Dynamics and double Hopf bifurcations of the rosehindmarsh model with time delay

International Journal of Bifurcation and Chaos

Volumn 19, Issue 11, 2009, Pages 3733-3751

  Ma, Suqi ^a   Feng, Zhaosheng ^b   Lu, Qishais ^c  

^a CHINA AGRICULTURAL UNIVERSITY   (China)

^b UNIVERSITY OF TEXAS PAN AMERICAN   (United States)

^c BEIHANG UNIVERSITY   (China)

Author keywords

Autonomous system; Center manifold method; Elliptic function; Hopf bifurcation; Normal form; RoseHindmarsh model

Indexed keywords

BIFURCATION (MATHEMATICS); TIME DELAY; TIMING CIRCUITS;

ASYMPTOTICALLY STABLE; AUTONOMOUS SYSTEMS; CENTER MANIFOLD; CHARACTERISTIC EQUATION; DOUBLE HOPF BIFURCATION; ELLIPTIC FUNCTIONS; NORMAL FORM; UNIVERSAL UNFOLDING;

HOPF BIFURCATION;

EID: 77649109705     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127409025080     Document Type: Article

References (22)

1. Buono, P. L. & Bélair, J. [2003] "Restrictions and unfolding of double Hopf bifurcation in functional differential equations," J. Diff. Eqs. 189, 234-266.
2. Campell, S. A. & Bélair, J. [1995a] "Analytical and symbolically-assisted investigation of Hopf bifurcations in delay-differential equations," Canad. Appl. Math. Quart. 3, 137-154.
3. Campell, S. A., Bélair, J., Ohira, T. & Milton, J. [1995b] "Limit cycles, tori, and complex dynamics in a second-order differential equations with delayed negative feedback," J. Dyn. Diff. Eqs. 7, 213-236.
4. Elphick, C., Tiraopegui, E., Brachet, M. E., Coullet, P. & Iooss, G. [1987] "A simple global characterization for normal forms of singular vector fields," Phys. D 29, 95-127.
5. Grebogi, C., Ott, E., Romeiras, F. & Yorke, J. A. [1986] "Critical exponent of chaotic transients in nonlinear dynamical systems," Phys. Rev. Lett. 57, 1284-1287.
6. Grebogi, C., Ott, E., Romeiras, F. & Yorke, J. A. [1987] "Critical exponents for crisis-induced intermittency," Phys. Rev. A 36, 5365-5380.
7. Guckenheimer, J. M. & Holmes, P. J. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, NY).
8. Hindmarsh, J. L. & Rose, R. M. [1982] "A model of the nerve impulse using two first-order differential equations," Nature 296, 162-164.
9. Hindmarsh, J. L. & Rose, R. M. [1984] "A model of neuronal bursting using three coupled first-order differential equations," Proc. Roy. Soc. Lond. B 221, 87-102.
10. Holden, A. V. & Fan, Y. S. [1992a] "From simple to simple bursting oscillatory behavior via chaos in the Rose-Hindmarsh model for neuronal activity," Chaos Solit. Fract. 2, 221-236.
11. Holden, A. V. & Fan, Y. S. [1992b] "From simple to complex oscillatory behavior via intermittent chaos in the Rose-Hindmarsh model for neuronal activity," Chaos Solit. Fract. 2, 346-369.
12. Ma, S. Q., Lu, Q. S. & Hogan, S. J. [2007] "Double Hopf bifurcation for Stuart-Landau system with nonlinear delay feedback and delay dependant parameters," Adv. Compl. Syst. 10, 1-27.
13. Ma, S. Q., Feng, Z. & Lu, Q. S. [2008a] "A two-patch ecological system with nonlinear transfer rate and noise effect," Dyn. PDEs 5, 281-298.
14. Ma, S. Q., Lu, Q. S. & Feng, Z. [2008b] "Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control," J. Math. Anal. Appl. 338, 993-1007.
15. Pomeau, Y. & Manneville, P. "Intermittent transition to turbulence in dissipative dynamic system," Commun. Math. Phys. 74, 189-197.
16. Shi, X., Lu, Q. S. & Chen, G. [2007] "Anti-phase synchronization of inhibitorily coupled neurons," Int. J. Bifurcation and Chaos 17, 4355-4364.
17. Wang, Z. & Hu, H. Y. [1999] "Delay independent stability of retarded dynamic systems of multiple degrees of freedom," J. Sound Vibr. 226, 57-81.
18. Wang, Z. & Hu, H. Y. [2000] "Stability switches of timedelayed dynamic systems with unknown parameters," J. Sound Vibr. 233, 215-233.
19. Xu, J. & Lu, Q. S. [1999] "Hopf bifurcation of time-delay Liénard equations," Int. J. Bifurcation and Chaos 9, 939-951.
20. Yang, L., Hou, X. R. & Zeng, Z. B. [1996a] "A complete discrimination system for polynomial," Sci. in China (Series E) 26, 424-441.
21. Yang, L., Hou, X. R. & Zeng, Z. B. [1996b] Nonlinear Algebraic Equation Systems and Automated Theory (in Chinese) (Shanghai Scientific and Technological Education Press, Shanghai).
22. Zhao, X. Q. & Zou, X. [2001] "Threshold dynamics in a delayed SIS epidemic model," J. Math. Anal. Appl. 257, 282-291.