Single direction, grid and spatial multi-scroll attractors in Hopfield neural network with the variable number memristive self-connected synapses

Due to the synapse-like nonlinearity and memory characteristics, the memristor is often used to simulate the biological neural synapse. In this paper, a family of three-neuron Hopfield neural network (HNN) models based on the variable number memristive self-connected synapses is proposed. Firstly, a single memristive self-connected synapse (SMSCS) HNN model is constructed, which can generate a single direction multi-scroll attractor controlled by the memristor parameters. Meanwhile, its dynamic behaviors including equilibrium points, multiple coexisting attractors and controllable n-scroll chaotic attractors are analyzed. Secondly, based on the above SMSCS HNN model, two types of multiple memristive self-connected synapse (MMSCS) HNN models are constructed. By changing the control parameters of the memristors, these MMSCS HNN models can not only generate the different scroll numbers of grid and spatial multi-scroll attractors, but also can produce the spatial initial-offset coexisting attractors. The above three HNN models utilizing the variable number memristors to simulate one to three self-connected synapses can generate a class of complex chaotic attractors, which include single direction, grid and spatial multi-scroll attractors. Finally, the feasibility of the proposed HNN models is verified by the FPGA platform.