Nonautonomous Differential Systems Research Articles

Stability analysis of homogeneous cooperative positive differential systems with time-varying delays and its generalization

This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.

Best Ulam constants for two‐dimensional nonautonomous linear differential systems

AbstractThis study deals with the Ulam stability of nonautonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two‐dimensional nonautonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on , but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for nonautonomous systems other than periodic systems.

Caputo-fabrizio fractional-order systems: periodic solution and stabilization of non-periodic solution with application to gunn diode oscillator

Fractional-order autonomous systems do not possess any non-constant periodic solutions, and to the best of our knowledge, there are no existing results regarding the existence of the periodic solution for fractional-order non-autonomous systems. The main objective of this work is to fill the above gap by studying the existence of a periodic solution of the Caputo-Fabrizio fractional-order system and also to find ways to stabilize a non-periodic solution. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo-Fabrizio system cannot admit a non-constant periodic solution. Under a similar assumption as the one for an integer-order differential system, and by using the properties of the Caputo-Fabrizio derivative, the existence of a periodic solution of a non-autonomous Caputo-Fabrizio fractional-order differential system is established. The main result is utilized in constructing and finding the periodic solution of the linear non-homogeneous Caputo-Fabrizio system. By using the result on linear systems, we derive a periodic solution of a fractional-order Gunn diode oscillator under a periodic input voltage, and observe that the diameter of the periodic orbit keeps reducing as the fractional-order continuously increases. In the end, by using the result on a linear non-homogeneous system, and by constructing a suitable linear feedback control, the solution of the linear non-homogeneous fractional-order system is stabilized to a periodic solution. An example is presented to support the obtained result. The main advantage of the proposed method over others is the simple considerations like the concept of equilibrium point and the utilization of the property of the Caputo-Fabrizio derivatives instead of other types of fractional derivatives.

Asymptotic Boundedness of Conformable Fractional Delay Differential Systems

This article focuses on the problem of asymptotic boundedness for non-autonomous conformable fractional delay differential systems. Firstly, by utilizing the definition of conformable fractional derivative and fractional comparison principle, we propose non-autonomous conformable fractional Halanay inequalities. And then, based on the Halanay inequalities, sufficient conditions for the asymptotic boundedness are constructed for the systems. Finally, a numerical example shows the applicability of the proposed technique.

Exploring the impact of temperature on the efficacy of replacing a wild Aedes aegypti population by a Wolbachia-carrying one

A non-autonomous time-delayed differential system, with time-varying delay, is proposed to reproduce the competitive dynamics of Wolbachia-infected and non-infected mosquito populations in several scenarios that differ by daily environmental temperature, the bacterial strain carried by the mosquito, and the guidelines for release of infected mosquitoes. Both mosquito entomological parameters and infection traits depend on temperature, which per se depends on time. Therefore, inspired by the literature on insect populations, functional forms are proposed to describe the rates of birth, development, and survival (or mortality) of Ae. aegypti as a function of temperature, as well as the rate of Wolbachia-infection loss. Numerical results showed that: (i) multiple releases were more efficient than a single one, (ii) when the mosquito population is high is the best time to implement the release of infected mosquitoes, (iii) strains that produce both high levels of cytoplasmic incompatibility and maternal inheritance boost the efficacy of the technique, and (iv) high temperature can jeopardize the efficacy of the technique.

Existence and approximate controllability results for second-order impulsive stochastic neutral differential systems

This paper focuses on the existence and approximate controllability of second-order non-autonomous impulsive stochastic neutral differential systems in Hilbert spaces. First, by using Schauder's fixed-point theorem, stochastic analysis theory, and evolution operators, a new set of sufficient conditions are formulated, and we prove the existence of mild solutions of second-order non-autonomous impulsive stochastic neutral differential systems. Further, the result is extended to study the approximate controllability results for second-order non-autonomous impulsive stochastic neutral differential systems. Then, a set of sufficient conditions are derived for the approximate controllability of second-order non-autonomous impulsive stochastic neutral differential system by assuming the associated linear system is approximately controllable. The results are obtained with the help of Krasnoselskii's fixed point theorem. Finally, an example is given to illustrate our obtained results.

On the Faedo–Galerkin Method for Non-autonomous Nonlinear Differential Systems

On the Faedo–Galerkin Method for Non-autonomous Nonlinear Differential Systems

Stability and robust stability of non-autonomous linear differential equations with infinite delay

By the idea of comparison principle, we first introduce novel explicit criteria for the $ \varrho $-stability of general linear non-autonomous functional differential equations with infinite delay. Some characterizations for the time-independent exponential stability are also investigated. We then present a sufficient condition for the exponential stability of the equations subject to time-varying structured perturbations. These results lead to applications to the linear non-autonomous differential system with unbounded delay and the linear non-autonomous integro-differential equations with delay. Some examples are given to illustrate the obtained results.

Large time behavior of nonautonomous differential systems modeling antibiotic‐resistant bacteria in rivers

In this paper, we study a general nonautonomous model for bacterial dynamics in rivers. The mathematical model is represented by a nonautonomous system of nonlinear ordinary differential equations. We show the existence of a bounded positive invariant and attracting set. By using the Lyapunov function method, we establish global stability of steady‐state solutions of the associated autonomous system. Second, the existence of positive periodic solutions of the nonautonomous system is proven using a continuation theorem based on coincidence degree theory.

A Lanczos-type procedure for tensors

The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real-world problems confirm the effectiveness of our approach.

Non-autonomous Differential Systems with Delays: A Global Attraction Analysis

In this paper, we derive criteria of global attractivity of a (possibly constant) positive periodic solution in non-autonomous systems of delay differential equations. Our approach can be viewed as the extension for non-autonomous systems of the folkloric connection between discrete dynamics and scalar delay differential equations. It is worth mentioning that we provide delay-dependent criteria of global attraction that cover the best delay independent conditions. We apply our results to non-autonomous variants of several classical models such that Nicholson’s blowfly equation, Goodwin’s model oscillator, the Mackey–Glass equation and systems with patch structure.

Positive Solutions of a Generalized Nonautonomous Fractional Differential System

In this work, we investigate the existence and uniqueness of positive solutions to a system of nonautonomous fractional differential equations. The fractional derivative of the system at hand is ψ‐ Riemann–Liouville fractional derivative. Some significant properties of this derivative that assist in the analysis of our results substantial are proved. The analysis is based on Guo‐Krasnoselskii’s and Banach’s fixed point theorem. Finally, we give two examples to verify our results.

Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms

In this paper, we obtain sufficient conditions for the persistence and permanence of a family of nonautonomous systems of delay differential equations. This family includes structured models from mathematical biology, with either discrete or distributed delays in both the linear and nonlinear terms, and where typically the nonlinear terms are nonmonotone. Applications to systems inspired by mathematical biology models are given.

Rectifiability of orbits for two-dimensional nonautonomous differential systems

Rectifiability of orbits for two-dimensional nonautonomous differential systems

Controllability of nonlocal non-autonomous neutral differential systems including non-instantaneous impulsive effects in 𝕉 n

Abstract This manuscript involves a class of first-order controllability results for nonlocal non-autonomous neutral differential systems with non-instantaneous impulses in the space 𝕉 n . Sufficient conditions guaranteeing the controllability of mild solutions are set up. Concept of evolution family and Rothe’s fixed point theorem are employed to achieve the required results. A model is investigated to delineate the adequacy of the results.

Approximate controllability of second-order non-autonomous stochastic impulsive differential systems

This paper is concerned with existence, uniqueness, and approximate controllability of mild solutions to second-order non-autonomous stochastic impulsive differential systems. The impulsive differential systems may provide mathematical models to the phenomena having discontinuous jumps. The main results are carried out by using the generalized Banach contraction principle, evolution family, and stochastic analysis. Finally, an example is given to illustrate the results.

Stability for Nonautonomous Linear Differential Systems with Infinite Delay

We study the stability of general $n$-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first situation, the effect of the delays is dominated by non-delayed diagonal negative feedback terms, and sufficient conditions for both the asymptotic and the exponential asymptotic stability of the system are given. In the second case, the stability depends on the size of some bounded diagonal delays and coefficients, although terms with unbounded delay may co-exist. Our results encompass DDEs with discrete and distributed delays, and enhance some recent achievements in the literature.

Nonuniform almost reducibility of nonautonomous linear differential equations

We prove that a linear nonautonomous differential system with nonuniform hyperbolicity on the half line can be written as diagonal system with a perturbation which is small enough. Moreover we show that the diagonal terms are contained in the nonuniform exponential dichotomy spectrum. For this purpose we introduce the concepts of nonuniform almost reducibility and nonuniform contractibility which are generalizations of these notions originally defined in a uniform context.

Boundedness analysis of non-autonomous stochastic differential systems with Lévy noise and mixed delays

The present research studies the boundedness issue of Levy driven non-autonomous stochastic differential systems with mixed discrete and distributed delays. A set of sufficient conditions of the $p$th moment globally asymptotical boundedness is obtained by combining the Lyapunov function method with the inequality technique. The proposed results reveal that the convergence rate $\lambda$ and the coefficients of the estimates for Lyapunov function $W$ and Ito operator $\mathcal {L}W$ can determine the upper bound for the solution. The presented results are demonstrated by an illustrative example.

Development and analysis of a malaria transmission mathematical model with seasonal mosquito life‐history traits

AbstractIn this paper, we develop and analyze a malaria model with seasonality of mosquito life‐history traits: periodic‐mosquitoes per capita birth rate, ‐mosquitoes death rate, ‐probability of mosquito to human disease transmission, ‐probability of human to mosquito disease transmission, and ‐mosquitoes biting rate. All these parameters are assumed to be time dependent leading to a nonautonomous differential equation system. We provide a global analysis of the model depending on two threshold parameters and (with ). When , then the disease‐free stationary state is locally asymptotically stable. In the presence of the human disease‐induced mortality, the global stability of the disease‐free stationary state is guarantied when . On the contrary, if , the disease persists in the host population in the long term and the model admits at least one positive periodic solution. Moreover, by a numerical simulation, we show that a sub‐critical (backward) bifurcation is possible at . Finally, the simulation results are in accordance with the seasonal variation of the reported cases of a malaria‐epidemic region in Mpumalanga province in South Africa.