Recent research on multi-double-scroll memristive Hopfield neural networks has attracted considerable attention, and significant progress has been made. However, most existing models rely on a single memristor for regulation, yielding unidirectional multi-double-scroll chaotic attractors. Moreover, current systems are typically constructed using polynomial-based memristor models, in which the number of scrolls varies with the number of polynomial terms. Although such approaches allow programmable control of scroll count, their complex mathematical formulations hinder further application.
To overcome this limitation, this study proposes a new memristor model based on piecewise functions, which is incorporated into a Hopfield neural network as both a self-synaptic and an inter-synaptic weight. Theoretical analysis shows that the proposed memristive Hopfield neural network can generate an arbitrary number of multi-double-scroll attractors, and the scroll count can be flexibly adjusted via the memristor’ s control parameters. In addition, the model can be extended by adding more memristive synapses to construct higher-dimensional networks, demonstrating flexibility and generality.
The nonlinear dynamics of the system are investigated using bifurcation diagrams, Lyapunov exponent spectra, phase portraits, and basins of attraction. Results indicate that the system can produce grid-style multi-double-scroll chaotic attractors, where the total number of scrolls equals the product of the scroll counts generated along each of the two directions. Further analysis reveals that the system exhibits initialoffset-enhanced coexisting attractors: varying only the initial condition of the memristor yields multiple chaotic attractors with identical shapes but shifted positions. The number of these coexisting attractors can also be controlled via parameters, indicating the presence of super-multistability.
The chaotic sequences generated by the network pass all 15 tests of the NIST SP 800-22 statistical suite, with p-values above 0.01, confirming their randomness and unpredictability for cryptographic applications such as image encryption and secure communication. Comparative results with recent studies show that the proposed system achieves p-values closer to 0.5 and relatively high pass rates across multiple tests, demonstrating satisfactory randomness.
Finally, the proposed system is implemented on an FPGA platform using SOPC technology and the Euler discretization method. Oscilloscope measurements agree well with MATLAB numerical simulations, verifying its physical realizability and feasibility.