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Annali di Matematica Pura ed Applicata

, Volume 145, Issue 1, pp 33–128 | Cite as

Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation

  • John Mallet-Paret
  • Roger D. Nussbaum
Article

Summary

The singularly perturbed differential-delay equation
$$\varepsilon \dot x(t) = - x(t) + f(x(t - 1))$$
is studied. Existence of periodic solutions is shown using a global continuation technique based on degree theory. For small ɛ these solutions are proved to have a square-wave shape, and are related to periodic points of the mappingf:RR.Whenfis not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series.

Keywords

Asymptotic Behaviour Periodic Solution Fourier Series Periodic Point Degree Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    R.Bellman - K. L.Cooke,Differential-Difference Equations, Academic Press, 1963.Google Scholar
  2. [2]
    S. P. Blythe -R. M. Nisbet -W. S. C. Gurney,Instability and complex dynamic behaviour in population models with long time delays, Theor. Pop. Biol.,2 (1982), pp. 147–176.Google Scholar
  3. [3]
    S. A. Chapin -R. D. Nussbaum,Asymptotic estimates for the periods of periodic solutions of a differential-delay equation, Mich. Math. J.,31 (1984), pp. 215–229.Google Scholar
  4. [4]
    S.-N. Chow,Existence of periodic solutions of autonomous functional differential equations, J. Diff. Eq.,15 (1974), pp. 350–378.Google Scholar
  5. [5]
    S.-N. Chow -O. Diekmann -J. Mallet-Paret,Stability, multiplicity, and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation, Japan Jour. Appl. Math.,2 (1985), pp. 433–469.Google Scholar
  6. [6]
    S.-N.Chow - D.Green Jr.,Some results on singular delay-differential equations, Chaos, Fractals, and Dynamics, ed. by P. Fischer and W. R. Smith, Marcel Dekker, 1985, pp. 161–182.Google Scholar
  7. [7]
    S.-N. Chow -J. Mallet-Paret,Integral averaging and bifurcation, J. Diff. Eq.,26 (1977), pp. 112–159.Google Scholar
  8. [8]
    S.-N. Chow -J. Mallet-Paret,The Fuller index and global Hopf bifurcation, J. Diff. Eq.,29 (1978), pp. 66–85.Google Scholar
  9. [9]
    S.-N.Chow - J.Mallet-Paret,Singularly perturbed delay-differential equations, Coupled Nonlinear Oscillators, ed. by J. Chandra and A. C. Scott, North Holland Math. Studies,80 (1983), pp. 7–12.Google Scholar
  10. [10]
    P.Collet - J.-P.Eckmann,Iterated Maps on the Interval as Dynamical Systems, Birk-häuser, 1980.Google Scholar
  11. [11]
    K. L. Cooke,The condition of regular degeneration for singularly perturbed linear differential-difference equations, J. Diff. Eq.,1 (1965), pp. 39–94.Google Scholar
  12. [12]
    K. L. Cooke -K. R. Meyer,The condition of regular degeneration for singularly perturbed systems of linear differential-difference equations, J. Math. Anal. Appl.,14 (1966), pp. 83–106.Google Scholar
  13. [13]
    M. W. Derstine -H. M. Gibbs -F. A. Hopf -D. L. Kaplan,Bifurcation gap in a hybrid optically bistable system, Phys. Rev. A,26 (1982), pp. 3720–3722.Google Scholar
  14. [14]
    M. W. Derstine -H. M. Gibbs -F. A. Hopf -D. L. Kaplan,Alternate paths to chaos in optical bistability, Phys. Rev. A,27 (1983), pp. 3200–3208.Google Scholar
  15. [15]
    J. Dugundji,An extension of Tietse's theorem, Pacific J. Math.,1 (1951), pp. 353–367.Google Scholar
  16. [16]
    L. E.El'sgol'ts - S. B.Norkin,Introduction to the theory and application of differential equations with deviating arguments, Mathematics in Science and Engineering, Vol.105 Academic Press, 1973.Google Scholar
  17. [17]
    J. D. Farmer,Chaotic attractors of an infinite-dimensional dynamical system, Physica D,4 (1982), pp. 366–393.Google Scholar
  18. [18]
    M. Feigenbaum,Quantitative universality for a class of nonlinear transformations, J. Statist. Phys.,19 (1978), pp. 25–52.Google Scholar
  19. [19]
    M. Feigenbaum,The universal metric properties of nonlinear transformations, J. Statist. Phys.,21 (1979), pp. 669–706.Google Scholar
  20. [20]
    A. C. Fowler,An asymptotic analysis of the delayed logistic equation when the delay is large, IMA J. Appl. Math.,28 (1982), pp. 41–49.Google Scholar
  21. [21]
    H. M. Gibbs -F. A. Hopf -D. L. Kaplan -R. L. Shoemaker,Observation of chaos in optical bistability, Phys. Rev. Lett.,46 (1981), pp. 474–477.Google Scholar
  22. [22]
    L. Glass -M. C. Mackey,Pathological conditions resulting from instabilities in physiological control systems, Ann. New York Acad. Sci.,316 (1979), pp. 214–235.Google Scholar
  23. [23]
    W. S. C. Gurney -S. P. Blythe -R. M. Nisbet,Nicholson's blowflies revisited, Nature,287 (1980), pp. 17–21.Google Scholar
  24. [24]
    K. P. Hadeler -J. Tomiuk,Periodic solutions of difference-differential equations, Arch. Rat. Mech. Anal.,65 (1977), pp. 82–95.Google Scholar
  25. [25]
    J. K.Hale,Theory of Functional Differential Equations, Springer-Verlag, 1977.Google Scholar
  26. [26]
    U. an der Heiden -M. C. Mackey,The dynamics of production and destruction: anal- ytic insight into complex behaviour, J. Math. Biol.,16 (1982), pp. 75–101.Google Scholar
  27. [27]
    U. an derHeiden - M. C.Mackey - H.-O.Walther,Complex oscillations in a simple deterministic neuronal network, Lecture Notes in Appl. Math.,19, pp. 355–360; Amer. Math. Soc., Providence, 1981.Google Scholar
  28. [28]
    U. an der Heiden -H.-O. Walther,Existence of chaos in control systems with delayed feedback, J. Diff. Eq.,47 (1983), pp. 273–295.Google Scholar
  29. [29]
    F. A. Hopf -D. L. Kaplan -H. M. Gibbs -R. L. Shoemaker,Bifurcation to chaos in optical bistability, Phys. Rev. A,25 (1982), pp. 2172–2182.Google Scholar
  30. [30]
    F. C. Hoppensteadt,Mathematical theories of population: demographics, genetics and epidemics, Regional Conference Series in Applied Mathematics, Vol.120, SIAM, Philadelphia, 1975.Google Scholar
  31. [31]
    K. Ikeda,Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. Comm.,30 (1979), pp. 257–261.Google Scholar
  32. [32]
    K. Ikeda -H. Daido -O. Akimoto,Optical turbulence: chaotic behaviour of transmitted light from a ring cavity, Phys. Rev. Lett.,45 (1980), pp. 709–712.Google Scholar
  33. [33]
    K. Ikeda -K. Kondo -O. Akimoto,Successive higher-harmonic bifurcations in systems with delayed feedback, Phys. Rev. Lett.,49 (1982), pp. 1467–1470.Google Scholar
  34. [34]
    G. S. Jones,Periodic motions in Banach space and applications to functional differential equations, Contrib. Diff. Eq.,3 (1964), pp. 75–106.Google Scholar
  35. [35]
    J. L. Kaplan -J. A. Yorke,On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal.,6 (1975), pp. 268–282.Google Scholar
  36. [36]
    J. L. Kaplan -J. A. Yorke,On the nonlinear differential delay equation x′(t)==-f(x(t), x(t-l)), J. Diff. Eq.,23 (1977), pp. 293–314.Google Scholar
  37. [37]
    A. Lasota,Ergodic problems in biology, Astérisque,50 (1977), pp. 239–250.Google Scholar
  38. [38]
    M. C. Mackey,Periodic auto-immune hemolytic anemia: an induced dynamical disease, Bull. Math. Biol.,41 (1979), pp. 829–834.Google Scholar
  39. [39]
    M. C. Mackey -L. Glass,Oscillation and chaos in physiological control systems, Science,197 (1977), pp. 287–289.Google Scholar
  40. [40]
    M. C. Mackey -U. an der Heiden,Dynamical diseases and bifurcations: understanding functional disorders in physiological systems, Funkt. Biol. Med., no. 1,156 (1982), pp. 156–164.Google Scholar
  41. [41]
    J. Mallet-Paret,Morse decompositions and global continuation of periodic solutions for singularly perturbed delay equations, Systems of Nonlinear Partial Differential Equations, ed. by J. M. Ball, D. Reidel, Dordrecht, 1983, pp. 351–365.Google Scholar
  42. [42]
    J.Mallet-Paret,Morse decompositions for delay-differential equations, submitted for publication.Google Scholar
  43. [43]
    J. Mallet-Paret -R. D. Nussbaum,Global continuation and complicated trajectories for periodic solutions of a differential-delay equation, Proceedings of Symposia in Pure Mathematics, American Mathematical Society,45 (1986), Part 2, pp. 155–167.Google Scholar
  44. [44]
    J.Mallet-Paret - R. D.Nussbaum,A bifurcation gap for a singularly perturbed delay equation, Chaotic Dynamics and Fractals, ed. by M. F. Barnsley and S. G. Demko, Academic Press (1986), pp. 263–286.Google Scholar
  45. [45]
    J.Mallet-Paret - R. D.Nussbaum,A differential-delay equation arising in optics and physiology, in preparation.Google Scholar
  46. [46]
    M. Martelli -K. Schmitt -H. Smith,Periodic solutions of some nonlinear delay-differential equations, Jour. Math. Anal. Appl.,74 (1980), pp. 494–503.Google Scholar
  47. [47]
    R. D. Nussbaum,Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl.,101 (1974), pp. 263–306.Google Scholar
  48. [48]
    R. D. Nussbaum,Periodic solutions of some nonlinear autonomous functional differential equations II, J. Diff. Eq.,14 (1973), pp. 360–394.Google Scholar
  49. [49]
    R. D. Nussbaum,A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis,19 (1975), pp. 319–338.Google Scholar
  50. [50]
    R. D. Nussbaum,Periodic solutions of nonlinear autonomous functional differential equations, Springer Lecture Notes in Math.,730 (1979), pp. 283–325.Google Scholar
  51. [51]
    R. D. Nussbaum,Asymptotic analysis of some functional differential equations, Dynamical Systems II: Proceedings of a University of Florida International Symposium, ed. by A. R. Bednarek and L. Cesari, Academic Press, New York, 1982, pp. 277–301.Google Scholar
  52. [52]
    R. D.Nussbaum,Boundary layer phenomena for a differential-delay equation, in preparation.Google Scholar
  53. [53]
    R. D. Nussbaum,Circulant matrices and differential-delay equations, J. Diff. Eq.,60 (1985), pp. 201–217.Google Scholar
  54. [54]
    R.O'Malley, Jr.,Introduction to Singular Perturbations, Academic Press, 1974.Google Scholar
  55. [55]
    H. Peters,Comportement chaotique d'une équation différentielle retardée, C. R. Acad. Sci. Paris, Ser. A,290 (1980), pp. 1119–1122.Google Scholar
  56. [56]
    D. Saupe,Global bifurcation of periodic solutions to some autonomous differential delay equations, Appl. Math. Comput.,13 (1983), pp. 185–211.Google Scholar
  57. [57]
    D.Saupe,Accelerated PL-continuation methods and periodic solutions of parametrized differential-delay equations, Ph. D. dissertation (in German), Univ. Bremen, 1982.Google Scholar
  58. [58]
    H. -O. Walther,On instability, ω-limit sets and periodic solutions to nonlinear autonomous differential delay equations, Springer Lecture Notes in Math.,730 (1979), pp. 489–503.Google Scholar
  59. [59]
    H. -O. Walther,On density of slowly oscillating solutions of x(t)=−f(x(t−1)), J. Math. Anal. Appl.,79 (1981), pp. 127–140.Google Scholar
  60. [60]
    H. -O. Walther,Homoclinic solution and chaos in x(t)=f(x(t−1)), Nonlin. Anal. Theory Meth. Appl.,5 (1981), pp. 775–788.Google Scholar
  61. [61]
    M. Wazewska-Czyzewska -A. Lasota,Mathematical models of the red cell system, (in Polish), Matematyka Stosowana,6 (1976), pp. 25–40.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1985

Authors and Affiliations

  • John Mallet-Paret
    • 1
  • Roger D. Nussbaum
    • 2
  1. 1.Division of Applied Mathematics (Box F)Brown UniversityProvidence
  2. 2.Department of MathematicsRutgers UniversityNew Brunswick

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