Since the Lorenz chaotic system was discovered in 1963, the construction of chaotic systems with complex dynamics has been a research hotspot in the field of chaos. Recently, memristive Hopfield neural networks (MHNNs) offer great potential in the design of complex, chaotic systems because of their special network structures, hyperbolic tangent activation function, and memory property. Many chaotic systems based on MHNNs have been proposed and exhibit various complex dynamical behaviors, including hyperchaos, coexisting attractors, multistability, extreme multistability, multi-scroll attractors, multi-structure attractors, and initial-offset coexisting behaviors. A comprehensive review of the MHNN-based chaotic systems has become an urgent requirement. In this review, we first briefly introduce the basic knowledge of the Hopfiled neural network, memristor, and chaotic dynamics. Then, different modeling methods of the MHNN-based chaotic systems are analyzed and discussed. Concurrently, the pioneering works and some recent important papers related to MHNN-based chaotic systems are reviewed in detail. Finally, we survey the progress of MHNN-based chaotic systems for application in various scenarios. Some open problems and visions for the future in this field are presented. We attempt to provide a reference and a resource for both chaos researchers and those outside the field who hope to apply chaotic systems in a particular application.

Memristors, with their unique nonlinear characteristics, are highly suitable for construction novel neural models with rich dynamic behaviors. In this paper, a memristor with piecewise nonlinear state function is introduced into the tabu learning neuron model, resulting in a novel memristive tabu learning neuron model capable of generating a double-wing chaotic butterfly. By modulating the state function of the memristor, we can effectively and easily alter the number of wings of the chaotic butterfly. Equilibrium points analysis further elucidates the mechanism behind the generation of multi-wing chaos. Various numerical simulation techniques, including phase portraits, bifurcation diagrams, Lyapunov exponent spectra, and local attraction basins, are employed to illustrate the dynamical behaviors of the proposed model. Moreover, the newly constructed neuron model is validated using FPGA hardware, with the results aligning with numerical simulations, thereby offering a dependable foundation for a memristor digital circuit-based brain-like neuron model. Lastly, an image encryption application based on the multi-wing chaotic butterfly is developed to demonstrate the potential application of the model.

Multiscroll hidden attractors have attracted extensive research interest in recent years. However, the previously reported multiscroll hidden attractors belong to only one category of hidden attractors, namely, the hidden attractors without equilibrium points. Up to now, multiscroll hidden attractors with stable equilibrium points have not been reported. This paper proposes a multiscroll chaotic system with two equilibrium points. The number of scrolls can be increased by adding breakpoints of a nonlinear function. Moreover, the two equilibrium points are stable node-foci equilibrium points. According to the classification of hidden attractors, the multiscroll attractors generated by a novel system are the hidden attractors with stable equilibrium points. The dynamical characteristics of the novel system are studied using the spectrum of Lyapunov exponents, a bifurcation diagram, and a Poincaré map. Furthermore, the novel system is implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.