A Novel Fractional-Order Cascade Tri-Neuron Hopfield Neural Network: Stability, Bifurcations, and Chaos

In this paper, we propose a novel Caputo-type fractional-order cascade tri-neuron Hopfield neural network (HNN) taking no connection between the first and third neuron. We analyse the symmetry and dissipativity of the system using divergence and transformations. The stability of the equilibrium points is checked by fixing the synaptic weights. To further analyse the dynamics of the HNN system, we derive a numerical solution by using the Adams–Bashforth–Moulton method along with its stability analysis. We performed several graphical simulations, considering two synaptic weights as adjustable variables, and explored the fact that the HNN system shows various periodic and chaotic attractors. The reason for proposing a fractional-order HNN is that such a system has limitless memory, which can improve the system’s controllability for a wide range of real-world phenomena with important applications. Also, the proposed fractional-order HNN shows better convergence compared to the integer-order case.