The neural network model coupled with memristors has been extensively studied due to its ability to more accurately represent the complex dynamic characteristics of the biological nervous system. Currently, the mathematical model of memristor used to couple neural networks mainly focuses on primary function, absolute value function, hyperbolic tangent function, etc. To further enrich the memristor-coupled neural network model and take into account the motion law of particles in some doped semiconductors, a new compound exponential local active memristor is proposed and used as a coupling synapse in the Hopfield neural network. Using the basic dynamic analysis method, the system’s dynamic behaviors are studied under different parameters and the coexistence of multiple bifurcation modes under different initial values. In addition, the influence of frequency change of external stimulation current on the system is also studied. The experimental results show that the internal parameters of memristor synapses regulate the system, and the system has a rich dynamic behavior, including symmetric attractor coexistence, asymmetric attractor coexistence, large-scale chaos as shown in attached figure, and bursting oscillation. Finally, the hardware of the system is realized by the STM32 microcontroller, and the experimental results verify the realization of the system.