Threshold Dynamics of Scalar Linear Periodic Delay-Differential Equations

  • Yuming Chen
  • Jianhong Wu
Part of the Fields Institute Communications book series (FIC, volume 64)


We consider the scalar linear periodic delay-differential equation \(\dot{x}(t) = -x(t) + ag(t)x(t - 1)\), where \(g : [0,\infty ) \rightarrow (0,\infty )\) is continuous and periodic with the minimal period ω>0. We show that there exists a positive a+ such that the zero solution is stable if \(a \in (0,{a}^{+})\) and unstable if a>a+. Examples and preliminary analysis suggest the challenge in obtaining analogous results when a<0.



Yuming Chen was supported in part by NSERC and the Early Researcher Award program of Ontario. Jianhong Wu was supported in part by CRC, MITACS and NSERC.

Received 4/4/2009; Accepted 2/10/2010


  1. [1].
    Y. Chen, Y.S. Huang, J. Wu, Desynchronization of large scale delayed neural networks. Proc. Amer. Math. Soc. 128, 2365–2371 (2000)MathSciNetCrossRefGoogle Scholar
  2. [2].
    Y. Chen, T. Krisztin, J. Wu, Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system. J. Differ. Equation 163, 130–173 (2000)MathSciNetCrossRefGoogle Scholar
  3. [3].
    S.N. Chow, H.O. Wlather, Characteristic multipliers and stability of symmetric periodic solutions of \(\dot{x}(t) = g(x(t - 1))\). Trans. Amer. Math. Soc. 307, 124–142 (1988)MathSciNetzbMATHGoogle Scholar
  4. [4].
    J. Hale, Theory of Functional Differential Equations (Springer, New York, 1977)CrossRefGoogle Scholar
  5. [5].
    Y.S. Huang, Desynchronization of delayed neural networks. Discrete Contin. Dyn. Syst. 7, 397–401 (2001)MathSciNetCrossRefGoogle Scholar
  6. [6].
    T. Krisztin, H.O. Walther, J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback (American Mathematical Society, Providence, 1999)Google Scholar
  7. [7].
    J. Mallet-Paret, G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J. Differ. Equat. 125, 385–440 (1996)MathSciNetCrossRefGoogle Scholar
  8. [8].
    X. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equations with bounded nonlinearity. J. Dyna. Differ. Equat. 3, 515–540 (1991)MathSciNetCrossRefGoogle Scholar
  9. [9].
    X. Xie, The multiplier equation and its application to S-solutions of a delay equation. J. Differ. Equat. 102, 259–280 (1992)Google Scholar
  10. [10].
    X. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity. J. Differ. Equat. 103, 350–374 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations