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The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag. (English) Zbl 1014.34062
The author considers the singularly perturbed delay differential system: $\varepsilon y'(t,\varepsilon) = f(t,y(t,\varepsilon),y(t-\tau(t),\varepsilon)), \quad t\geq t_0,\qquad y(t,\varepsilon)=\varphi(t), \quad t\leq t_0,$ with $$f:[0,+\infty)\times \mathbb{C}^s\times \mathbb{C}^s \to \mathbb{C}^s$$ and $$y(t,\varepsilon):\mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{C}^s$$.
Sufficient conditions are provided to ensure that any solution to this system with a bounded lag is exponentially stable uniformly for sufficiently small $$\varepsilon>0$$. As a preliminary result, a generalized Halanay inequality is derived.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K26 Singular perturbations of functional-differential equations 34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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##### References:
 [1] Derstine, M.W.; Gibbs, H.M.; Hopf, F.A.; Kaplan, D.L., Bifurcation gap in a hybrid optical system, Phys. rev. A, 26, 3720-3722, (1982) [2] Mackey, M.C.; Glass, L., Oscillation and chaos in physiological control systems, Science, 197, 287-289, (1977) · Zbl 1383.92036 [3] Longtin, A.; Milton, J., Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. biosci., 90, 183-199, (1988) [4] Ikeda, K., Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. commun., 30, 257, (1979) [5] Gibbs, H.M.; Hopf, F.A.; Kaplan, D.L.; Shoemaker, R.L., Observation of chaos in optical bistability, Phys. rev. lett., 46, 474-477, (1981) [6] Halanay, A., Differential equations, stability, oscillations, time lags, (1966), Academic Press New York · Zbl 0144.08701 [7] C.T.H. Baker, A. Tang, Generalized Halanay inequalities for Volterra functional differential equations and the discrete versions, Volterra Centennial Meeting, Arlington, June 1996 [8] Cooke, K.L., The condition of regular degeneration for singularly perturbed linear differential-difference equations, J. differential equations, 1, 39-94, (1965) · Zbl 0151.10303 [9] Hale, J.K.; Lunel, S.M.Verduyn, Introduction to functional differential equations, (1993), Springer-Verlag New York [10] Smith, D.R., Singular perturbation theory, (1985), Cambridge University Press [11] H. Tian, Numerical treatment of singularly perturbed delay differential equations, Ph.D. thesis, University of Manchester (2000) [12] Tian, H.; Kuang, J., The stability analysis of θ-methods for delay differential equations, J. comput. math., 14, 203-212, (1996) · Zbl 0857.65081 [13] Torelli, L., A sufficient condition for GPN-stability for delay differential equations, Numer. math., 25, 311-320, (1991) · Zbl 0712.65079
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