Sufficient Conditions for the Stability of Trivial Solutions of a Certain Class of Nonlinear Delay Differential Equations
Keywords:
Stability, Trivial solution, nonlinear delay differential equations, (DDES) Perturbation, linear and homogeneous systemAbstract
In this paper  we study  the stability of a trivial solution of  certain  nonlinear delay differential equations of the form where for us to improves on the existence literature, equation (1.1) was re-written as a perturbation of the linear homogeneous system of the form and and we use existences an uniqueness theorem of a linear system to establish sufficient conditions that guarantee the stability of the trivial solutions of a certain class of nonlinear delay differential equations. The goal of this paper is to give a simple criterion for the stability of (1.1) when re-written as a perturbation of a linear homogeneous system of the form (1.3) and (1.4).References
. An der Heiden U. Mackey M.C; The Dynamics of Production and Destruction. Analytic Insight into Complex Behavior, J.Math. Biology, 16(1982),75-101
. An der Heiden U. Mackey M.C; The Dynamics of Recurrent Inhibition, J. Math. Biology, 19(1984),211-225
. Bellman, R. ands Cooke, K. (1963). Difference Equations. Academic Press, New York.
. Balachandran B. Nonlinear dynamics of milling processes, Phil, Trans. R. Soc. Lond. A, 359 (2001) 793-819
. Cushing J. (1977), Integro differential Equations and Delay models in Population Dynamics, Volume 20 of Lecture Notes in Biomathematics. Springer –Verlag, Berlin; New York.
. Drive, R (1977) ordinary and Delay Differential Equations. Springer –Verlag, Berlin; New York.
.Edwards, C. and Penney, D. (2000), Differential Equations and Boundary Value Problems. Prentice Hall, Upper Saddle River, N.J.
.El’sgol’ts, Introduction to the theory of differential equations with deviating argument’ Holden-Day,1966
.Ebiendele E.P. (2010) on the Boundedness and Stability of Solutions of certain third-order non-linear Differential Equations. Archives of Applied Science Research, 2010, 2(4); 329-337.
.Glass L.Mackey M.C, Pathological conditions Resulting From Instabilities in Physiological control system, Annals of the New York Academy of sciences 316(1979),214-235
. Glass L.Mackey M.C “From clocks to Chaos’’ Princeton university press, 1988
. Gilsinn D.E. Discrete Fourier Series approximation to periodic solutions of autonomous delay differential equations, proceedings of IDETC/CIE 2005; ASME 2005 International Design Engineering Technical Conferences and Computer and Information in Engineering Conference, September 24-25, Lung Beach, CA (CD Rom).
. Hauie C; D.C. Dale, Rudnicki R, M.C Mackey, Modeling Complex Neutrophil Dynamics in The Grey Collie, J.THEOR.biol.(in press)(2000)
.Mac Donald N. Biological Delay Systems; linear stability theory, Cambridge University Press. Cambridge, 1989.
.Paul C.A.H, Developing a delay differential solver, Applied Numerical Mathematics 9(1992) 403-414.
. Tlusty J. and Polacek M. The Stability of Machine tool against self-excited vibration in Machining, Proc. Conf. on International Research in Production Engineering, Pittsburgg, PA, USA, (1963), 465-474.
]Tobias S.A. Machine-Tool Vibration, Wiley, 1965.
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