between two successive passages of a body through pericenters is called the apsidal angle. In this paper we prove that for central forces of the form $f(r)\sim λr^{-(α+1)}$ with $α<2$ the apsidal angle is a monotonous function of the energy, or equivalently of the orbital eccentricity.

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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.

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arXiv2017

Bifurcation of relative equilibria generated by a circular vortex path in a circular domain

We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number.

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