Dynamical Bifurcations in a Fractional-Order Neural Network with Nonidentical Communication Delays

The article delves into the complex dynamics of fractional-order neural networks (FONNs) with nonidentical communication delays, a topic of growing interest due to the enhanced dynamics these systems exhibit compared to classical integer-order neural networks. The study begins by highlighting the significant progress made in the applications of neural networks across various fields, such as optimization, control, and memory systems. The authors then introduce the concept of fractional-order calculus, which allows for more flexible and rich dynamics in neural networks. The main focus is on the bifurcations that occur due to time delays in these systems, a phenomenon that is crucial for understanding the stability and behavior of neural networks. The authors present a general fractional-order four-neuron structured neural network model with two delays and analyze the bifurcations that arise from these delays. They derive conditions for stability and Hopf bifurcations, providing a theoretical framework that can be extended to n-dimensional FONNs with nonidentical communication delays. The results are validated through numerical simulations, showcasing the advantages of fractional-order models in improving stability and postponing the onset of bifurcations. The article concludes by suggesting future research directions, including the investigation of Hopf bifurcations in general n-dimensional systems and the handling of multiple leakage delays in fractional-order neural networks.