Hopf bifurcation of time-delay lienard equations

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 9, Issue 5, 1999, Pages 939-951

  Xu, Jian ^a   Lu, Qi Shao ^b  

^a HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^b BEIHANG UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033130661     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127499000675     Document Type: Article

References (28)

1. Alfredo, S. S. [1978] "Periodic solutions of the sunflower equation: ẍ+a/rẋ+(br) sin x(t-r) = 0," Quart. Appl. Math. 4, 468-478.
2. Baptistini, M. Z. [1996] "On the existence and global bifurcation of periodic solutions to planar differential delay equations," J. Diff. Eq. 127, 391-425.
3. Belair, J. & Champell, S. A. [1994] "Stability and bifurcations of equilibria in a multiple delay differential equation," SIAM J. Appl. Math. 54, 1402-1424.
4. Belair, J., Champell, S. A. & van den Driessche, P. [1996] "Frustration, stability and delay-induced oscillations in a neural network model," SIAM J. Appl. Math. 56, 245-255.
5. Beuter, A., Belair, J. & Labrie, C. [1993] "Feedback and delays in neurological diseases: A modeling study using dynamical systems," Bull. Math. Biol. 55, 525-541.
6. Chafee, N. [1971] "A bifurcation problem for a functional differential equation of finitely retarded," J. Math. Anal. Appl. 35, 312-348.
7. Cooke, K. L. & Grossman, Z. [1982] "Discrete delay, distributed delay and stability switches," J. Math. Anal. Appl. 86, 592-627.
8. Cooke, K. L. & van den Driessche, P. [1986] "On zeros of some transcendental equations," Funkcialaj Ekvacioj 29, 77-90.
9. Chow, S. & Mallet-Paret, J. [1977] "Integral averaging and bifurcations," J. Diff. Eq. 26, 112-159.
10. Chukwu, E. N. [1992] Stability and Time-Optimal Control of Hereditary Systems (Academic Press, Boston).
11. Faria, T. & Magalhaes, L. T. [1995] "Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation," J. Diff. Eq. 122, 181-200.
12. Freedman, H. I. & Kuang, Y. [1991] "Stability switches in linear scalar neutral delay equations," Funkcialaj Ekvacioj 34, 187-209.
13. Gricorieva, E., Kashckenko, S., Loiko, N. & Samson, A. [1992] "Nonlinear dynamics in a laser with a negative delayed feedback," Physica D59, 297-319.
14. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, NY).
15. Hairer, E. & Norsett, S. P. [1987] Solving Ordinary Differential Equations I (Springer-Verlag, Berlin).
16. Hale, J. K. [1977] Functional Differential Equations (Springer-Verlag, Berlin).
17. Hale, J. K. [1979] "Nonlinear oscillation in equations with delays," in Nonlinear Oscillations in Biology, ed. Hoppenstadt, K. (AMS, Providence, RI) pp. 157-185.
18. Hale, J. K. [1985] "Flows in center manifold for scalar functional differential equations," Proc. R. Soc. Edinburgh A101, 193-201.
19. Hale, J. K. & Ivanov, A. F. [1993] "On a high order differential delay equation," J. Math. Anal. Appl. 170, 505-514.
20. Hassard, B. D. [1997] "Counting roots of the characteristic equation for linear delay-differential systems," J. Diff. Eq. 136, 222-235.
21. Kolmanovskii, V. B. & Nosov, V. R. [1986] Stability of Functional Equations (Academic Press, San Diego).
22. Kuang, Y. [1993] Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston).
23. Kulenovic, M. R. S. & Ladas, G. [1988] "Oscillations of the sunflower equation," Quart. Appl. Math. 46, 23-28.
24. Mallet-Paret, J. & Sell, G. R. [1996] "Systems of differential delay equations: Floquet multiplies and discrete Lyapunov functions," J. Diff. Eq. 125, 385-440.
25. Nayfeh, A. H. [1993] Method of Normal Forms (John Wiley, NY).
26. Pyragas, K. [1992] "Continuous control of chaos by self-controlling feedback," Phys. Lett. A170, 421-428.
27. Wei, J. J. [1996] "Hopf bifurcation of sunflower equation," Acta Math. Appl. Sinica 19, 73-79.
28. Xu, J., Lu, Q. S. & Huang, K. L. [1997] "Stability and Hopf bifurcation in a van del Pol equation with time-delay," in Modern Mechanics and Science Advances - Conf. Fortieth Anniversary of Mechanics Society of China, Vol. 2, ed. Zhuang, F. G. (Tsinghua University Press, Beijing), pp. 1551-1554.