The Influence of Time-delay for the Bifurcation of the Fractional Neuron Network

Abstract

Bifurcation is a special phenomenon of a nonlinear system, which will cause the system to lose its original structural stability. Most of the existing researches are aimed at the neuron network systems with integer order, but few research on fractional order neuron system. Especially, it is more difficult to study the case of self-leakage delay at the same time. In this paper, the effect of delay on the bifurcation point of fractional neural network is studied, not only considering the delay between neurons, but also considering the leakage delay of each neuron's self-feedback. Considering the case of self-feedback information with both time delay and non-time delay, the complexity and difficulty of problem analysis are increased. And by changing the order of the system, the influence of the order of the system on the bifurcation point is analyzed. This paper uses fractional Laplace transform and variable substitution method to analyze the characteristic equation and determine the critical value of bifurcation. The results show that the proposed method can effectively calculate the bifurcation point in the system. When the time delay does not exceed this point, the system is stable, and vice versa. The effects of time delay and fractional order on the dynamical behavior of neuron network were investigated. Different system order and number of system neurons with leakage time delay have different effects on bifurcation. The stability and bifurcation of the neuron system are influenced by the interaction of system order, delay size and neuron parameters.