The nonresonant double Hopf bifurcation in delayed neural network

Abstract

A two-neuron network with self/neighbour-delayed-connections has been investigated. The self-connections are proposed by excitatory synapses and the neighbour ones by excitatory and inhibitory synapses. It is found that the excitatory self-connections can lead to non-resonant double Hopf bifurcations, for example, 1:√2 double Hopf bifurcation, since the double Hopf bifurcation disappears without self-delayed connections. Various dynamic behaviours are classified in the neighbourhood of the double Hopf bifurcation point, and the bifurcating solutions are computed in an approximate form by the centre manifold reduction. It follows from the approximate solutions that the network of two identical neurons with neighbour-connection cannot be synchronized completely. However, generalized synchronization can occur in the network under consideration. The time delay can also lead to the disappearance of periodic motion. These results may have some potential applications, such as controlling associative memory and designing neural networks.

Keywords:

• Bifurcation computation
• Delayed neural network
• Delayed differential equation
• Double Hopf bifurcation
• Synchronization

AMS Subject Classifications :

• 92B20
• 34K17
• 70K50
• 37L10

Acknowledgements

This work is supported by National Nature and Science Foundation of China under grant No. 10472083 and No. 10532050. The senior author acknowledges the support of the Outstanding Young Scientists Foundation under grant No. 10625211. The constructive comments by the anonymous reviewers are gratefully acknowledged.