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.

Memristors can serve as variable synaptic weights between neurons for adaptive neural network regulation. Inspired by this, a discrete memristive Hopfield neural network (DM-HNN) is constructed utilizing an adaptive memristor weight instead of a fixed resistor weight. It has a line fixed point set with stability strongly related to the memristor initial state. On this basis, chaotic/hyperchaotic attractors with bifurcation dynamics are explored. Further, the memristor initial-boosting mechanism is examined and the memristor initial-boosted homogeneous attractors are elucidated. The results present that DM-HNN can exhibit chaotic/hyperchaotic attractors with intricate structures and memristor initial-boosted homogeneous attractors. Notably, the coexisting homogeneous sequences with excellent performance indices can be toggled by the memristor initial state, well reflecting the adaptive regulation of the memristor. Additionally, kinetic experiments on Field Programmable Gate Array (FPGA) verify the hardware implementability of DM-HNN, based on which an innovative memristor-state-based image encryption scheme is proposed, enabling resource-constrained scenarios and demonstrating excellent encryption performance.

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.

Nonlinear activation function is the most essential factor that enables Hopfield neural networks (HNNs) to generate complex dynamics. However, less attention has been paid to the activation function. In this article, we propose a simple discrete model of self-connectionless HNN in which two heterogeneous neurons have different activation functions of sine and hyperbolic tangent. By theoretical and numerical methods, we explore coexisting bifurcation behaviors induced by Neimark-Sacker bifurcations and multifolded hyperchaotic attractors caused by multiple fixed points with different stability. We also evaluate the randomness of these multifolded hyperchaotic sequences and implement the discrete model on field programmable gate array (FPGA) hardware device. In brief, this discrete model with simple algebraic equations has the simplest connection structure under the two-heterogeneous-neuron HNN framework, but it can generate various multifolded hyperchaotic attractors with high randomness. Additionally, based on FPGA hardware device, a chaos-based hardware Poisson encoder is developed to implement the reconstruction of gray image.

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.

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.

Designing low-dimensional discrete maps with initial-dependent coexisting property is an attractive but challenging task. The coexisting property of a discrete map can be featured by initial-offset-control dynamics. To this end, this article proposes a two-dimensional discrete neuron model with sine activation function. The mechanisms of initial-offset-control coexisting dynamics are theoretically investigated and the homogenous coexisting behaviors are numerically revealed. The results show that the homogenous coexisting attractors are controlled along one direction by one initial-offset and along two directions by two initial-offsets. The former makes it model own finite invariant points, while the latter makes it own infinite invariant points. The homogenous coexisting hyperchaotic attractors are experimentally acquired on field programmable gate array digital platform. Besides, eight pseudorandom number generators (PRNGs) are designed using the proposed model under different parameter and initial settings, and the test results by TestU01 test suite show the high randomness of these PRNGs without chaos degradation.

Memristor is a special nonlinear circuit component with internal state and can lead to excellent chaos complexity in its constructed discrete system. To enhance the chaos complexity of a memristor-based discrete system, this article proposes a 2-D sine-transform-based (STB) memristive model. The model has line fixed point and its stability is dependent on memristor initial state. Complex dynamics with quasi-periodic bifurcation and multistability are demonstrated using numerical methods. For different control parameters, chaotic and hyperchaotic attractors are emerged and their complicated fractal structures and outstanding performance indicators are exhibited. A hardware prototype is developed and these attractors are experimentally captured therein. Besides, six pseudorandom number generators (PRNGs) are designed using the proposed model under different control parameters and the test results by the TestU01 standard show that these PRNGs have high randomness without chaos degradation. In brief, the proposed 2-D STB memristive model is flexible to generate chaos and hyperchaos with high performance.

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.

The magnetic induction effects have been emulated by various continuous memristive models but they have not been successfully described by a discrete memristive model yet. To address this issue, this article first constructs a discrete memristor and then presents a discrete memristive Rulkov (m-Rulkov) neuron model. The bifurcation routes of the m-Rulkov model are declared by detecting the eigenvalue loci. Using numerical measures, we investigate the complex dynamics shown in the m-Rulkov model, including regime transition behaviors, transient chaotic bursting regimes, and hyperchaotic firing behaviors, all of which are closely relied on the memristor parameter. Consequently, the involvement of memristor can be used to simulate the magnetic induction effects in such a discrete neuron model. Besides, we elaborate a hardware platform for implementing the m-Rulkov model and acquire diverse spiking-bursting sequences. These results show that the presented model is viable to better characterize the actual firing activities in biological neurons than the Rulkov model when biophysical memory effect is supplied.

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.

Chaotic systems are widely employed in many practical applications for their significant properties. Existing chaotic systems may suffer from the drawbacks of discontinuous chaotic ranges and frail chaotic behaviors. To solve this issue, this paper proposes a two-dimensional (2D) sine chaotification system (2D-SCS). 2D-SCS can not only significantly enhance the complexity of 2D chaotic maps, but also greatly extend their chaotic ranges. As examples, this paper applies 2D-SCS to two existing 2D chaotic maps to obtain two enhanced chaotic maps. Performance evaluations show that these two enhanced chaotic maps have robust chaotic behaviors in much larger chaotic ranges than existing 2D chaotic maps. A microcontroller-based experiment platform is also designed to implement these enhanced chaotic maps in hardware devices. Furthermore, to investigate the application of 2D-SCS, these two enhanced chaotic maps are applied to design a pseudorandom number generator. Experiment results show that these enhanced chaotic maps can produce better random sequences than the existing 2D and several state-of-the-art one-dimensional (1D) chaotic maps.

Memristors can be employed to mimic biological neural synapses or to describe electromagnetic induction effects. To exhibit the threshold effect of electromagnetic induction, this paper presents a threshold flux-controlled memristor and examines its frequency-dependent pinched hysteresis loops. Using an electromagnetic induction current generated by the threshold memristor to replace the external current in 2-D Hindmarsh-Rose (HR) neuron model, a 3-D memristive HR (mHR) neuron model with global hidden oscillations is established and the corresponding numerical simulations are performed. It is found that due to no equilibrium point, the obtained mHR neuron model always operates in hidden bursting firing patterns, including coexisting hidden bursting firing patterns with bistability also. In addition, the model exhibits complex dynamics of the actual neuron electrical activities, which acts like the 3-D HR neuron model, indicating its feasibility. In particular, by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem, the bifurcation mechanisms of hidden bursting firings are expounded. Finally, circuit experiments on hardware breadboards are deployed and the captured results well match with the numerical results, validating the physical mechanism of biological neuron and the reliability of electronic neuron.

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.