<p>Grid multi-attractors, characterized by intricate topology and enhanced dynamics, have been widely investigated in chaotic systems. However, discrete neural network maps with controllable grid multi-structure hyperchaotic attractors remain rarely reported. This paper presents a method for constructing a discrete multi-structure chaotic system based on the memristive neural network. Firstly, a three-neuron discrete Hopfield neural network incorporating a discrete multi-segment state function memristor is constructed, which initially yields multi-structure hyperchaotic attractors along a single direction. Significantly, there is no relevant research on the influence of non-autonomous multi-level logic pulses on discrete chaotic systems. By introducing multi-level logic pulse stimulation, it is found that the system’s attractors can be replicated and expanded along the directions of neuronal state variables, enabling the emergence of controllable grid and spatial lattice multi-structure hyperchaotic attractors. The offset-control behavior of the attractors is also studied, revealing the position regulation of a unit attractor by the initial values and external constant bias. The above dynamics are studied through various numerical simulations such as equilibrium point analysis, phase diagrams, bifurcation diagrams, and Lyapunov exponent spectra. Furthermore, an FPGA-based digital circuit is implemented, and the experimental results demonstrate consistency with numerical simulations. Finally, a pseudorandom number generator (PRNG) is developed based on the proposed model.</p>