Hopf and resonant codimension two bifurcation in van der Pol equation with two time delays

Chaos, Solitons and Fractals

Volumn 23, Issue 3, 2005, Pages 857-871

  Liao, Xiaofeng ^a  

^a CHONGQING UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; DAMPING; OSCILLATIONS; RESONANCE; SET THEORY; SYSTEM STABILITY;

HOPF BIFURCATIONS; NONLINEAR DAMPING; TIME DELAYS; VAN DER POL EQUATIONS;

BIFURCATION (MATHEMATICS);

EID: 4644254239     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.chaos.2004.05.048     Document Type: Article

References (25)

1. Guckenheimer J., Holmes P.J. Nonlinear oscillations, dynamical system, and bifurcations of vector fields. 1983;Springer, Berlin.
2. Schuster H.G. Deterministic chaos. 1984;Physik Verlag, Weunheim.
3. Venkatasubramanian, Vaithianathan Singularity induced bifurcation and the van der Pol oscillator. IEEE Trans. CAS. 41:1994;765-769.
4. Frey D.R. A Class of relaxation algorithms for finding the periodic steady-state solution. IEEE Trans. CAS. 45(6):1998;659-663.
5. Buonomo A. The periodic solution of Van der Pol's equation. SIAM J. Appl. Math. 59:1998;156-171.
6. Murakaimi K. Bifurcated periodic solutions for delayed van der pol equation. Neural Parallel Sci. Comput. 7:1999;1-16.
7. Cooke K.L., Yorke J.A. Some equations modeling growth processes and Gonorrhea epidemics. Math. Biosci. 16:1973;75-101.
8. MacDonald N. Biological delay systems: linear stability theory. 1989;Cambridge University Press, New York.
9. Belair J., Campbell S.A. Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54:1994;1402-1424.
10. MacDonald N. An activation-inhibition model of cyclic granulopoiesis in chronic granulocytic leukemia. Math. Biosci. 54:1980;61-70.
11. MacDonald N. Cyclical neutropenia: models with two cell types and two time lags. Valleron A.J., MacDonald P.D.M. Biomathematics and cell kinetics. 1979;287-295 North Holland, Amsterdam.
12. Beuter A., Belair J., Labrie C. Feedback and delays in neurological diseases: a modeling study using dynamical systems. Bull. Math. Biol. 55:1993;525-541.
13. Mizuno M., Ikeda K. An unstable mode selection rule: Frustrated optical instability due to two competing boundary conditions. Physica D. 36:1989;327-342.
14. Marriott C., Vallee R., Delisle C. Analysis of a first order differential-delay equation containing two delays. Phys. Rev. A. 40:1989;3420-3428.
15. Haller G., Stepan G. Codimension two bifurcation in an approximate model for delayed robot control. Seydel R., Schneider F.W., Kupper T., Troger H. Bifurcation and chaos: analysis, algorithms, applications. 1991;155-159 Birkhauser, Basel.
16. Howroyd T.D., Russell A.M. Cournot oligopoly models with time lags. J. Math. Econ. 13:1984;97-103.
17. Marcus C.M., Westervelt R.M. Stability of analog neural network with delay. Phys. Rev. A. 39:1989;347-359.
18. Liao X., Yu J. Robust stability of interval Hopfield neural network with time delay. IEEE Trans. Neural Networks. 9(5):1998;1042-1046.
19. Liao X., Wu Z., Yu J. Stability Switches, Bifrucations and stability of periodic solutions for neural network with continuous time delay. IEEE Trans. Syst. Man Cybernet., Part A. 29(6):1999;692-696.
20. Liao X., Wong K.-W., Leung C.-S., Wu Z. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function. Chaos, Solitons & Fractals. 21:2001;1535-1547.
21. Stepan G. Retarded dynamical system: stability and characteristic functions. Pitman Research Notes in Mathematics Series. vol. 201:1989;Academic Press, New York.
22. Marsden J.E., McCracken M. The Hopf bifurcation and its applications. Appl. Math. Sci. vol. 19:1976;Springer-Verlag.
23. Hassard B.D., Kazarinoff N.D., Wan Y.H. Theory and applications of Hopf bifurcation. 1981;Cambridge University Press, Cambridge.
24. Hale J., Verduyn Lunel S. Introduction to functional differential equations. 1993;Springer, New York.
25. Iooss G., Joseph D.D. Elementary stability and bifurcation theory. 2nd ed. 1989;Springer-Verlag, New York.