Resonant hopf-hopf interactions in delay differential equations

Journal of Dynamics and Differential Equations

Volumn 10, Issue 2, 1998, Pages 327-346

  Campbell, Sue Ann ^a,b   LeBlanc, Victor G ^c  

^a UNIVERSITY OF WATERLOO   (Canada)

^b MCGILL UNIVERSITY   (Canada)

^c UNIVERSITY OF OTTAWA   (Canada)

Author keywords

Delay differential equations; Normal forms; Resonant double hopf bifurcation

Indexed keywords

EID: 54649083001     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1023/A:1022622101608     Document Type: Article

References (18)

1. an der Heiden, U. (1979). Delays in physiological systems. J. Math. Biol. 8, 345-364.
2. Arnold, V. I. (1983). Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York.
3. Boe, E., and Chang, H.-C. (1989). Dynamics of delayed systems under feedback control. Chem. Eng. Sci. 44, 1281-1294.
4. Campbell, S., and Bélair, J. (1995). Resonant codimension two bifurcation in the harmonic oscillator with delayed forcing. Preprint.
5. Campbell, S., Bélair, J., Ohira, T., and Milton, J. (1995). Limit cycles, tori, and complex dynamics in a second-order differential equations with delayed negative feedback. J. Dynam. Diff. Eq. 7, 213-236.
6. Campbell, S., Bélair, J., Ohira, T., and Milton, J. (1995). Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback. Chaos 5(4), 640-645.
7. Chung, L., Reinhorn, A., and Soong, T. (1988). Experiments on active control of seismic structures. J. Eng. Mech. 114, 241-256.
8. Cooke, K. L., and Grossman, A. (1982). Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86, 592-627.
9. Diekmann, O., van Gils, S., Lunel, S. V., and Walther, H.-O. (1995). Delay Equations, Springer-Verlag, New York.
10. Golubitsky, M., and Schaeffer, D. G. (1985). Singularities and Groups in Bifurcation Theory, Vol. 1, Springer-Verlag, New York.
11. Hale, J., and Lunel, S. V. (1993). Introduction to Functional Differential Equations, Springer Verlag, New York.
12. Langford, W. F., and Iooss, G. (1980). Interactions of the Hopf and pitchfork bifurcations. In Bifurcation Problems and Their Numerical Solutions (International Series on Numerical Mathematics 54), pp. 103-134.
13. LeBlanc, V. (1995). On the 1:2 Resonant Hopf Bifurcation, Ph.D. thesis, University of Waterloo, Waterloo.
14. LeBlanc, V., and Langford, W. (1996). Classification and unfoldings of 1:2 resonant Hopf bifurcation. Arch. Ration. Mech. Anal. 136, 305-357.
15. Milton, J., and Longtin, A. (1990). Evaluation of pupil constriction and dilation from cycling measurements. Vision Res. 30, 515-525.
16. Milton, J., and Ohira, T. (1993). Dynamics of neuromuscular control with delayed displacement-dependent feedback. In Differential-Delay Equations and Applications to Biology and Medicine (World Congress of Nonlinear Analysts).
17. Takens, F. (1973). Applications of Global Analysis I, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, Utrecht.
18. Wischert, W., Wunderlin, A., Pelster, A., Olivier, M., and Groslambert, J. (1994). Delay-induced instabilities in nonlinear feedback systems. Phys. Rev. E 49, 203-219.