Global Existence of Periodic Solutions in a Delayed Tri–neuron Network
In this paper, a tri–neuron network model with time
delay is investigated. By using the Bendixson-s criterion for high–
dimensional ordinary differential equations and global Hopf bifurcation
theory for functional differential equations, sufficient conditions
for existence of periodic solutions when the time delay is sufficiently
large are established.
[1] Xiaoming Liu, Xiaofeng Liao. Necessary and sufficient conditions for
Hopf bifurcation in tri-neuron equation with a delay. Chaos, Solitons
and Fractals, 40(2009), 481-490.
[2] P D Gupta, N C Majee, A B Roy. Stability, bifurcation and global
existence of a Hopf-bifurcating periodic solution for a class of three-
neuron delayed network models. Nonlinear Analysis, 67(2007), 2934-
2954.
[3] Bingji Xu, Xinzhi Liu, Xiaoxin Liao. Global asymptotic stability of high-
order Hopfield type neural networks with time delays. Computers and
Mathematics with Applications, 45(2003), 1729-1737.
[4] Qiang Zhang, Xiaopeng Wei, Jin Xu. Global asymptotic stability of
Hopfield neural networks with transmission delays. Physics Letters A,
318(2003), 399-405.
[5] Bingwen Liu, Lihong Huang. Existence and exponential stability of
periodic solutions for cellular neural networks with time-varying delays.
Physics Letters A, 349(2006), 474-483.
[6] Bingwen Liu, Lihong Huang. Existence and exponential stability of
almost periodic solutions for cellular neural networks with time-varying
delays. Physics Letters A, 341(2006), 135-144.
[7] Chunrui Zhang, Baodong Zheng. Hopf bifurcation in numerical approximation
of a n−dimension neural network model with multi-delays.
Chaos, Solitons and Fractals, 25(2005), 129-146.
[8] M Y Li, J Muldowney. On Bendixson-s criterion. Journal of Differential
Equations, 106(1993), 27-39.
[9] Jianhong Wu. Symmetric functional differential equations and neural
networks with memory. Transactions of the AMS, 350(1998), 4799-4838.
[1] Xiaoming Liu, Xiaofeng Liao. Necessary and sufficient conditions for
Hopf bifurcation in tri-neuron equation with a delay. Chaos, Solitons
and Fractals, 40(2009), 481-490.
[2] P D Gupta, N C Majee, A B Roy. Stability, bifurcation and global
existence of a Hopf-bifurcating periodic solution for a class of three-
neuron delayed network models. Nonlinear Analysis, 67(2007), 2934-
2954.
[3] Bingji Xu, Xinzhi Liu, Xiaoxin Liao. Global asymptotic stability of high-
order Hopfield type neural networks with time delays. Computers and
Mathematics with Applications, 45(2003), 1729-1737.
[4] Qiang Zhang, Xiaopeng Wei, Jin Xu. Global asymptotic stability of
Hopfield neural networks with transmission delays. Physics Letters A,
318(2003), 399-405.
[5] Bingwen Liu, Lihong Huang. Existence and exponential stability of
periodic solutions for cellular neural networks with time-varying delays.
Physics Letters A, 349(2006), 474-483.
[6] Bingwen Liu, Lihong Huang. Existence and exponential stability of
almost periodic solutions for cellular neural networks with time-varying
delays. Physics Letters A, 341(2006), 135-144.
[7] Chunrui Zhang, Baodong Zheng. Hopf bifurcation in numerical approximation
of a n−dimension neural network model with multi-delays.
Chaos, Solitons and Fractals, 25(2005), 129-146.
[8] M Y Li, J Muldowney. On Bendixson-s criterion. Journal of Differential
Equations, 106(1993), 27-39.
[9] Jianhong Wu. Symmetric functional differential equations and neural
networks with memory. Transactions of the AMS, 350(1998), 4799-4838.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:50563", author = "Kejun Zhuang and Zhaohui Wen", title = "Global Existence of Periodic Solutions in a Delayed Tri–neuron Network", abstract = "In this paper, a tri–neuron network model with time
delay is investigated. By using the Bendixson-s criterion for high–
dimensional ordinary differential equations and global Hopf bifurcation
theory for functional differential equations, sufficient conditions
for existence of periodic solutions when the time delay is sufficiently
large are established.", keywords = "Delay, global Hopf bifurcation, neural network, periodicsolutions.", volume = "5", number = "7", pages = "950-3", }