Global existence of periodic solutions in a tri-neuron network model with delays

Physica D: Nonlinear Phenomena

Volumn 198, Issue 1-2, 2004, Pages 106-119

  Wei, Junjie ^a   Li, Michael Y ^b  

^a HARBIN INSTITUTE OF TECHNOLOGY   (China)

^b UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Delay systems; Hopf bifurcations; Neural networks; Periodic solutions.

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTER SIMULATION; DELAY CONTROL SYSTEMS; ORDINARY DIFFERENTIAL EQUATIONS; THEOREM PROVING; TOPOLOGY;

DELAY SYSTEMS; HOPF BIFURCATION; PERIODIC SOLUTIONS;

NEURAL NETWORKS;

EID: 6444228136     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2004.08.023     Document Type: Article

References (21)

1. Y. Chen, J. Wu, Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network, Physica D 134 (1999) 185-199.
2. Y. Chen, J. Wu, Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential Integral Equations 14 (2001) 1181-1236.
3. Y. Chen, J. Wu, Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. Math. Anal. Appl. 259 (2001) 188-208.
4. T. Faria, On a planar system modelling a neuron network with memory, J. Differential Equations 168 (2000) 129-149.
5. F. Giamnakopoulos, A. Zapp, Bifurcation in a planar system of differential delay equations modelling neural activity, Physica D 159 (2001) 215-232.
6. L. Olien, J. Bélair, Bifurcation stability and monotonicity properties of a delayed neural network model, Physica D 102 (1997) 349-363.
7. K. Gopalsamy, I. Leung, Delay induced periodicity in a neural network of excitation and inhibition, Physica D 89 (1996) 395-426.
8. N.C. Majee, A.B. Roy, Temporal dynamics of a two-neuron continuous network model with time delay, Appl. Math. Modeling 21 (1997) 673-679.
9. X. Liao, K. Wong, Z. Wu, Bifurcation analysis on a two-neuron system with distribution delays, Physica D 149 (2001) 123-141.
10. J. Wei, S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D 130 (1999) 255-272.
11. P. Baldi, A. Atiya, How delays affect neural dynamics and learning, IEEE. Tran. NN 5 (1994) 610-621.
12. J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Am. Math. Soc. 350 (1998) 4799-4838.
13. S.A. Campbell, Stability and bifurcation of a simple neural network with multiple time delays, Fields Inst. Commun. 21 (1999) 65-79.
14. S.A. Campbell, S. Ruan, J. Wei, Qualitative analysis of a neural network model with multiple time delays, Int. J. Bifu. Chaos 9 (1999) 1585-1595.
15. J. Wei, M. Velarde, Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos (2004), in press.
16. S. Ruan, J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretions, IMA. J. Math. Appl. Medi. Biol. 18 (2001) 41-52.
17. M.Y. Li, J. Muldowney, On Bendixson's criterion, J. Differential Equations 106 (1994) 27-39.
18. E. Beretta, F. Solimano, Y. Tacheuchi, Negative criteria for the existence of periodic solutions in a class of delay- differential equations, Nonlinear Anal. 50 (2002) 941-966.
19. M. Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech. Math. J. 99 (1974) 392-402.
20. J.S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990) 857-871.
21. 1-degree and global Hopf bifurcation theory of functional differential equations, J. Differential Equations 98 (1992) 277-298.