Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two time delays

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 11, Issue 8, 2001, Pages 2105-2121

  Liao, Xiaofeng ^b   Chen, Guanrong ^a  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b CHONGQING UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035622326     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127401003425     Document Type: Article

References (26)

1. Algaba, A., Friere, E., Gamero, E. & Rodriguez-Luis, A. J. [2000] "A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol-Duffing oscillator," Nonlin. Dyn. 22, 249-269.
2. Belair, J. & Campbell, S. A. [1994] "Stability and bifurcations of equilibria in a multiple-delayed differential equation," SIAM J. Appl. Math. 54, 1402-1424.
3. 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.
4. Buonomo, A. [1998] "The periodic solution of van der Pol's equation," SIAM J. Appl. Math. 59, 156-171.
5. Cooke, K. L. & Yorke, J. A. [1973] "Some equations modeling growth processes and gonorrhea epidemics," Math. Biosci. 16, 75-101.
6. Guckenheimer, J. & Holmes, P. J. [1983] Nonlinear Oscillations, Dynamical System, and Bifurcations of Vector Fields (Springer, Berlin).
7. Hale, J. & Lunel, S. V. [1993] Introduction to Functional Differential Equations (Springer, NY).
8. Haller, G. & Stepan, G. [1991] "Codimension two bifurcation in an approximate model for delayed robot control," in Bifurcation and Chaos: Analysis, Algorithms, Applications, eds. Seydel, R., Schneider, F. W., Kupper, T. & Troger, H. (Birkhauser, Basel), pp. 155-159.
9. Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. [1981] Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge).
10. Holmes, P. [1985] "Dynamics of a nonlinear oscillator with feedback control I: Local analysis," Trans. ASME J. Dyn. Syst. Measurement Contr. 107, 159-165.
11. Howroyd, T. D. & Russell, A. M. [1984] "Cournot oligopoly models with time lags," J. Math. Econ. 13, 97-103.
12. Iooss, G. & Joseph, D. D. [1989] Elementary Stability and Bifurcation Theory, 2nd edition (Springer-Verlag, NY).
13. Liao, X. F. & Yu, J. B. [1998] "Robust stability of interval Hopfield neural network with time delay," IEEE Trans. Neural Networks 9(5), 1042-1046.
14. Liao, X. F., Wu, Z. F. & Yu, J. B. [1999] "Stability switches, bifrucations and stability of periodic solutions for neural network with continuous time delay," IEEE Trans. Syst. Man Cybern. Part A 29(6), 692-696.
15. Liao, X. F., Wong, K. W., Leung, C. S. & Wu, Z. F. [2001] "Hopf bifurcaiton and chaos in a single delayed neuron equation with non-monotonic activation function," Chaos Solit. Fract. 12, 1535-1547.
16. MacDonald, N. [1979] "Cyclical neutropenia: models with two cell types and two time lags," in Biomathematics and Cell Kinetics, eds. Valleron, A. J. & MacDonald, P. D. M. (North Holland, Amsterdam), pp. 287-295.
17. MacDonald, N. [1980] "An activation-inhibition model of cyclic granulopoiesis in chronic granulocytic leukemia," Math. Biosci. 54, 61-70.
18. MacDonald, N. [1989] Biological Delay Systems: Linear Stability Theory (Cambridge University Press, NY).
19. Marcus, C. M. & Westervelt, R. M. [1989] "Stability of analog neural network with delay," Phys. Rev. A39, 347-359.
20. Marriott, C., Vallee, R. & Delisle, C. [1989] "Analysis of a first order differential-delay equation containing two delays," Phys. Rev. A40, 3420-3428.
21. Marsden, J. E. & McCracken, M. [1976] The Hopf Bifurcation and Its Applications (Springer-Verlag, NY).
22. Mizuno, M. & Ikeda, K. [1989] "An unstable mode selection rule: Frustrated optical instability due to two competing boundary conditions," Physica D36, 327-342.
23. Moiola, J. L. & Chen, G. [1996] Hopf Bifurcation Analysis: A Frequency Domain Approach (World Scientific, Singapore).
24. Murakaimi, K. [1999] "Bifurcated periodic solutions for delayed van der Pol equation," Neural Parall. Scient. Comput. 7, 1-16.
25. Schuster, G. H. [1984] Deterministic Chaos (Physik Verlag, Weunheim).
26. Stepan, G. [1989] Retarded Dynamical System: Stability and Characteristic Functions (Academic Press, NY).