With the widespread application of audio communication in the internet of things, secure and efficient audio data transmission has become increasingly critical. However, existing chaos-based encryption schemes often face challenges in flexibly generating complex chaotic attractors and achieving a practical balance between security, flexibility, and efficiency. To address these limitations, this paper proposes a novel universal method for generating multi-structure chaotic attractors based on a quaternion rotation matrix transformation. This method enables flexible control over the position, scale, and number of wing units in attractors by configuring quaternion parameters and multi-level pulse function. Building upon this, we design a high-performance chaos-based audio encryption algorithm, and implement an encryption system for wireless transmission using a microcontroller and LoRa module. The system integrates audio recording, encryption, wireless transmission, and decryption. Comprehensive security analyses demonstrate the proposed algorithm achieves robust performance. The successful integration of flexible chaotic attractor design, robust cryptographic security, and hardware realization confirms the practical viability of our scheme for information transmission in resource-constrained environments.

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.

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.

An autonomous mobile robot with chaotic navigation, named chaotic mobile robot (CMR), has shown significant potential applications for dangerous or special duties in Internet of Things. As the control source of CMR, the application of novel chaotic dynamical systems with good design flexibility, topological transitivity, and sensitivity to initial conditions, will favor analyzing the whole workplace. To this purpose, a novel dual-route fractal process (DRFP) is proposed based on Julia and advanced Julia functions, which implements the automatic design of multifolded chaotic prototype models for given seed chaotic system. In comparison with the traditional Julia fractal process, the DRFP shows significant advantage in filling in the chaotic trajectory in empty inner space, which is beneficial to design CMR with high performance. The simulation and experiment are presented to validate the design feasibility of the DFRP. Finally, a two-wheeled differential drive system coupled with the chaotic model is designed to observe its navigation performances in either open region or bounded region with obstacles.

By integrating memristors into a Hopfield neural network (HNN), a diverse range of dynamical behavior can be generated, which has significant implications for modeling and biomimetic applications of artificial neurons. However, research on the firing dynamics of HNNs remains relatively limited. In response, a memristive tri-neurons Hopfield neural network (MTN-HNN) was constructed, with the synapse of the second neuron replaced by the proposed memristor. A theoretical and experimental investigation of the dynamics of this neural network was conducted using general analytical tools, such as phase diagrams, Lyapunov exponents, bifurcation diagrams, and others. Experimental results indicate that the dynamics of the MTN-HNN is influenced by the internal parameters of the memristor, enabling the network to extend attractors in up to two directions and thereby form grid multi-scrolls. Notably, the MTN-HNN exhibits various firing modes, including periodic and chaotic bursting. Finally, an encryption scheme was proposed to demonstrate the potential of the MTN-HNN, and both the custom digital circuits and the encryption scheme were successfully implemented on a Field-Programmable Gate Array (FPGA).

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.

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 generation of complicated grid hyperchaotic hidden attractors is an interesting but challenging work. Up to now, no grid hyperchaotic system with stable equilibria has been reported and implemented yet. This paper presents a two-step design scheme for constructing novel grid hyperchaotic system with only stable equilibria. First, the simple variable boosting is applied to a single-stable-equilibrium hyperchaotic system to generate offset boosted hyperchaotic hidden attractors in two directions. At this point, it is associated with the increase of the number of equilibria in corresponding directions. To solve this problem, the configuration of two specific sawtooth waveform functions implements the offset switching in two directions but only extends the number of the equilibria in a single direction. Meanwhile, all these equilibria have the same stability conditions that will not change with the increase of the grid. Consequently, various grid hyperchaotic hidden attractors are generated. Numerical simulations are presented to intuitively show the phase portraits and cross sections. Finally, hardware experiments using analog circuit implementation are presented to verify the feasibility of the grid hyperchaotic system.

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.

As a paradigm bridging the mathematical chaos to physical one, Chua’s circuit and its dimensionless Chua’s system have been widely studied and applied. From the completeness point of view, it is an interesting issue under investigation that how many potential Chua’s circuit variants exist within the framework of a single Chua’s system. Follow the duality principle in electrical circuits, this paper presents a Chua’s circuit employing ideal charge-controlled memristive model, which complements the circuit genealogy of the classical Chua’s system to four, i.e., Chua’s circuit, dual Chua’s circuit, memristive Chua’s circuit, and dual memristive Chua’s circuit (DMCC). To study the initial condition-related dynamical behaviors of the DMCC, we further present the flux-charge analysis. Some complex coexisting dynamics that cannot be typically observed in the voltage-current-domain have been investigated in the flux-charge-domain. Based on the dimensionless equations of the DMCC, an equivalent circuit is implemented for verification.

Bionic firing circuits are beneficial for bionic hardware applications. It is an attractive but challenging task to design a simple bionic firing circuit to generate spiking and bursting behaviors. To hit this aim, this paper newly proposes a universal configuration framework for mem-element-emulator-based bionic firing circuit, which deploys the charge-controlled memcapacitor and voltage-controlled locally active memristor (LAM) to characterize the electrophysiological behaviors of liquid bilayer and ion channels of a neuronal membrane, respectively. The configuration framework only contains one memcapacitor, LAM, DC voltage, and current stimulus. Then, a charge-controlled memcapacitor and an N-type LAM are employed as a case to verify the effectiveness of the configuration framework. Numerical simulations, PSpice circuit simulations, and discrete component-based hardware experiments are performed, which display that the bionic firing circuit can generate abundant spiking and bursting behaviors. One key feature is that the configuration framework can employ different charge-controlled memcapacitors and voltage-controlled LAMs to build more potential bionic firing circuits. To the authors’ knowledge, this configuration framework is extendable and sheds new light on the design of bionic firing circuits.

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.

The systematic control of chaotic phase orbits in desired location and orientation favors of the generation of multiwing/scroll attractors for potential industrial applications, which is an attractive but challenging work. In this article, we present a universal matrix transformation method for generating multistructure chaotic attractors. First, several basic transformation matrices or their composite forms are chosen to implement multiplex control of a seed. On this foundation, applying it in a general 3-D system and further obtaining the universal transformed system. Taking the Lorenz system, Chua's system, and Rössler system as the seeds, we construct various multistructure attractors with custom-shaped distributions via stringing together multiple separate units. The novelties are that the method is universal, has wide applicability, and does not introduce any extra variables for state-extension. Numerical phase orbits and hardware implementations verified the feasibility of the method. A pseudorandom number generator with qualified performance is implemented to support its potential applications.

Analog circuit implementation of neuron model is an essential category of neuromorphic circuit since it can reproduce neuron firing patterns and assist in exploring neuron-based applications. The neuron models built by electrophysiological ion transport mechanism can effectively mimic the neuron firing patterns. However, they are rather difficult to implement on analog level because these neuron models involve complex exponential nonlinearities for characterizing the ion channels. Thanks to the superiorities of locally active memristor in constructing artificial neuron circuit, two locally active memristors are employed to characterize the sodium and potassium ion channels and then a memristive neuromorphic circuit is successfully built in this paper. Numerical simulations demonstrate that the memristive neuromorphic circuit can generate abundant firing patterns with respect to memristor- and stimulus-related parameters. Moreover, quasi-periodic bifurcation behavior and coexisting firing patterns are triggered by different memristor initial conditions. Employing off-the-shelf discrete circuit components, a PCB-based analog circuit is manually constructed and hardware experiments are executed. Experimental results captured from hardware measurements satisfactorily verify the numerically simulated ones and effectively show the feasibility of the memristive neuromorphic circuit in generating neuron firing patterns.

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.

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.

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.