Abstract
This paper is concerned with the stability and Hopf bifurcation of fractional-order neural networks with discrete and distributed delays. The novelty of this paper is to take into account the discrete time delay and the distributed time delay for fractional-order systems. By introducing two virtual neurons to the original network, a new four-neuron network only involving discrete delays is formed. The sum of discrete delays is adopted as the bifurcation parameter to demonstrate the existence of Hopf bifurcation. It is found that the critical value of bifurcation can be effectively manipulated by choosing appropriate system parameters and order. Finally, numerical simulations are executed to substantiate the theoretical results and describe the relationships between the parameters and the onset of bifurcation.
Highlights
As is known to all, the neuronal system is a complex nonlinear dynamic system
Xu et al [39] considered a double-neuron fractional-order neural networks (FNNs) with two discrete delays, discussed four possible cases with delays, and revealed the effects of different time delays on the stability of networks
(2) there have been some results on the dynamics of stability and bifurcation for delayed fractional-order neural networks, only discrete delays are considered
Summary
As is known to all, the neuronal system is a complex nonlinear dynamic system. And the neuron is considered the basic processing unit that has simplicity and simulation. Several studies have developed low dimensional systems into high dimensional ones concerning merely discrete delays [16], [17] These works only dealt with integer-order models of neural networks. Xu et al [39] considered a double-neuron FNNs with two discrete delays, discussed four possible cases with delays, and revealed the effects of different time delays on the stability of networks. (2) there have been some results on the dynamics of stability and bifurcation for delayed fractional-order neural networks, only discrete delays are considered. If distributed delays are added, the imbalance of delays should be discussed in transmitting information In this paper, both discrete delays and distributed delays are taken into account at the same time in fractional-order neural networks.
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