Harmonic components dynamics of symmetric coexisting attractors and pitchfork bifurcation in memristive Hopfield neural networks

As an artificial neural network paradigm capable of generating complex dynamical behaviors, the Hopfield neural network (HNN) has been widely applied in modern medicine and artificial intelligence. Significantly, memristors have demonstrated considerable potential in enhancing the complex dynamic characteristics of HNN. Consequently, this study proposes a three-neuron memristive Hopfield neural network (MHNN) with memristive coupling synaptic weights, which exhibits complex periodic motions and coexisting attractors. To thoroughly explore the intricate dynamical behaviors, this paper first adopts the discrete mapping method to systematically analyze the MHNN. Furthermore, the introduction of the finite Fourier series expands the research into the frequency domain. Specifically, the amplitude–frequency characteristics of the MHNN are comprehensively revealed by specifically analyzing the constant terms and harmonic amplitude characteristics within the pitchfork bifurcation. Additionally, the relationships between the harmonic amplitudes and phases of the symmetric coexisting attractors effectively reflect the harmonic components dynamics. Finally, the simulation results are validated by a field-programmable gate array (FPGA) digital circuit, which the high-precision implementation of complex functions is achieved using the piecewise linear method. This study introduces a novel perspective for investigating the dynamic behaviors in memristive neural networks, particularly regarding complex periodic oscillations, which contributes to the functional design of neuromorphic computing systems.