arXiv subjects

Manuel Pinto

Publications and source records attributed to Manuel Pinto.

23 records · Page 2Linked to original sources

arXiv2015

A Grobman-Hartman theorem for a differential equation with piecewise constant generalized argument

We obtain sufficient conditions ensuring the existence of a uniformly continuous and Hölder continuous homeomorphism between the solutions of a linear system of differential equations with piecewise constant argument of generalized type and the solutions of the quasilinear corresponding system. We use a definition (recently introduced by M. Akhmet) of exponential dichotomy for those systems combined with technical assumptions on the nonlinear part. Our result generalizes a previous work of G. Papaschinopoulos.

math.CA↗

arXiv2014

Asymptotic Integration of a Linear Fourth Order Differential Equation of Poincaré Type

This article deals with the asymptotic behavior of fourth order differential equation where the coefficients are perturbations of linear constant coefficient equation. We introduce a change of variable and deduce that the new variable satisfies a third order differential equation of Riccati type. We assume three hypothesis. The first is the following: all roots of the characteristic polynomial associated to the fourth order linear equation has distinct real part. The other two hypothesis are related with the behavior of the perturbation functions. Under this general hypothesis we obtain four main results. The first two results are related with the application of fixed point theorem to prove that the Riccati equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the Riccati equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise formulas for the asymptotic behavior of the linear fourth order differential equation.

math.CA↗

arXiv2014

Existence and Stability of Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Argument

This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of that kind satisfying an exponential dichotomy on a difference equation. The stability of that solution in a semi-axis studied.

math.DS↗

arXiv2013

Traveling waves for a model of the Belousov-Zhabotinsky reaction

Following J.D. Murray, we consider a system of two differential equations that models traveling fronts in the Noyes-Field theory of the Belousov-Zhabotinsky (BZ) chemical reaction. We are also interested in the situation when the system incorporates a delay $h\geq 0$. As we show, the BZ system has a dual character: it is monostable when its key parameter $r \in (0,1]$ and it is bistable when $r >1$. For $h=0, r\not=1$, and for each admissible wave speed, we prove the uniqueness of monotone wavefronts. Next, a concept of regular super-solutions is introduced as a main tool for generating new comparison solutions for the BZ system. This allows to improve all previously known upper estimations for the minimal speed of propagation in the BZ system, independently whether it is monostable, bistable, delayed or not. Special attention is given to the critical case $r=1$ which to some extent resembles to the Zeldovich equation.

math.CA↗

arXiv2011

Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = Δu(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.

math.AP↗