Stability and asymptotic behavior of differential equations. We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use them to stabilize numerical calculations. Stability and asymptotic behavior of difference equations. By establishing two lemmas, the exponential stability and the asymptotical stability for mild solution to the secondorder neutral stochastic partial differential equations with infinite delay are obtained. Asymptotic stability and asymptotic solutions of second. Asymptotic stability of a nonlinear volterra integro differential system, bull. Diamandescu, y bounded solutions for a lyapunov matrix differential equation, electronic journal of qualitative theory of differential equations. Using razumikhintype techniques and liapunovs direct method, they establish conditions to ensure the ultimate boundedness and the global attractivity of solutions of, and when ft 0, the asymptotic stability. Asymptotic behavior of difference equations springerlink. It presents papers on the theory of the dynamics of differential equations ordinary differential equations, partial differential equations, stochastic differential equations, and functional differential equations. Autonomous equations stability of equilibrium solutions.
On the asymptotic behavior of solutions of certain third. We then recast our equations in mollified form thereby obtaining stability. Several avenues are available for members of the uva community needing library resources, including hathitrusts newlyreleased trove of ed digital material, open educational resources, online journals, databases, and ebooks. Coppel, stability and asymptotic behavior of differential equations, d. Stability and boundedness are two of the most important topics in the study of stochastic functional differential equations sfdes. Burton1 and tetsuo furumochi2 1northwest research institute 732 caroline st. Levinson, the asymptotic nature of solutions of linear systems of differential equations, duke math. Suppose that we have a set of autonomous ordinary differential equations, written in vector form.
Asymptotic behavior and exponential stability criteria for. We study an atherosclerosis model described by a reactiondiffusion system of three equations, in one dimension, with homogeneous neumann boundary conditions. Chapter eleven stability theory and asymptotic behavior. Asymptotic behavior and stability problems in ordinary differential equations. We prove asymptotical stability and instability results for a general secondorder di. Journal of dynamics and differential equations home. On the uniform asymptotic stability of certain linear. Also, the assumption onis sufficient to preserve the strong stability from 1. Coupling functions, asymptotic instability analysis, discretization, qualitative theory of differential equations, bifurcation.
Graduate texts in mathematics readings in mathematics, vol 182. Ball, on the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, j. To prove asymptotic stability of linear secondorder di. In this paper the monotonicity and asymptotic behavior of solutions for a class of secondorder nonlinear di. Astashova, asymptotic behavior of solutions of certain nonlinear differential equations, reports of extended session of a seminar of the i. The aim of this work is to study asymptotic properties of a class of fourthorder delay differential equations. Artstein, limiting equations and stability of nonautonomous ordinary differential equations, appendix to j. Our method follows classical analysis for firstorder systems and higherorder scalar equations where growth behavior. Linearized asymptotic stability for fractional differential equations 3 2 preliminaries we start this section by brie. Asymptotic stability of nsolitons in the cubic nls equation. Asymptotic behavior and stability problems in ordinary. Hovhannisyan, asymptotic stability for secondorder differential equations with complex coef.
Imprint boston, heath 1965 physical description viii, 166 p. Pdf asymptotic behavior of nonoscillatory solutions of. In this paper we use fixed point method to prove asymptotic stability results of the zero solution of the totally nonlinear neutral difference equation with variable delay. The behaviour of solutions to certain second order nonlinear delay differential equations with variable deviating arguments is discussed. A scalar delay differential equation with diffusion term in one space dimension, where the diffusivity d is a bifurcation parameter, is considered. The notion of exponential stability guarantees a minimal rate of decay, i. In general the stability analysis depends greatly on the form of the function ft. We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach.
Asymptotic behavior and stability of the solutions of functional differential equations in hilbert space article pdf available in nonlinear dynamics and systems theory january 2002 with 36 reads. In this paper, the existence and uniqueness, the stability analysis for the global solution of highly nonlinear stochastic differential equations with timevarying delay and markovian switching are. Necessary and sufficient conditions for the asymptotic. Stability and asymptotic behaviour of differential equations. Port angeles, wa 98362 2department of mathematics shimane university matsue, japan 6908504 abstract. Linearized asymptotic stability for fractional differential. We refer the reader to the books 5,6 for more details. New numerical methods for this kind of equations are constructed. Liapunov functions and functionals one of the most effective methods for treating stability problems in the theory of differential equations or of differentialdifference equations is the socalled direct method. This volume presents several important and recent contributions to the emerging field of fractional differential equations in a selfcontained manner. Asymptotic behavior of nonlinear compartmental systems. Special attention is paid in proving sufficient conditions ensuring almost sure asymptotic stability. Asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations mizukami, masatsugu, naito, manabu, and usami, hiroyuki, hiroshima mathematical journal, 2002. Our results in this paper not only generalize some previous results, but also.
This paper mainly discusses the almost sure asymptotic stability and the boundedness of nonlinear sfdes satisfying the local lipschitz condition but not the linear growth condition. Abstract pdf 2736 kb 1979 qualitative behavior of nonlinear differential equations describing adaptive filters using nonideal multipliers. The main procedure lies in the properties of a complete lyapunov. Asymptotic stability of a nonlinear volterra integrodifferential system, bull. Diamandescu, y bounded solutions for a lyapunov matrix differential equation, electronic journal of qualitative theory of differential equations, no.
Stability and asymptotic properties of a linear fractional. Asymptotic methods in the theory of linear differential equations by. Criteria are established for uniform asymptotic stability, boundedness, uniform ultimate boundedness and asymptotic behaviour of solutions of certain. Stability analysis for systems of differential equations. The convergence and asymptotic stability of the methods for this kind of equations. The simplest numerical method, eulers method, is studied in chapter 2.
The stability of constant steadystate solutions and the asymptotic behavior of the timedependent solutions are studied. Journal of differential equations 12,236255 1972 asymptotic behavior and exponential stability criteria for differential delay equations steven e. Stability and asymptotic behavior of differential equations by w. The mathematical theory of the linear system is simpler and more complete than that of the nonlinear system.
The idea of lyapunov stability can be extended to infinitedimensional manifolds, where it is known as structural stability, which concerns the behavior of different but nearby solutions to differential equations. Pdf asymptotic behavior of nonlinear compartmental systems. Pdf asymptotic behavior and stability of the solutions. Pseudoregularly varying functions and generalized renewal processes, 345393.
This paper discusses qualitative properties of the twoterm linear fractional difference equation. This chapter discusses the stability theory and asymptotic behavior for nonlinear differential difference equations. Stability and asymptotic behavior of differential equations c by w. Monotonicity and asymptotic behavior of solutions for secondorder nonlinear differential equations lianwen wang and je. The journal of dynamics and differential equations answers the research needs of scholars of dynamical systems. Asymptotic stability of nsolitons in the cubic nls. The almost sure asymptotic stability and boundedness of. We give sufficient conditions for a differential equation in a banach space to possess asymptotic equilibrium. This paper is concerned with a class of linear impulsive delay differential equations. Chapter eleven stability theory and asymptotic behavior for. In the last few decades the theory of ordinary differential equations has grown rapidly under the action of forces which have been working both from within and without. Stability behavior of the zero solution for nonlinear damped vectorial second order differential. False asymptotic instability behavior at iterated functions. Asymptotic behavior of nonoscillatory solutions of nonlinear differential equations with forcing term.
Aug 31, 2017 the aim of this work is to study asymptotic properties of a class of fourthorder delay differential equations. The concept of hstability is studied and compared with the classical stabilities. In this matter, singh discussed the asymptotic behavior of solutions of the thirdorder linear di. Consider a system of real differential equations for i v v j j ivtc j j 1. Nonoscillation and stability article pdf available in ieee transactions on circuits and systems cas256. To prove asymptotic stability of secondorder differential equations, we establish stability. Here we assume that the coefficients of sfdes are polynomial or dominated by. Basically, the hstability is applied to obtain a uniform treatment for the concept of stability in difference equations. On asymptotic behavior of solutions of certain nonlinear differential equations. Pdf asymptotic stability in totally nonlinear neutral. Stability and asymptotic behaviour of solutions is then discussed using as candidates the norm and a lyapunovlike function. Grossman department of mathematics, technzon, hazfa, israel presently at coppin state college, baltzmore, md. In 1973, driver, sasser and slater 4 studied asymptotic behavior, oscillation and stability of. Some properties of the solutions of third order linear ordinary differential equations.
Jul 04, 2007 asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations mizukami, masatsugu, naito, manabu, and usami, hiroyuki, hiroshima mathematical journal, 2002 zero noise limit of a stochastic differential equation involving a local time kuwada, kazumasa and matsumura, taro, osaka journal of mathematics, 2018. Lasalle, the stability of dynamical systems, regional conference series in applied mathematics 25, siam, 1976. Autonomous equations stability of equilibrium solutions first order autonomous equations, equilibrium solutions, stability, longterm behavior of solutions, direction fields, population dynamics and logistic equations autonomous equation. Criteria are established for uniform asymptotic stability, boundedness, uniform ultimate boundedness and asymptotic behaviour of solutions of certain third order nonlinear di. Since stable and unstable equilibria play quite different roles in the dynamics of a system, it is useful to be able to classify equilibrium points based on their stability. The study of the cauchy problem for differential equations in a banach space has taken two directions. Jul 17, 2006 asymptotic behavior of solutions of stochastic differential equations. Asymptotic stability of a class of impulsive delay. Some results on the pathwise asymptotic stability of solutions to stochastic partial differential equations are proved. In this paper, the existence and uniqueness, the stability analysis for the global solution of highly nonlinear stochastic differential equations with timevarying delay and markovian switching are analyzed under a locally lipschitz condition and a monotonicity condition. Stability and asymptotic behaviour of differential equations w a coppel download bok. Examples are considered to elucidate the main results. Bifurcation and asymptotic behavior of solutions of a. Astashova, asymptotic behavior of solutions of certain nonlinear differential equations.
On the asymptotic behavior of highly nonlinear hybrid. Asymptotic stability of analytic solutions of this kind of equations is studied by the property of delay differential equations without impulsive perturbations. Our results in this paper not only generalize some previous results, but also improve the earlier ones. Pinto, stability of nonlinear difference equations, proceedings of dynamic systems and applications 2 1996, 397404. Dec 14, 2009 stability and asymptotic behavior of differential equations by w. For this purpose, we show that this fractional equation is the volterra equation of convolution type. Asymptotic behavior and stability of second order neutral. Lamberto cesari in the last few decades the theory of ordinary differential equations has grown. On qualitative properties and asymptotic behavior of.485 500 1459 899 255 1120 551 1456 451 477 1254 980 1097 1175 85 779 36 487 1009 870 236 685 1270 427 1219 1111 873 1197 302 380