Journal of Dynamics and Differential Equations

, Volume 22, Issue 2, pp 253–284

Global Continua of Rapidly Oscillating Periodic Solutions of State-Dependent Delay Differential Equations



We apply our recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with state-dependent delay. We give sufficient geometric conditions to ensure the uniform boundedness of periodic solutions, obtain an upper bound of the period of non-constant periodic solutions in a connected component of Hopf bifurcation, and establish the existence of rapidly oscillating periodic solutions.


Differential equations State-dependent delay Hopf bifurcation Global continuation Upper bound of period 

Mathematics Subject Classification (2000)

34K18 46A30 


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  1. 1.
    Chow S.N., Mallet-Paret J.: The Fuller index and global Hopf bifurcation. J. Differ. Equ. 29, 66–85 (1978)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Diliberto, S.P.: Bounds for periods of periodic solutions. In: Lefschetz, S. (ed.) Contributions to the Theory of Nonlinear Oscillations, vol. 3, pp. 269–275. Annals of Mathematics Studies, no. 36. Princeton University Press, Princeton, NJ (1956)Google Scholar
  3. 3.
    Fuller F.B.: On the surface of section and periodic trajectories. Am. J. Math. 87(2), 473–480 (1965)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fuller F.B.: Bounds for the periods of periodic orbits. In: Auslander, J., Gottschalk, W.H. (eds) Topological Dynamics: An International Symposium, pp. 205–215. W. A. Benjamin, Inc., New York (1968)Google Scholar
  5. 5.
    Gustafson G.B., Schmit K.: A note on periodic solutions for delay-differential systems. Proc. Am. Math. Soc. 42(1), 161–166 (1974)MATHGoogle Scholar
  6. 6.
    Hu, Q., Wu, J.: Second order differentiability with respect to parameters of solutions of differential equations with state-dependent delay. Front. Math. China (accepted) (2010)Google Scholar
  7. 7.
    Hu, Q., Wu, J.: Global Hopf bifurcation of differential equations with state-dependent delay. J. Differ. Equ. (accepted) (2010)Google Scholar
  8. 8.
    Hu, Q., Wu, J., Zou, X.: Global continua of slowly oscillating periodic solutions of state-dependent delay differential equations (preprint) (2009)Google Scholar
  9. 9.
    Lau P.J.: Bounds for the lengths and periods of closed orbits of two-dimensional autonomous systems of differential equations. J. Differ. Equ. 3, 330–342 (1967)MATHCrossRefGoogle Scholar
  10. 10.
    Munkres J.: Topology, 2nd edn. Prentice Hall, Upper Saddle River, NJ (1975)MATHGoogle Scholar
  11. 11.
    Nussbaum R.D.: Circulant matrices and differential-delay equations. J. Differ. Equ. 60, 201–217 (1985)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nussbaum R.D.: The range of periods of periodic solutions of x′(t) = −α f(x(t−1)). J. Math. Anal. Appl. 58, 280–292 (1977)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Rudin, W.: Functional Analysis. McGraw-Hill Science (1991)Google Scholar
  14. 14.
    Smith, R.A.: Period bound for autonomous Liénard oscillations. Quart. Appl. Math. 27, 516–522 (1969/1970)Google Scholar
  15. 15.
    Schwartzman S.: Asymptotical cycles. Ann. math. 66(2), 270–284 (1957)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Wei J., Li Michael Y.: Global existence of periodic solutions in a tri-neuron network model with delays. Phys. D 198(1–2), 106–119 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wu J.: Global continua of periodic solutions to some differential equations of neutral type. Tôhoku Math J. 45(1), 67–88 (1993)MATHCrossRefGoogle Scholar
  18. 18.
    Wu J., Xia H.: Rotating waves in neutral partial functional-differential equations. J. Dynam. Differ. Equ. 11(2), 209–238 (1999)MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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