arXiv subjects

John A. Arredondo

Publications and source records attributed to John A. Arredondo.

13 recordsLinked to original sources

arXiv2025

Periodic solutions in a tumor-immune competition system with time-delay and chemotherapy effects

The main purpose of this paper is to analyze the dynamics of the system of time-delay differential equations (DDEs) \begin{equation*} \begin{split} \dot{T}(t)&=T(t) f(t,T(t))-γE(t)T(t),\\ \dot{E}(t)&=σ+ \frac{pE(t)T(t-τ_1)}{g+a T(t-τ_1)}-\frac{mE(t)T(t-τ_2)}{g+a T(t-τ_2)}-ηE(t), \end{split} \end{equation*} where $T=T(t)$ and $E=E(t)$ represent the concentrations of tumor and effector cells at the time $t$. The coefficients $σ$, $μ$, $γ$, and $η$ are all positive, and $f(t, T)$ represents the relative growth rate of tumor cells, corresponding to a generalized logistic growth function that describes periodic time chemotherapeutic effects. The parameter $τ_1 \in \mathbb{R}_{\ge 0}$ is the response time delay of the immune system (mediated by effector cells) to an invasion of tumor cells, while $τ_2 \in \mathbb{R}_{\ge 0}$ represents the time delay of tumor cells in response to the appearance of effector cells.

math.DS↗

arXiv2023

Periodic oscillations in electrostatic actuators under time delayed feedback controller

In this paper, we prove the existence of two positive $T$-periodic solutions of an electrostatic actuator modeled by the time-delayed Duffing equation $$\ddot{x}(t)+f_{D}(x(t),\dot{x}(t))+ x(t)=1- \dfrac{e \mathcal{V}^{2}(t,x(t),x_{d}(t),\dot{x}(t),\dot{x}_{d}(t))}{x^2(t)}, \qquad x(t)\in\,]0,\infty[ $$ where $x_{d}(t)=x(t-d)$ and $\dot{x}_{d}(t)=\dot{x}(t-d),$ denote position and velocity feedback respectively, and $$ \mathcal{V}(t,x(t),x_{d}(t),\dot{x}(t),\dot{x}_{d}(t))=V(t)+g_{1}(x(t)-x_{d}(t))+g_{2}(\dot{x}(t)-\dot{x}_{d}(t)),$$ is the feedback voltage with positive input voltage $V(t)\in C(\mathbb{R}/T\mathbb{Z})$ for $e\in \mathbb{R}^{+}, g_{1},g_{2}\in \mathbb{R}$, $d\in [0,T[$. The damping force $f_{D}(x,\dot{x})$ can be linear, i.e., $f_{D}(x,\dot{x}) = c\dot{x}$, $c\in\mathbb{R}^+$ or squeeze film type, i.e., $f_{D}(x,\dot{x}) = γ\dot{x}/x^{3}$, $γ\in\mathbb{R}^+$. The fundamental tool to prove our result is a local continuation method of periodic solutions from the non-delayed case $(d=0)$. Our approach provides new insights into the delay phenomenon on microelectromechanical systems and can be used to study the dynamics of a large class of delayed Liénard equations that govern the motion of several actuators, including the comb-drive finger actuator and the torsional actuator. Some numerical examples are provided to illustrate our results.

math.OC↗

arXiv2022

On the fiber product over infinite-genus Riemann surfaces

Considering non-constant holomorphic maps $β_{i}:S_{i}\to S_{0}$, $i\in\{1,2\}$, between non-compact Riemann surfaces for which it is associated its fiber product $S_{1}\times_{(β_{1},β_{2})}S_{2}$. With this setting, in this paper we relate the ends space of such fiber product to the ends space of its normal fiber product. Moreover, we provide conditions on the maps $β_{1}$ and $β_{2}$ to guarantee connectednes on the fiber product. From these conditions, we link the ends space of fiber product with the topological type of the Riemann surfaces $S_{1}$ and $S_{2}$. We also study the fiber product over infinite hyperelliptic curves and discuss its connectedness and ends space.

math.AG↗

arXiv2022

Plane polynomials and Hamiltonian vector fields determined by their singular points

Let $Σ(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. Our goal is to study the critical point map $\mathfrak{S}_d$, by sending polynomials $f$ of degree $d$ to their critical points $Σ(f)$ . Very roughly speaking, a polynomial $f$ is essentially determined when any other $g$ sharing the critical points of $f$ satisfies that $f= λg$; here both are polynomials of at most degree $d$, $λ\in \mathbb{K}^*$. In order to describe the degree $d$ essentially determined polynomials, a computation of the required number of isolated critical points $δ(d)$ is provided. A dichotomy appears for the values of $δ(d)$; depending on a certain parity the space of essentially determined polynomials is an open or closed Zariski set. We compute the map $\mathfrak{S}_3$, describing under what conditions a configuration of four points leads to a degree three essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree three non essential determined polynomials. The quotient space of essentially determined polynomials of degree three up to the action of the affine group $\hbox{Aff}(\mathbb{K}^2)$ determines a singular surface over $\mathbb{K}$.

math.AG↗

arXiv2022

Periodic oscillations in a 2N-body problem

Hip-Hop solutions of the $2N$-body problem are solutions that satisfy at every instance of time, that the $2N$ bodies with the same mass $m$, are at the vertices of two regular $N$-gons, each one of these $N$-gons are at planes that are equidistant from a fixed plane $Π_0$ forming an antiprism. In this paper, we first prove that for every $N$ and every $m$ there exists a family of periodic hip-hop solutions. For every solution in these families the oriented distance to the plane $Π_0$, which we call $d(t)$, is an odd function that is also even with respect to $t=T$ for some $T>0.$ For this reason we call solutions in these families, double symmetric solutions. By exploring more carefully our initial set of periodic solutions, we numerically show that some of the branches stablished in our existence theorem have bifurcations that produce branches of solutions with the property that the oriented distance function $d(t)$ is not even with respect to any $T>0$, we call these solutions single symmetry solutions. We prove that no single symmetry solution is a choreography. We also display explicit double symmetric solutions that are choreographies.

math.DS↗

arXiv2021

Parabolicity of zero-twist tight flute surfaces and uniformization of the Loch Ness monster

We study the zero-twist flute surface and we associate to each one of them a sequence of positive real numbers $\mathbf{x}=(x_{n})_{n\in\mathbb{N}_{0}}$, with a torsion-free Fuchsian group $Γ_{\mathbf{x}}$ such that the convex core of $\mathbb{H}^2/Γ_{\mathbf{x}}$ is isometric to a zero-twist tight flute surface $S_{\mathbf{x}}$. Moreover, we prove that the Fuchsian group $Γ_{\mathbf{x}}$ is of the first kind if and only if the series $\sum x_{n}$ diverges. As consequence of the recent work of Basmajian, Hakobian and {Š}ari{ć}, we obtain that the zero-twist flute surface $S_{\mathbf{x}}$ is of parabolic type if and only $\sum x_{n}$ diverges. In addition, we present an uncountable family of hyperbolic surfaces homeomorphic to the Loch Ness Monster. More precisely, we associate to each sequence $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}$, where $y_{n}=(a_{n},b_{n},c_{n},d_{n},e_{n})\in \mathbb{R}^5$ and $a_n\leq b_n \leq c_n \leq d_n \leq e_n \leq a_{n+1}$, a Fuchsian group $G_{\mathbf{y}}$ such that $\mathbb{H}^2/G_{\mathbf{y}}$ is homeomorphic to the Loch Ness Monster.

math.DG↗

arXiv2021

Symmetry Type Graphs on 4-Orbit maps

It is well known that there exist twenty two symmetry type graphs associated to 4-orbit maps. For this ones we give the feasible values taken by the degree of the vertices and the number appropriate of edges in the boundary of each face of the map, by introducing the concepts of vertex type graph, face type graph and characteristic system.

math.CO↗

arXiv2019

Geometric Schotkky groups and non compact hyperbolic surface with infinite genus

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface $S_{m,s}$ with $s$ ends and exactly $m$ of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\leq s$, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group $Γ_{m,s}$ such that the quotient space $\mathbb{H}/ Γ_{m, s}$ is homeomorphic to $S_{m,s}$ and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

math.DG↗

arXiv2018

On Infinitely generated Fuchsian groups of some infinite genus surfaces

In this paper, for a non compact and orientable surface $S$ been either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we construct explicitly an infinitely generated Fuchsian group $Γ<PSL(2,\mathbb{R})$, such that the quotient $\mathbb{H}/Γ$ is a hyperbolic surface homeomorphic to $S$.

math.DG↗

arXiv2017

On the Infinite Loch Ness monster

In this paper we present in a topological way the construction of the orientable surface with only one end and infinite genus, called \emph{The Infinite Loch Ness Monster}. In fact, we introduce a flat and hyperbolic construction of this surface. We discuss how the name of this surface has evolved and how it has been historically understood.

math.GT↗

arXiv2015

On infinite regular and chiral maps

We prove that infinite regular and chiral maps take place on surfaces with at most one end. Moreover, we prove that an infinite regular or chiral map on an orientable surface with genus can only be realized on the Loch Ness monster, that is, the topological surface of infinite genus with one end.

math.CO↗

arXiv2014

Relative equilibria in quasi-homogeneous three body problems

In this paper we find the families of relative equilibria for the three body problem in the plane, when the interaction between the bodies is given by a quasi-homogeneous potential, which is the sum of two homogeneous functions. The number of the relative equilibria depends of the values of the masses and of the size of the system, measured by the moment of inertia.

math.DS↗

arXiv2014

Dynamics in the Schwarzschild isosceles three body problem

The Schwarzschild potential, defined as U(r)=-A/r-B/r^3, where r is the distance between two mass points and A,B>0, models astrophysical and stellar dynamics systems in a classical context. In this paper we present a qualitative study of a three mass point system with mutual Schwarzschild interaction where the motion is restricted to isosceles configurations at all times. We retrieve the relative equilibria and provide the energy-momentum diagram. We further employ appropriate regularization transformations to analyse the behaviour of the flow near triple collision. We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for non-zero total angular momenta (e.g., when two of the mass points spin infinitely many times around the centre of mass). This phenomenon is known in celestial mechanics as the "black-hole effect" and it is understood as an analogue in the classical context of the behaviour near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits which are neither homothetic, nor homographic.

math.DS↗