Memristive Hopfield neural networks (MHNN) integrate the non-volatile memory of memristors and feedback dynamics of neural networks, showing great potential in chaotic information security. Nevertheless, most existing MHNNs adopt the conventional \(\tanh\) activation function, which suffers from vanishing gradients and low dynamical complexity. Besides, studies on activation function optimization for Hopfield neural networks (HNN) are still insufficient. Therefore, a novel hybrid exponential Tanh unit function (HETUF) activation function is proposed in this work. Based on the HETUF activation function, a compact two-neuron HETUF-memristive Hopfield neural network (HETUF-MHNN) is constructed. Dynamical analyses and quantitative comparisons with the tanh-MHNN are conducted by using phase portraits, 0–1 test, Lyapunov exponents and bifurcation diagrams. A new phenomenon named derived chaotic decay with time-dependent attenuation is discovered, and its essential difference from traditional multi-stability is clarified. Additionally, a coordinate-constrained attractor capturing strategy is presented, and the corresponding chaotic sequences also pass the NIST test. Finally, a low-complexity spatial domain HETUF-MHNN-based image encryption scheme is designed and validated.
