Some Applications of the Extended Bendixson-Dulac Theorem

During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work is to present an unified approach of some of these results, together with their corresponding proofs. We also provide several applications.Comment: 19 pages, 3 figure

Further considerations on the number of limit cycles of vector fields of the form X(v) = Av + f(v) Bv

AbstractIn Gasull, Llibre, and Sotomayor. (J. Differential Equations, in press) we studied the number of limit cycles of planar vector fields as in the title. The case where the origin is a node with different eigenvalues, which then resisted our analysis, is solved in this paper

Preprint arXiv: 2209.11253 Submitted on 22 Sep 2022

We study the interplay between symmetry representations of the physical and virtual space on the class of tensor network states for critical spins systems known as field tensor network states (fTNS). These are by construction infinite dimensional tensor networks whose virtual space is described by a conformal field theory (CFT). We can represent a symmetry on the physical index as a commutator with the corresponding CFT current on the virtual space. By then studying this virtual space representation we can learn about the critical symmetry protected topological properties of the state, akin to the classification of symmetry protected topological order for matrix product states. We use this to analytically derive the critical symmetry protected topological properties of the two ground states of the Majumdar-Ghosh point with respect to the previously defined symmetries

The local period function for Hamiltonian systems with applications

In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations.Comment: 23 page

Aplicación del algoritmo de Kalman al estudio de la ritmicidad ultradiana en el comportamiento expontáneo de sujetos humanos aislados

Este trabajo explora la posible existencia de ritmos ultradianos (período comprendido entre 30 minutos y 20 horas) en el comportamiento espontáneo vigil del hombre, analizando espectralmente las señales con el filtra do adaptativo de Kalman. En cuatro sujetos, cada uno de los cuales permaneció aislado en una habitación, se registraron los siguientes comportamientos: deambulación. exploración, autocontacto y alimentación. De forma simultánea se contabilizó la motilidad empleando un procedimiento telemétrico. La información se totalizó de manera acumulativa a intervalos de 5 minutos y se analizó utilizando el algoritmo de Kalman. El método de estimación espectral empleado obedece a un sistema adaptativo basado en predicción lineal, dando lugar a un cálculo recursivo de los parámetros del filtro predictor. La optimización de este procedimiento se pone en evidencia al compararlo con el de Widrow, en el que la ganancia permanece en un valor previamente fijado (~ constante). Todos los tipos de comportamiento (excepto la alimentaciÓn) muestran picos destacados en la banda de frecuencia de 10-20 ciclos por día (período medio 96 minutos) habitual en muchos otros parámetros biológicos del hombre, asi aomo ritmicidades de periodos más cortos.Peer ReviewedPostprint (published version

New advances on the Lyapunov constants of some families of planar differential systems

This note presents some advances regarding the Lyapunov constants of some families of planar polynomial differential systems, as a first step toward the resolution of the center and cyclicity problems. First, a parallelization approach is computationally implemented to achieve the 14th Lyapunov constant of the complete cubic family. Second, a technique based on interpolating some specific quantities so as to reconstruct the structure of the Lyapunov constants is used to study a Kukles system, some fifth-degree homogeneous systems, and a quartic system with two invariant lines

Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences.Comment: 25 page

Interaction of cochlin and mechanosensitive channel TREK-1 in trabecular meshwork cells influences the regulation of intraocular pressure.

This work was funded by National Institute of Health Grants R01 EY016112, EY015266, and EY014801 and an unrestricted grant to the University of Miami's Bascom Palmer Eye Institute from Research to Prevent Blindness. Financial support from Fight for Sight is gratefully acknowledged. Funding to XG was provided by Instituto de Salud Carlos III, Spain (FIS PI14/00141 and RETIC RD12/0034/0003) and Generalitat de Catalunya (2014SGR1165). In the eye, intraocular pressure (IOP) is tightly regulated and its persistent increase leads to ocular hypertension and glaucoma. We have previously shown that trabecular meshwork (TM) cells might detect aqueous humor fluid shear stress via interaction of the extracellular matrix (ECM) protein cochlin with the cell surface bound and stretch-activated channel TREK-1. We provide evidence here that interaction between both proteins are involved in IOP regulation. Silencing of TREK-1 in mice prevents the previously demonstrated cochlin-overexpression mediated increase in IOP. Biochemical and electrophysiological experiments demonstrate that high shear stress-induced multimeric cochlin produces a qualitatively different interaction with TREK-1 compared to monomeric cochlin. Physiological concentrations of multimeric but not monomeric cochlin reduce TREK-1 current. Results presented here indicate that the interaction of TREK-1 and cochlin play an important role for maintaining IOP homeostasis