A fractional-order four-layer neural network involving mixed time delays: Hopf bifurcation analysis

As is known to us that appropriately increasing the number of layers in a neural network can enhance the adaptability of the model to complex tasks and thus has important applications in some fields such as medical diagnosis, deep learning, image recognition. Moreover, the bidirectional transmission of information is very common in ecosystem and other real-world systems. Consequently, a novel fractional-order four-layer delayed neural network (FOFLDNN) with bidirectional information interflow is proposed in this paper. First, the stability and Hopf bifurcation are investigated for the proposed FOFLDNN without leakage delay by selecting the sum of the involved transmission delay and feedback delay as the bifurcation parameter. Then leakage delay is further taken into account due to its inevitable existence in the non-instantaneous self-decay process of a neuron during the performing of hardware, and some leakage delay dependent Hopf bifurcation criteria are given. The discriminate criteria of stability for fractional-order dynamical systems, Hopf bifurcation theory as well as Coates’ flow-graph method are effectively applied in the Hopf bifurcation analysis, and the critical bifurcation points are determined with the help of maple or matlab toolbox. Finally, the application and validity of our presented theoretical results are illustrated through a numerical example, from which it is discovered that the leakage delay have greater impact on the occurrence of Hopf bifurcation for systems with smaller order of the fractional derivative.