Publications

My publications in chronological order.

Preprints

Averaging and tracking of local attractors in slowly varying systems with two time scales

Carmen Núñez, Rafael Obaya, and Jorge Rodríguez

Abs arXiv

The paper analyzes to what extent the dynamics of a nonautonomous n-dimensional dynamical system with two time scales, formulated in the slow time as dx/dt=f(t/ɛ, t, x), can be approximated for small values of ɛ by the dynamics of the averaged system dz/dt=F(t,z). Assuming that the skewproduct flow associated with the averaged system admits a local attractor A, we prove that the solutions of the original system whose initial data lie in the basin of attraction of A track the fibers of the inflated attractor for all positive times. If the fiber map of A is continuous, inflation is no longer required. Alternative tracking results with a more classical formulation are also presented, under assumptions involving uniformly asymptotically stable solutions or uniform local attractors for the nonautonomous process, rather than for the skewproduct flow. Several examples illustrate the scope and applicability of the results. The twofold extension of the classical averaging results (to the doubly nonautonomous setting and to the whole positive halfline) is expected to be relevant to a broad range of application

2026

A note on the averaging principle for ordinary differential equations depending on the slow time

Carmen Núñez, Rafael Obaya, and Jorge Rodríguez

Applied Mathematics Letters, 2026

Abs DOI

The work presents a “doubly” nonautonomous version of the averaging principle, applicable to equations that depend on a small parameter ɛ and on (fast) time τ, but also on slow time t=ɛτ. The objectives are to establish optimal conditions on the dependence of the coefficients of the equations on t under which the averaging principle can be extended and to provide good estimates of the distance between the solutions of the initial equation and those of the averaged equation, always with τ varying in intervals of length proportional to 1/ɛ. The applicability of these results is based on the fact that the estimates obtained are uniform with respect to the initial time at which the solutions of both equations coincide.

2025

Métodos dinámicos y numéricos de la teoría del promedio con aplicación en problemas de osciladores

Jorge Rodríguez

2025

Abs Website

This work aims to demonstrate various bounds for the difference between the solution of a system of differential equation where two time-scales are present and its averaged system, using a modern and rigorous approach. The concept of averaging is defined, and properties of KBM functions are analyzed, establishing bounds for the difference between solutions on a time scale of 1/ε. Additionally, UKBM functions are introduced to generalize results to unbounded intervals. The theory of averaging for periodic functions is presented in its classical version, studying a change of variables that transforms the original equation into the averaged one, and generalizing to higher-order approximations. Finally, practical applications are presented, such as the analysis of the Kapitza pendulum and the theoretical basis and implementation of the of stroboscopic averaging numerical methods, which allow for the efficient integration of oscillator problems.

2024

Construcción de un pequeño radiotelescopio para la detección de la lı́nea del hidrógeno neutro

Jorge Rodríguez

2024

Abs Website

This work describes, from basic principles, how to build and operate a radio telescope for detecting the neutral hydrogen line at 21 cm. It covers the design and construction of the antenna, the operation of the electronics, and the subsequent data processing. Finally, it presents the spectra obtained through the described process.
Puntos de inflexión de superficies algebraicas, un enfoque moderno del trabajo de Salmon

Jorge Rodríguez

2024

Abs Website

This work focuses on providing an updated theoretical framework for Clebsch’s calculation, addressing limit cases and formalizing the symbolic method used by Clebsch and Salmon. The symbolic method is formalized using multilinear algebra, properties of determinants in rings are reviewed, and concepts of algebraic geometry are introduced to study inflection points. Finally, a rigorous proof of Clebsch’s calculation is presented to obtain a closed expression for Salmon’s polynomial.