Chaos in a two dimensional fractional discrete Hopfield neural network and its control

A novel two-dimensional fractional discrete Hopfield neural network is presented in this study, which is based on discrete fractional calculus. This network incorporates both constant and variable orders, and its behavior is examined using phase plots, time evolution, bifurcation, Lyapunov exponents, and complexity analysis. Compared to integer and constant fractional orders, the numerical simulations demonstrate that the proposed variable-order fractional HNN exhibits more complex characteristics, and by selecting different fractional variable orders, novel attractors with chaotic behavior can be obtained. Additionally, a control scheme is proposed to stabilize the suggested neural network by utilizing the stability theorem for fractional discrete time systems. This control scheme is applied to both states in the study.