Stability Of Nonlinear Differential Equations Research Articles

On stability of nonlinear difference equations and some of their applications

The issues of stability in solving nonlinear difference equations were considered. Based on a generalized difference analog of the well-known Bihari lemma, stability conditions for a trivial solution based on initial data were obtained, and an a priori estimate of stability under permanent disturbances was determined. The results were used to study the stability of solving explicit and implicit difference schemes approximating nonlinear parabolic equations.

A New Discriminant Method for the Stability of Fourth-Order Differential Equation

The stability of nonlinear differential equations has been widely studied based on various methods. In this paper, we transform a class of fourth-order differential equations into a perturbation system and study the stability and asymptotic stability of zero solutions. By utilizing Lyapunov’s second method, we give sufficient conditions for zero-solution stability and asymptotic stability and provide the construction method of the zero-solution asymptotically stable Lyapunov function of such fourth-order differential equations. In addition, this paper removes the requirement of negative definite when the general differential equation is zero solution stable and obtains a looser stability condition. This paper provides an alternative approach to the research of stability of differential systems.

Nagumo-type uniqueness and stability for nonlinear differential equations on semi-infinite intervals

We prove a Nagumo-type uniqueness result for a nonlinear differential equation on a semi-infinite interval. A typical example of such boundary value problems is a recently derived nonlinear model for the ocean flow in arctic gyres. Successive approximations are constructed to converge uniformly to the unique solution. A stability result is also proved.

Ulam-Hyers-Rassias Stability of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

Fractional derivatives are used to model the transmission of many real world problems like COVID-19. It is always hard to find analytical solutions for such models. Thus, approximate solutions are of interest in many interesting applications. Stability theory introduces such approximate solutions using some conditions. This article is devoted to the investigation of the stability of nonlinear differential equations with Riemann-Liouville fractional derivative. We employed a version of Banach fixed point theory to study the stability in the sense of Ulam-Hyers-Rassias (UHR). In the end, we provide a couple of examples to illustrate our results. In this way, we extend several earlier outcomes.

Asymptotic behavior of a class of perturbed differential equations

UDC 517.9 This paper deals with the problem of stability of nonlinear differential equations with perturbations. Sufficient conditions for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained. The asymptotic behavior is studied in the sense that the trajectories converge to a small ball centered at the origin. Furthermore, an illustrative example in the plane is given to verify the effectiveness of the theoretical results.

On stability functional differential equation with delay variable by using fixed point-theory

I will explain how to use the Banach fixed point theory in the asymptotic stability of nonlinear differential equations; I will obtain appropriate generalizations and strong forms of some of the results in [11]. Specifically, in the above-mentioned paper, asymptotic stability is achieved, while I will discuss how to achieve a asymptotic stability as well as stability by making a simple observation, also circulate the previous asymptotic stability results to the Functional Differential Equations systems, not only on the scalar Functional Differential Equations as is the case in the mentioned paper. This raises the question of how much this particular method can afford us, and what are the limitations of this technique. I will refer to the important limitation of the fixed point theory on the uniqueness of solutions only within the complete metric space area where they are not specified. If the metric space onto which the contraction mapping principle is applied is very small, i do not get a satisfactory result. I will discuss this in detail below.

Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation

Linearization is a useful tool for analyzing the stability of nonlinear differential equations. Unfortunately, the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations. General results giving conditions for when stability (or instability) of the linearized equation implies the same for the nonlinear equation are given here. These results are applied to stability and stabilization of the Kuramoto--Sivashinsky equation, a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibrium solutions depends on the value of a positive parameter $\nu$. It is shown that if $\nu>1$, then the set of constant equilibrium solutions is globally asymptotically stable. If $\nu<1$, then the equilibria are unstable. It is also shown that stabilizing the linearized equation implies local exponential stability of the equation. Stabilization of the Kuramoto--Sivashinsky equation using a single distributed control is considered and it is described how to use a finite-dimensional approximation to construct a stabilizing controller. The results are illustrated with simulations.

Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses

This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0,T], for a class of higher‐order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley &amp; Sons, Ltd.

New Conditions for the Exponential Stability of Nonlinear Differential Equations

We develop a method for proving local exponential stability of nonlinear nonautonomous differential equations as well as pseudo-linear differential systems. The logarithmic norm technique combined with the “freezing” method is used to study stability of differential systems with slowly varying coefficients and nonlinear perturbations. Testable conditions for local exponential stability of pseudo-linear differential systems are given. Besides, we establish the robustness of the exponential stability in finite-dimensional spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. We illustrate the application of this test to linear approximations of the differential systems under consideration.

A Fixed Point Approach to Hyers-Ulam-Rassias Stability of Nonlinear Differential Equations

In this paper we use the fixed point approach to obtain sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear differential. Some illustrative examples are given.

ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS

We investigate the stability of nonlinear differential equations of the form <TEX>$y^{(n)}(x)=F(x,y(x),y^{\prime}(x),{\cdots},y^{(n-1)}(x))$</TEX> with a Lipschitz condition by using a fixed point method. Moreover, a Hyers-Ulam constant of this differential equation is obtained.

Stability of Nonlinear Differential Equations with Stochastic Methods

In the paper, the stability of a stochastically perturbed Lotka-Volterra-type nonlinear system is studied by computer techniques.

Optimal linear stability condition for scalar differential equations with distributed delay

Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean delay, the linear equation with distributed delay is asymptotically stable if the associated differential equation with a discrete delay is asymptotically stable. We illustrate this criterion on a compartment model of hematopoietic cell dynamics to obtain sufficient conditions for stability.

GENERALIZED DISCRETE HALANAY INEQUALITIES AND THE ASYMPTOTIC BEHAVIOR OF NONLINEAR DISCRETE SYSTEMS

In this paper, some new generalized discrete Halanay inequalities are established. On the basis of these new established inequalities, we obtain the attracting set and the global asymptotic stability of the nonlinear discrete systems. Our results established here extend the main results in [R. P. Agarwal, Y. H. Kim, and S. K. Sen, New discrete Halanay inequalities: stability of difference equations, Commun. Appl. Anal. 12 (2008), no. 1, 83-90] and [S. Udpin and P. Niamsup, New discrete type inequalities and global stability of nonlinear difference equations, Appl. Math. Lett. 22 (2009), no. 6, 856-859].

Stability of nonlinear time-varying perturbed differential equations

The goal of this paper is twofold. The first part presents a converse Lyapunov theorem for the notion of uniform practical exponential stability of nonlinear differential equations in presence of small perturbation. This class of nonlinear differential equations can be viewed as parametric differential equations. The second part provides the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The practical asymptotic validity on the approximate is established on infinite-time interval. Finally, we give a numerical example to prove the validity of our methods.

Nonlinear Differential Equations with “Maxima”: Parametric Stability in Terms of two Measures

This paper investigates parametric stability for nonlinear differential equations with maxima. Several sufficient conditions for parametric stability as well as uniform parametric stability are obtained based on the Razumikhin method. Two different types of Lyapunov functions have been applied. A comparison with scalar ordinary differential equations is offered.

Hyers-Ulam Stability of Linear and Nonlinear Differential Equations of Second Order

In this paper we established the Hyers-Ulam stability of a nonlinear differential equation of second order with initial condition. We also proved the Hyers -Ulam stability of a linear differential equation of second order with initial condition.

Averaging techniques without requiring a fast time-varying differential equation

An averaging result is presented for local uniform asymptotic stability of nonlinear differential equations without requiring a fast time-varying vectorfield. The nonlinearity plays a crucial role: close to the origin, the trajectories vary slowly compared to the time dependence of the vectorfield. The result generalises averaging results which prove stability properties for systems having a homogeneous vectorfield with positive order. The result is illustrated with several examples.

Exponential stability of nonlinear delay equation with constant decay rate via perturbed system method

This paper studies the exponential stability of nonlinear differential equations with constant decay rate under the assumption that the corresponding crisp equation (without delay, simply, nondelay equation) is exponentially stable. Different from most publications dealing with delay systems by applying Lyapunov-type methods, the perturbed system method is used in this paper. It shall be shown that the considered equations will remain exponentially stable provided the time lag is small enough. Moreover, we formulate and estimate the threshold of delay ensuring exponential stability when a constant decay rate appears explicitly in system model, which is better than the existing results.

Study on the stability of nonlinear differential equations with initial time difference

Abstract This paper establishes several criteria of stability for nonlinear differential dynamical systems relative to initial time difference by employing perturbing Lyapunov functions. The reported novel results are more easy to be tested than the existing results. As an application, these results are applied to a nonlinear dynamical system to get the stability properties relative to initial time difference.