arXiv subjects

César M. Silva

Publications and source records attributed to César M. Silva.

At least 19 recordsLinked to original sources

arXiv2024

Convergence of asymptotic systems in Cohen-Grossberg neural network models with unbounded delays

In this paper, we investigate the convergence of asymptotic systems in non-autonomous Cohen--Grossberg neural network models, which include both infinite discrete time-varying and distributed delays. We derive stability results under conditions where the non-delay terms asymptotically dominate the delay terms. Several examples and a numerical simulation are provided to illustrate the significance and novelty of the main result.

math.DS↗

arXiv2023

Generalized non-autonomous Cohen-Grossberg neural network model

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincide degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen-Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and a numerical simulation to illustrate the effectiveness of some of our results.

q-bio.NC↗

arXiv2023

Nonuniform $μ$-dichotomy spectrum and kinematic similarity

For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate $μ$ in a large family of growth rates, we consider a notion of spectrum, named nonuniform $μ$-dichotomy spectrum. This family of spectrums contain the nonuniform dichotomy spectrum as the very particular case of exponential growth rates. For each growth rate $μ$, we describe all possible forms of the nonuniform $μ$-dichotomy spectrum, relate its connected components with adapted notions of Lyapunov exponents, and use it to obtain a reducibility result for nonautonomous linear differential equations. We also give an illustrative examples where the spectrum is obtained, including a situation where a normal form is obtained for polynomial behavior.

math.DS↗

arXiv2022

Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory(BAM).

math.CA↗

arXiv2021

An eco-epidemiological model with general functional response of predator to prey

We consider a nonautonomous eco-epidemiological model with general functions for predation on infected and uninfected preys as well as general functions associated to the vital dynamics of the susceptible prey and predator populations. We obtain persistence and extinction results for the infected prey based on assumptions on systems related to the dynamics in the absence of infected preys. We apply our results to eco-epidemiological models constructed from predator-prey models existent in the literature. Some illustrative simulation is undertaken.

math.DS↗

arXiv2020

Admissibility and generalized nonuniform dichotomies for discrete dynamics

We obtain characterizations of nonuniform dichotomies, defined by general growth rates, based on admissibility conditions. Additionally, we use the obtained characterizations to derive robustness results for the considered dichotomies. As particular cases, we recover several results in the literature concerning nonuniform exponential dichotomies and nonuniform polynomial dichotomies as well as new results for nonuniform dichotomies with logarithmic growth.

math.DS↗

arXiv2020

Random perturbations of an eco-epidemiological model

We consider random perturbations of a general eco-epidemiological model. We prove the existence of a global random attractor, the persistence of susceptibles preys and provide conditions for the simultaneous extinction of infectives and predators. We also discuss the dynamics of the corresponding random epidemiological $SI$ and predator-prey models. We obtain for this cases a global random attractor, prove the prevalence of susceptibles/preys and provide conditions for the extinctions of infectives/predators.

math.DS↗

arXiv2020

Dynamics of a discrete eco-epidemiological model with disease in the prey

Using Mickens nonstandard method, we obtain a discrete family of nonautonomous eco-epidemiological models that include general functions corresponding to the predation of the infected and uninfected preys. We obtain results on the persistence and extinction of the infected preys assuming that the bi-dimensional predator-prey subsystem that describes the dynamics in the absence of the infection satisfies some assumptions. Some examples and simulations are undertaken to illustrate our results.

math.DS↗

arXiv2018

Optimal control of non-autonomous SEIRS models with vaccination and treatment

We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.

math.OC↗

arXiv2017

Integral conditions for nonuniform $μ$-dichotomy on the half-line

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

math.DS↗

arXiv2017

Cosmic infinity: A dynamical system approach

Dynamical system techniques are extremely useful to study cosmology. It turns out that in most of the cases, we deal with finite isolated fixed points corresponding to a given cosmological epoch. However, it is equally important to analyse the asymptotic behaviour of the universe. On this paper, we show how this can be carried out for 3-forms model. In fact, we show that there are fixed points at infinity mainly by introducing appropriate compactifications and defining a new time variable that washes away any potential divergence of the system. The richness of 3-form models allows us as well to identify normally hyperbolic non-isolated fixed points.

gr-qc↗

arXiv2017

A nonautonomous epidemic model with general incidence and isolation

We obtain conditions for eradication and permanence of infection for a nonautonomous SIQR model with time-dependent parameters, that are not assumed to be periodic. The incidence is given by functions of all compartments and the threshold conditions are given by some numbers that play the role of the basic reproduction number. We obtain simple threshold conditions in the autonomous, asymptotically autonomous and periodic settings and show that our thresholds coincide with the ones already established. Additionally, we obtain threshold conditions for the general nonautonomous model with mass-action, standard and quarantine-adjusted incidence.

math.DS↗

arXiv2016

Existence of periodic orbits for periodic eco-epidemic models with disease in the prey

For an eco-epidemic model with disease in the prey and periodic coefficients it is conjectured in [Xingge Niu, Tailei Zhang, Zhidong Teng, The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey, Applied Mathematical Modelling 35, 457-470 (2011)] that, when the infected prey is permanent, there is a positive periodic orbit that is globally asymptotically stable in the interior of $(\R^+)^3$. In this paper we prove the existence part of the conjecture.

math.DS↗

arXiv2015

Existence of periodic solutions of a periodic SEIRS model with general incidence

For a family of periodic SEIRS models with general incidence, we prove the existence of at least one endemic periodic orbit when R_0>1. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when R_0<1. In particular, our main result establishes a sharp threshold between existence and non-existence of endemic periodic orbits for this family of models.

math.DS↗

arXiv2015

Nonuniform behavior and stability of Hopfield neural networks with delay

We obtain a result on the behavior of the solutions of a general nonautonomous Hopfield neural network model with delay, assuming some general bound for the product of consecutive terms in the sequence of neuron charging times and some conditions to control the nonlinear part of the equations. We then apply this result to improve some existent results on the stability of Hopfield models. Our results are based on a new abstract result on the behavior of nonautonomous delayed equations.

math.DS↗

arXiv2015

A mathematical model for the customer dynamics based on marketing policy

We consider a compartmental model to study the evolution of the number of regular customers and referral customers in some corporation. Transitions between compartments are modeled by parameters depending on the social network and the marketing policy of the corporation. We obtain some results on the asymptotic number of regular customers and referral customers in several particular scenarios. Additionally we present some simulation that illustrates the behavior of the model and discuss its applicability.

math.DS↗

arXiv2013

Generalized nonuniform dichotomies and local stable manifolds

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.

math.DS↗

arXiv2013

A non-autonomous SEIRS model with general incidence rate

For a non-autonomous SEIRS model with general incidence, that admits [T. Kuniya and Y. Nakata, Permanence and extinction for a nonautonomous SEIRS epidemic model, Appl. Math. Computing 218, 9321-9331 (2012)] as a very particular case, we obtain conditions for extinction and strong persistence of the infectives. Our conditions are computed for several particular settings and extend the hypothesis of several proposed non-autonomous models. Additionally we show that our conditions are robust in the sense that they persist under small perturbations of the parameters in some suitable family. We also present some simulations that illustrate our results.

math.DS↗