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.

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.

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.

Discrete chaotic maps in the real number field have been widely investigated and applied to various applications. However, there has been limited focus on constructing discrete chaotic maps with complicated dynamics in the complex field. In light of this, this article proposes a 1-D complex-variable chaotic model (1-D-CCM), which can produce a multitude of 1-D complex-variable chaotic maps by combining unbounded analytic functions and locally bounded analytic functions. To illustrate the effectiveness of 1-D-CCM, we construct two 1-D complex-variable chaotic maps by combining inverse trigonometric functions and hyperbolic trigonometric functions. We provide theoretical proof of a new 1-D complex-variable chaotic map as one example to demonstrate that the generated chaotic maps satisfy the chaos definition in terms of Lyapunov exponent. Property analysis reveals distinct strange attractors and hyperchaotic behaviors for the two 1-D complex-variable chaotic maps. Performance evaluations show that the example maps of 1-D-CCM model can achieve a 0–1 test value of 1.0017, a$C_{0}$complexity of 0.7253, a correlation dimension of 2.0242, and a sample entropy of 0.8288. Experimental results demonstrate superior performance indicators compared to other representative chaotic maps. We construct a hardware platform using a microcontroller to implement the attractors of the two new complex-variable chaotic maps. Finally, we design pseudorandom number generators to demonstrate the potential applications of the two 1-D complex-variable chaotic maps.

To address the trade-off between security and efficiency in Industrial IoT, this article presents a memristive chaotic system and a feature-driven encryption architecture. First, a dynamic window decay (DWD) memristor incorporating a voltage-dependent window function is proposed. By coupling this memristor with the Ikeda map, a novel chaotic model termed the DWD-Ikeda map is developed. This model exhibits high-complexity chaotic behavior and serves as the pseudorandom source. Based on this, a 3-D plaintext-feature-driven encryption scheme is developed. Then, the architecture integrates fractional-wavelet-gradient feature extraction for dynamic parameter modulation and a feature-anchored chain reset diffusion mechanism to control error propagation. Finally, performance evaluations on standard benchmarks and the NEU-DET industrial dataset show a number of pixel change rate of 99.6% and a uniform average change intensity of 33.4%. For 256 × 256 images, the software encryption time is 0.20 s, while the FPGA hardware implementation achieves a throughput of 440 Mbps. These results verify the scheme’s suitability for secure real-time industrial image transmission.

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.

In this paper, by taking a locally active memristor (LAM) as the neuronal synapse to link two identical/disparate ReLU-type Hopfield neural networks (RHNNs), the memristive homogeneous and heterogeneous neural networks are presented and their coupling dynamics are discussed in succession. The homogeneous RHNN model consists of two 3D RHNNs with a LAM, in which the coupling strength-and initial value-induced synchronous transitions are revealed. Besides, the heterogeneous RHNN model is focused on which is constructed by using a LAM to couple a 3D and 2D RHNN, where complex and rich dynamics are revealed in numerical, including hyperchaos, chaos, quasi-period, and multi-stable patterns. Particularly, attractor control of offset boosting as well as color image encryption application are realized, indicating the high controllability and security of the LAM-coupled neural networks. Finally, the electrical neuron depending on the digital circuit is implemented and hardware experimental results verify the numerical measurements well.

Secure communication techniques can protect data confidentiality during transmission through public channels. Chaotic systems are commonly used in secure communication due to their random-like behavior, unpredictability, and ergodicity. However, existing chaos-based secure communication schemes have some drawbacks concerning the chaotic systems used and the communication structures, so they cannot achieve satisfactory performance to resist transmission channel noise. In light of this, in this paper, we propose a two-dimensional (2D) cyclic chaotic system (2D-CCS) and design a novel chaos-based secure communication scheme called noise-reduced orthogonal frequency division multiplexing based differential chaos shift keying (NR-OFDM-DCSK). The 2D-CCS is a general framework that can generate a large number of new 2D chaotic maps using existing one-dimensional (1D) chaotic maps as seed maps. Theoretical analysis and experiment results demonstrate its robust chaotic behaviors. The NR-OFDM-DCSK employs a new chaotic map generated by 2D-CCS as the chaos generator, and its structure exhibits a strong ability to resist channel noise, as demonstrated by formulaic analysis. Our extensive experiments show that our developed 2D chaotic maps are more suitable for secure communication applications than existing 2D chaotic maps, and our NR-OFDM-DCSK can achieve a lower bit-error-rate (BER) than state-of-the-art secure communication schemes.

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.

Chaotic systems have attracted extensive research due to their pseudorandomness, ergodicity, and unique properties. Most studies focus on chaotic systems in the real number domain, but recent research has explored the design of complex chaotic systems. However, the chaotic behaviors of previous complex chaotic systems can only be observed through experiments and lack theoretical proof. In this article, we construct a 2-D coupled complex chaotic (2D-CCC) map using two nonlinear functions in the complex number domain. We theoretically prove the robust and complex chaotic behavior of the 2D-CCC map using the Lyapunov exponent. In addition, we conduct extensive experiments to demonstrate the map's intricate dynamics and high performance indicators. Comparison results highlight its superiority over previous chaotic systems. We also implement our 2D-CCC map on a hardware platform to validate its implementation feasibility on hardware devices. Finally, we investigate the 2D-CCC map's application in pseudorandom number generation and the testing results validate the high degree of randomness in the generated pseudorandom numbers.

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).

Detection of hidden dynamics is of great value in model prediction and control engineering. To explore its effects and control methods in the memristive network model, this paper presents a memristor synapse-driven ReLU-type Hopfield neural network (MRHNN). The generalized Hamilton function is derived from Helmholtz’s theorem and the equilibrium points of the model are analyzed. It is found via numerical computations that because of no existence of equilibrium, the MRHNN model always unfolds hidden dynamics, including hidden bifurcation, hidden mode transition, hidden transient chaos, and hidden multistability. In addition, amplitude and offset boosting control of hidden attractors are executed, illustrating the flexibility of the attractor regulation. Finally, based on digital hardware devices, circuit experiments are deployed and their measurements well agree with the numerical results, certifying the dynamical effects and lossless control of the memristive neural network and physical reliability of the electronic neuron.

Designing chaotic maps and fractal maps with rich dynamics in the complex field presents an interesting and challenging research topic. In this paper, we propose a novel approach called the one-dimensional multi-valued model (1D-MVM) for generating 1D complex chaotic and fractal maps by levering both single-valued and multi-valued functions. To demonstrate the effectiveness of the 1D-MVM, we present two 1D complex chaotic maps and one 1D fractal map as specific examples. Theoretical analysis confirms that the chaotic maps generated by the 1D-MVM exhibit chaotic behavior, while property analysis reveals that these chaotic maps possess hyperchaotic strange attractors, with their associated Lyapunov exponents being determined by specific system parameters. We also conduct extensive experiments to demonstrate the intricate dynamics and high performance indicators of the newly generated complex chaotic maps. A hardware platform is developed and the principal value attractors of these chaotic maps are experimentally captured. In addition, we explore the application of the hyperchaotic sequences generated by these complex chaotic maps to the pseudo-random number generators. Rigorous testing results validate the high degree of randomness exhibited by the generated pseudo-random numbers. Finally, we leverage the second branch of the 1D fractal map to generate a diverse range of fractal structures, further demonstrating the versatility and potential applications of the proposed approach.

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.

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.

High-dimensional (HD) chaotic map has wide applications in various research fields such as neural networks and secure communication. Designing HD chaotic maps with expected dynamics and robust hyperchaotic behaviors is an interesting but challenging topic. In this article, we propose an$n$-dimensional hyperchaotic map$(n\text{D}$-HCM) generation method on the basis of the Gershgorin-type theorem. First, the general form of the proposed$n\text{D}$-HCM is built using$n$parametric polynomials. Then, the entity and coefficient parameter matrices are configured according to the Gershorin-type theorem. Theoretical analysis shows that the generated$n\text{D}$-HCM has$n$positive Lyapunov exponents and thus can show robust hyperchaotic behaviors. Two examples of hyperchaotic map with specified equations are provided and their properties are analyzed to show the availability of the proposed method. Performance evaluations display that our$n\text{D}$-HCM possesses abundant properties and complex behaviors, and it can outperform some representative HD chaotic maps. Moreover, to show the application of our$n\text{D}$-HCM, we apply it to a secure communication scheme and the experimental results exhibit that it shows much better performance than these representative HD chaotic maps in resisting transmission noise.

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.

Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an$n$-dimensional polynomial chaotic system ($n\text{D}$-PCS) that can generate$n\text{D}$chaotic maps with any desired LEs. The$n\text{D}$-PCS is constructed from$n$parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the$n\text{D}$-PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the$n\text{D}$-PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from$n\text{D}$-PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of$n\text{D}$-PCS, we developed a chaos-based secure communication scheme. Simulation results show that$n\text{D}$-PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.

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.

When high-dimensional chaotic systems are applied to many practical applications, they are required to have robust and complex hyperchaotic behaviors. In this article, we propose a novel$n$D chaotic system construction method using the Pascal-matrix theory. First, a parametric Pascal matrix is constructed. Then, an$n$D chaotic system can be generated by using the parametric Pascal matrix as the parameter matrix of the system. Theoretical analysis shows that the generated$n$D chaotic systems have robust and complex chaotic behaviors, and they become$n$D Arnold Cat maps by fixing the parameters as some special values. Performance evaluations demonstrate that the$n$D chaotic systems have more complex chaotic behaviors and better distribution of outputs compared with existing HD chaotic systems. A 4-D Arnold Cat map and a 4-D chaotic map with hyperchaotic behaviors are generated as two examples. The two chaotic maps are then simulated on a microcontroller-based hardware platform and the chaotic sequences are tested to show good randomness.

When used in engineering applications, most existing chaotic systems may have many disadvantages, including discontinuous chaotic parameter ranges, lack of robust chaos, and easy occurrence of chaos degradation. In this article, we propose a two-dimensional (2-D) parametric polynomial chaotic system (2D-PPCS) as a general system that can yield many 2-D chaotic maps with different exponent coefficient settings. The 2D-PPCS initializes two parametric polynomials and then applies modular chaotification to the polynomials. Setting different control parameters allows the 2D-PPCS to customize its Lyapunov exponents in order to obtain robust chaos and behaviors with desired complexity. Our theoretical analysis demonstrates the robust chaotic behavior of the 2D-PPCS. Two illustrative examples are provided and tested based on numeral experiments to verify the effectiveness of the 2D-PPCS. A chaos-based pseudorandom number generator is also developed to illustrate the applications of the 2D-PPCS. The experimental results demonstrate that these examples of the 2D-PPCS can achieve robust and desired chaos, have better performance, and generate higher randomness pseudorandom numbers than some representative 2-D chaotic maps.

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.

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.