Journal of Dynamics and Differential Equations

, Volume 7, Issue 1, pp 213–236 | Cite as

Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback

  • Sue Ann Campbell
  • Jacques Bélair
  • Toru Ohira
  • John Milton
Article

Abstract

We analyze a second-order, nonlinear delay-differential equation with negative feedback. The characteristic equation for the linear stability of the equilibrium is completely solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The bifurcations occurring as the linear stability is lost are investigated by the construction of a center manifold: The nature of Hopf bifurcations and more degenerate, higher-codimension bifurcations are explicitly determined.

Key words

Time delay negative feedback Hopf bifurcations center manifold invariant tori 

AMS(MOS) classification numbers

34K15 54F36 93D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bélair, J., and Campbell, S. A. (1994). Stability and bifurcations of equilibria in a multiple-delayed differential equation.SIAM J. Appl. Math. 54, 1402–1424.Google Scholar
  2. 2.
    Berger, B. S., Rokni, M., and Minis, I. (1993). Complex dynamics in metal cutting.Q. Appl. Math. 51, 601–612.Google Scholar
  3. 3.
    Beuter, A., Bélair, J., and Labrie, C. (1993). Feedback and delays in neurological diseases: a modeling study using dynamical systems.Bull. Math. Biol. 55, 525–541.Google Scholar
  4. 4.
    Bhatt, S. J., and Hsu, C. S. (1966). Stability criteria for second order dynamical systems with time lag.J. Appl. Mech. 33, 113–118.Google Scholar
  5. 5.
    Boe, E., and Chang, H.-C. (1989). Dynamics of delayed systems under feedback control.Chem. Eng. Sci. 44, 1281–1294.Google Scholar
  6. 6.
    Boe, E., and Chang, H.-C. (1991). Transition to chaos from a two-torus in a delayed feedback system.Int. J. Bifurc. Chaos 1, 67–82.Google Scholar
  7. 7.
    Boese, F. G. (1989). The stability chart for the linearized Cushing equation with a discrete delay and with gamma-distributed delays.J. Math. Anal. Appl. 140, 510–536.Google Scholar
  8. 8.
    Boese, F. G., and van den Driessche, P. (1994). Stability with respect to the delay in a class of differential-delay equations.Can. Appl. Math. Q. 2, 151–175.Google Scholar
  9. 9.
    Campbell, S. A., and Bélair, J. (1995). Analytical and symbolically-assisted investigation of Hopf bifurcations in delay-differential equations,Can. Appl. Math. Q. (in press).Google Scholar
  10. 10.
    Campbell, S. A., Bélair, J., Ohira, T., and Milton, J. (1994). Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback, (submitted for publication).Google Scholar
  11. 11.
    Chuma, J., and van den Driessche, P. (1980). A general second-order transcendental equation.Appl. Math. Notes 5, 85–96.Google Scholar
  12. 12.
    Cooke, K. L., and Grossman, Z. (1982). Discrete delay, distributed delay and stability switches,J. Math. Anal. Appl. 86, 592–627.Google Scholar
  13. 13.
    Gorelik, G. (1939). To the theory of feedback with delay.J. Tech. Phys. 9, 450–454.Google Scholar
  14. 14.
    Guckenheimer, J., and Holmes, P. J. (1983).Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.Google Scholar
  15. 15.
    Hale, J. K., and Verduyn Lunel, S. M. (1993).Introduction to Functional Differential Equations, Springer-Verlag, New York.Google Scholar
  16. 16.
    Hayes, N. (1950). Roots of the transcendental equation associated with a certain difference-differential equations.J. London Math. Soc. 25, 226–232.Google Scholar
  17. 17.
    an der Heiden, U. (1979). Delays in physiological systems.J. Math. Biol. 8, 345–364.Google Scholar
  18. 18.
    an der Heiden, U. (1979). Periodic solutions of a nonlinear second order differential equation with delay,J. Math. Anal. Appl. 70, 599–609.Google Scholar
  19. 19.
    an der Heiden, U., Longtin, A., Mackey, M. C., Milton, J. G., and Scholl, R. (1990). Oscillatory modes in a nonlinear second-order differential equation with delay.J. Dyn. Diff. Eq. 2, 423–449.Google Scholar
  20. 20.
    Longtin, A., and Milton, J. (1989). Modelling autonomous oscillations in the human pupil light reflex using delay-differential equations.Bull. Math. Biol. 51, 605–624.Google Scholar
  21. 21.
    MacDonald, N. (1989).Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge.Google Scholar
  22. 22.
    Marsden, J. E., and McCracken, M. (1976).The Hopf Bifurcation and Its Applications, Springer-Verlag, New York.Google Scholar
  23. 23.
    Milton, J. G., and Longtin, A. (1990). Evaluation of pupil constriction and dilation from cycling measurements.Vision Res. 30, 515–525.Google Scholar
  24. 24.
    Milton, J. G., and Ohira, T. (1993). Dynamics of neuro-muscular control with delayed displacement-dependent feedback.Proc. 1st World Congr. Nonl. Anal. (in press).Google Scholar
  25. 25.
    Reichardt, W., and Poggio, T. (1976). Visual control of orientation behavior of the fly.Q. Rev. Biophys. 9, 311–438.Google Scholar
  26. 26.
    Stépán, G. (1989).Retarded Dynamical Systems, Vol. 210. Pitman Research Notes in Mathematics, Longman Group, Essex.Google Scholar
  27. 27.
    Takens, F. (1974). Singularities of vector fields.Publ. Math. l'IHES 43, 47–100.Google Scholar
  28. 28.
    Vallée, R., Dubois, M., Coté, M., and Delisle, C. (1987). Second-order differential-delay equation to describe a hybrid bistable device.Phys. Rev. A 36, 1327–1332.Google Scholar
  29. 29.
    Waterloo Maple Software (1990).Maple V, University of Waterloo, Waterloo, Canada.Google Scholar
  30. 30.
    Wu, D. W., and Liu, C. R. (1985). An analytical model of cutting dynamics, Parts 1 and 2.ASMLE J. Eng. Indust. 17, 107–111.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Sue Ann Campbell
    • 1
    • 2
    • 3
  • Jacques Bélair
    • 1
    • 3
    • 6
  • Toru Ohira
    • 4
  • John Milton
    • 1
    • 5
  1. 1.Centre for Nonlinear Dynamics in Physiology and MedicineMcGill UniversityMontréalCanada
  2. 2.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  3. 3.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada
  4. 4.Department of PhysicsThe University of ChicagoChicago
  5. 5.Department of NeurologyThe University of ChicagoChicago
  6. 6.Département de mathématiques et de statistiqueUniversité de MontréalMontréalCanada

Personalised recommendations