Bifurcation Dynamics, Amplitude–Frequency Characteristics of Hopfield Neural Network and Its Application

The Hopfield neural network (HNN) has been widely used to simulate brain electrical activity due to its flexible topology and rich dynamical behaviors. Moreover, the spectral characteristics of neural activity serve as fundamental signal, offering potential to identify novel targets for neurological disorders and advancing next-generation neuromodulation therapies. However, the frequency-domain analyses of HNN have rarely been reported. To further explore the complex periodic motions induced by harmonic terms, this study employs discrete mapping method incorporating finite Fourier series to analyze bifurcation evolutions and coexistence of firing behaviors, providing the quantitative analysis of their amplitude-frequency characteristics for the first time. Additionally, by investigating the relationship between harmonic amplitudes and phases in coexisting attractors, new perspectives on the study of coexisting attractors are provided. Finally, the simulation results are verified through the hardware circuit built using printed circuit board and a high-performance pseudorandom number generator is designed by leveraging chaotic sequences derived from unstable periodic orbits. This research contributes a new approach for data encryption and communication security in Internet of Things.