The development of low-dimensional hyperchaotic maps with controllable dynamics and high randomness is both important and challenging. To address these issues, we propose a novel two-dimensional torus hyperchaotic map (2D-THM). This map is constructed through torus projections, which ensures global boundedness and produces diverse petal-like hyperchaotic attractors. Particularly, the torus parameters enable flexible control over attractor structure and pattern characteristics. Comparative analyses using numerical simulations and entropy metrics demonstrate that 2D-THM achieves superior complexity and robust hyperchaotic properties. Based on this map, a novel image encryption algorithm is presented. This algorithm utilizes attractor features to achieve high storage efficiency, making it particularly suitable for resource-constrained environments. Comprehensive security analyses confirm strong resistance to various attacks. Furthermore, this algorithm is implemented on FPGA platform, achieving an encryption time of 47 ms for a 1 MB image with low power consumption and efficient RAM usage, thereby validating its practical efficiency and hardware adaptability for real-world applications.

Leveraging the ability of locally active memristors to generate and flexibly modulate neuronal firing patterns with simple topologies, this paper proposes a dual-channel memristive neural circuit for scalable neuromorphic implementation. The proposed design employs two structurally identical branches, each incorporating a locally active memristor, enabling on-demand control of firing dynamics within a unified hardware framework. We systematically analyze the stability distribution of equilibrium points and investigate the underlying firing patterns under various stimulus conditions through numerical bifurcation analysis. The results demonstrate that the proposed neural circuit topology supports controllable transitions among resting, spiking, bursting, and chaotic states under undriven, DC-driven, and pulse-triggered conditions. These transitions are achieved through electrical modulation of key circuit parameters, namely the memristor bias voltages, external stimulation, and coupling resistance. PCB-level experimental measurements validate the feasibility of the theoretical and simulation analyses. Overall, this work provides fundamental guidance for designing LAM-based neural circuits and contributes to the development of essential building blocks for neuromorphic engineering.

Designing a universal framework for configuring bionic spiking circuits is an attractive topic for spike-based applications. This paper comprehensively considers the electrophysiological properties of a biological neuron and electrical features of mem-elements, then presents a new universal framework for configuring fully mem-element-based bionic spiking circuits. The universal framework contains a current-controlled locally active memristor (LAM), a charge-controlled memcapacitor, and a flux-controlled meminductor to respectively characterize the electrophysiological properties of ion channels, liquid bilayer membrane, and electromagnetic induction of a biological membrane. The universal framework only involves the three mem-elements and a DC current stimulus. Afterward, a case of a mem-element emulator-based bionic spiking circuit is configured employing the universal framework. Numerical explorations and experimental measurements based on analog hardware circuits are performed to validate the effectiveness of the universal framework in generating abundant bifurcation behaviors and spiking activities. Actually, the universal framework is available for employing different mem-element emulators or physical mem-elements to construct bionic spiking circuits. The universal framework is extensible and sheds new light on the configuration of fully mem-element-based bionic spiking circuits.

The synapse-like properties of memristors make them a popular choice for neuromorphic circuits, particularly in neural networks. However, most existing neural networks capable of generating complex dynamics are continuous-time models with high dimensionality, resulting in significant resource overhead for digital implementation. This study incorporates a memristor with an internal multisegment state function into two-neuron Hopfield neural network (HNN), thereby constructing a novel discrete-time memristive two-neuron HNN (DMT-HNN) capable of emerging complex multi-stripe/wave hyperchaos. DMT-HNN generates multistripe and multiwave hyperchaotic attractors, with the number of stripes/waves continuously expanding as the segment number of the memristor state function increases. By decreasing the memristor scaling factor, the multi-stripe/wave hyperchaotic attractors can be decomposed into varying numbers of coexisting hyperchaotic attractors. An FPGA hardware setup is designed based on DMT-HNN, and the multi-stripe/wave hyperchaotic attractors are successfully captured on an oscilloscope. Besides, hardware pseudorandom level generators are fabricated, and the results are confirmed by NIST randomness tests.

Neuromorphic circuits with diverse neuronal firing features are the foundation for implementing neuron-based intelligent tasks. The locally active memristor (LAM) is a good candidate for constructing such kind of artificial circuits. However, it is attractive and challenging to explore synchronicity of firing patterns between LAM-based neuromorphic circuits, especially on hardware level. To this end, a neuromorphic circuit is constructed with a newly designed LAM emulator, a capacitor, a DC voltage, and an external current stimulus. Abundant stimulus-relating firing patterns are numerically revealed, whose bifurcation mechanism are theoretically discussed. Besides, using a LAM emulator as the coupling synapse, the interactions between two firing neuromorphic circuits are explored under identical and nonidentical stimulus conditions. The results manifest that the amplitude and phase of the stimulus current, as well as the static potential of the LAM branch, all exert significant influences on the synchronization behaviors. Furthermore, PCB-based analog circuits are implemented to demonstrate the firing patterns of single and coupled neuromorphic circuits from the perspective of physical experiments. This research could provide a viable paradigm to theoretical research and engineering application of memristive neuromorphic circuits.

Compared with the continuous Hopfield neural network (HNN), the discrete HNN remains relatively under-explored in both academic and industrial domains. This paper proposes a simple two-dimensional (2-D) HNN-like map for neurons with specific internal decay. It is a discrete map with an infinite number of grid unstable points, leading to the appearance of grid homogeneous coexisting attractors. Theoretical analysis deduces the switching mechanism of initial-offsets, while numerical simulations disclose the grid homogeneous coexisting bifurcation behaviors and hyperchaotic attractors. The results manifest that 2-D HNN-like map can exhibit grid homogeneous coexisting hyperchaos in two dimensions and the highly random hyperchaotic sequences can be losslessly switched by two initial values. Additionally, hardware implementation on an STM32 platform validates the coexisting hyperchaotic attractors. Furthermore, using the initials-switched grid homogeneous coexisting hyperchaotic sequences, we develop a reliable and secure geolocation-based chaotic hardware encryptor. To our best knowledge, this is the first application of grid homogeneous coexisting hyperchaos in industrial field.

Designing low-dimensional chaotic maps with strong resistance to chaos degradation is of great significance to chaos theory and chaos-based applications. The famous Hopfield neural network (HNN) is an artificial neural network, which has been widely studied and applied. However, discrete HNNs, especially their complex dynamics and hyperchaotic attractors with complex structures, are rarely reported in the literature. To this end, this paper proposes a two-dimensional discrete model of bi-neuron HNN with sine activation functions. The fixed points with stability analysis are theoretically explored and complex dynamics with coexisting behaviors are numerically revealed. The discrete model has infinitely many fixed points and emerges various polyhedral chaotic/hyperchaotic attractors with marvelous fractal structures. Besides, using the model, we design four pseudorandom number generators (PRNGs) and test their randomness by TestU01 suite. The results manifest that the PRNGs have high randomness and strong resistance to chaos degradation. Lastly, an FPGA hardware platform is developed to implement the proposed discrete model, upon which the polyhedral chaotic/hyperchaotic attractors are experimentally acquired to validate the numerical results, and two hardware PRNGs are fabricated to provide the true pseudorandom numbers (PRNs).

Because of the special nonlinearity with inner states, memristors can induce the initial-condition-dependent extreme multistability in their constituent systems. However, when the memristor is taken as a coupler to synchronize two identical memristive systems, what synchronous behaviors could be achieved? So far, it has not been comprehensively concerned in literature. To this end, this paper presents a 9-D memristor-coupled system consisting of two homogenous memristive systems and studies its initial-boosted dynamics and synchronous behaviors. Dynamics evolutions related to the coupling memristor and two subsystem memristors are elaborated. Using Lyapunov’s stability theorem, the synchronicities of the initial-condition-sensitive dynamics are demonstrated theoretically. Furthermore, based on the integral transformation model, the synchronization parameter regions are estimated quantitatively using a newly proposed variable-parameter conversion method. Thereafter, various coexisting synchronous behaviors are discovered by tuning the memristor initial conditions and verified via the microcontroller-based hardware measurements. It is demonstrated that the memristor coupler provides a flexible control scheme for the synchronization of memristive systems.

As a classic paradigm of chaos generation, Chua’s circuit has been widely studied and applied in different fields. Various Chua’s circuits employing voltage-controlled Chua’s diodes have been designed and implemented, but rare such circuits employing current-controlled Chua’s diodes have been reported to date. Following the technical steps which Chua used to design the famous Chua’s circuit, this paper presents a systematic design procedure of the chaotic circuits employing current-controlled nonlinear resistors. Two simple circuit topologies are presented as paradigms, which respectively are dual to the classical Chua’s circuit and the canonical Chua’s circuit. In specific, we obtain two simple implementations of the dual Chua’s circuits consisting of two inductors, one capacitor, one passive/active linear resistor, and one piecewise-linear current-controlled Chua’s diode. The numerical simulations and the previously unreported physical implementations confirmed the feasibility of the design. Consequently, it enriches the genealogy of the Chua’s circuit family.

The design of hyperchaotic systems with complex dynamics and ultrawide parameter space has been a research hotspot. In this article, we propose a 2-D threshold discrete map model and reveal its bifurcation behaviors and coexisting behaviors using several numerical methods. The proposed model with provable boundedness has multiple fixed points with parameter-related stability, and can produce amplitude-controllable hyperchaotic attractors with complex fractal structures and exhibit hyperchaotic dynamics with ultrawide parameter space. The hyperchaotic attractors are verified by the developed STM32 hardware prototype. In addition, a novel chaos-based timed one-time password scheme is presented and its generator is implemented through both software and hardware platforms. Multiplatform experiments verify the consistency of the one-time password, and randomness test also verifies the reliability of chaos-based timed one-time password generator. In summary, the proposed model easily achieves hyperchaotic regimes with ultrawide parameter space and delivers reliable performance for implementing the chaos-based timed one-time password.

Multipliers are essential in implementing nonlinear neuron models, but they take huge implementation costs. Many multiplierless fitting schemes have been proposed to simplify the implementation of nonlinearities in neuron models. To optimize these schemes, this paper presents a nullcline-characteristics- based piecewise linear (NC-PWL) fitting scheme for multiplierless implementations of Hindmarsh-Rose (HR) neuron model. This NC-PWL fitting scheme uses as few line segments as possible to approximate the critical nonlinearity characteristics of the local nullclines. A NC-PWL HR neuron model that reproduces diverse firing patterns of the original one is successfully established. Using off-the-shelf low-cost components, an analog multiplierless circuit is designed for this fitting model and welded on print circuit board (PCB). Meanwhile, by logical shift method, a digital multiplierless circuit with low resource consumption is developed for this fitting model on field-programmable gate array (FPGA) platform. Experimental results of the analog and digital multiplierless hardware implementations verify the numerical simulations and show the simplicity and feasibility of the presented fitting scheme.

With the nonlinearity and plasticity, memristors are widely used as nonlinear devices for chaotic oscillations or as biological synapses for neuromorphic computations. But discrete memristors (DMs) and their coupling maps have not received much attention, yet. Using a DM model, this article presents a general three-dimensional discrete memristor-based (3-D-DM) map model. By coupling the DM with four 2-D discrete maps, four examples of 3-D-DM maps with no or infinitely many fixed points are generated. We simulate the coupling coefficient-depended and memristor initial-boosted bifurcation behaviors of these 3-D-DM maps using numerical measures. The results demonstrate that the memristor can enhance the chaos complexity of existing discrete maps and its coupling maps can display hyperchaos. Furthermore, a hardware platform is developed to implement the 3-D-DM maps and the acquired hyperchaotic sequences have high randomness. Particularly, these hyperchaotic sequences can be applied to the auxiliary classifier generative adversarial nets for greatly improving the discriminator accuracy.

Extreme multistability has seized scientists’ attention due to its rich diversity of dynamical behaviors and great flexibility in engineering applications. In this paper, a four-dimensional (4D) memcapacitive oscillator is built using four linear circuit elements and one nonlinear charge-controlled memcapacitor with a cosine inverse memcapacitance. The 4D memcapacitive oscillator possesses a line equilibrium set, and its stability periodically evolves with the initial condition of the memcapacitor. The 4D memcapacitive oscillator exhibits initial-condition-switched boosting extreme multistability due to the periodically evolving stability. Complex dynamical behaviors of period doubling/halving bifurcations, chaos crisis, and initial-condition-switched coexisting attractors are revealed by bifurcation diagrams, Lyapunov exponents, and phase portraits. Thereafter, a reconstructed system is derived via integral transformation to reveal the forming mechanism of the initial-condition-switched boosting extreme multistability in the memcapacitive oscillator. Finally, an implementation circuit is designed for the reconstructed system, and Power SIMulation (PSIM) simulations are executed to confirm the validity of the numerical analysis.

Regarding as a basic circuit component with special nonlinearity, memristor has been widely applied in chaotic circuits and neuromorphic circuits. However, discrete memristor (DM) has not been received much attention, yet. To this end, this paper reports a general DM model and its unified mapping model. Using the general DM model, four representations of DMs are given and their pinched hysteresis loops are exhibited. Based on the unified DM mapping model, four two-dimensional (2D) DM maps are generated and their parameter-relied and initials-relied behaviors are explored using multiple numerical measures. The results demonstrate that all the four 2D DM maps can generate hyperchaos with coexisting bi-stable or memristor initial-boosted behavior, and their sequences have excellent performance indictors. A hardware device is constructed to implement these maps and the analog voltage signals are experimentally acquired. Moreover, pseudo-random number generators (PRNGs) are designed using these DM maps and the test results show that the generated pseudo-random numbers (PRNs) have high randomness.

When chaotic sequences are used in engineering applications, their oscillating amplitudes need to be adjusted nondestructively. To accommodate this issue, this article presents a simple 2-D sine map. It can not only generate the chaotic sequences with high complexity, but also boost the oscillating amplitudes by switching their initial states. To show the complex dynamics of the sine map, this article investigates its control parameters-related dynamical behaviors and initials-boosted coexisting bifurcations using numerical methods. The results demonstrate that the oscillating amplitudes of chaotic sequences generated by the sine map can be nondestructively controlled by switching their initial states. This makes the sine map more suitable for many chaos-based engineering applications. Furthermore, we develop a microcontroller-hardware test platform to implement the sine map. The experimental results show that the platform synchronously outputs multichannel initials-controlled chaotic sequences. We also design a pseudorandom number generator to explore the application of the sine map.

A four-dimensional memristive system is constructed using a novel ideal memristor with cosine memductance. Due to the special memductance nonlinearity, this memristive system has a line equilibrium set (0, 0, 0, δ ) located along the coordinate of the inner state variable of the memristor, whose stability is periodically varied with a change of δ. Nonlinear and one-dimensional initial offset boosting behaviors, which are triggered by not only the initial condition of the memristor but also other two initial conditions, are numerically uncovered. Specifically, a wide variety of coexisting attractors with different positions and topological structures are revealed along the boosting route. Finally, circuit simulations are performed by Power SIMulation (PSIM) to confirm the unique dynamical features.