Discrete Map Research Articles

N-dimensional hyperchaotic discrete map with desired positive Lyapunov exponents and application to UART secure communication

n-dimensional hyperchaotic discrete map with desired positive Lyapunov exponents and application to UART secure communication

Investigating the dynamics of generalized discrete logistic map

Investigating the dynamics of generalized discrete logistic map

Identifying Arnold's tongue for digital oscillators through event-based control in phase-locked loops.

Digital phase-locked loops (PLLs) are essential feedback circuits for synchronizing signals in digital communication systems. While amplitude and phase vary continuously in analog oscillators, the amplitude remains constant in digital oscillators with dynamical variations manifesting exclusively through changes in the timing of signal transitions. In this work, we introduce a novel analytically solvable event-based model for phase-locking in digital PLLs that leverages the discrete nature of digital signals. By employing a sampled control strategy, we demonstrate one-to-one and higher ratios of frequency locking under positive and negative feedback. By discretizing the continuous control signal, we drive a discrete iterative map, which we then use to derive analytical expressions for bifurcation curves, analogous to Arnold's tongue in analog oscillators. This mathematical framework provides an analytical approach for the analysis of synchronization and phase-locking in digital oscillators. Furthermore, the event-based control presented in this work for digital oscillators paves the way for energy-efficient circuit design and optimized control strategies for future digital communication systems.

Hidden attractors in fractional-order discrete maps

Hidden attractors in fractional-order discrete maps

Dislocation-based finite element method for homogenized limit domain characterization of structured metamaterials

PurposeHeterogeneous and micro-structured materials have been the object of multiscale and homogenization techniques aimed at recognizing the elastic properties of the equivalent continuum. The proposed investigation deals with the mechanical characterization of the heterogeneous material structured metamaterials through analyzing the ultimate strength using the limit analysis of the Representative Volume Element (RVE). To get the desired material strength, a novel finite element formulation based on the derivation of self-equilibrated solutions through the finite elements devoted to calculating the lower bound theorem has been implemented together with the limit analysis in Melàn’s formulation.Design/methodology/approachThe finite element formulation is based on discrete mapping of Volterra dislocations in the structure using isoparametric representation. Using standard finite element techniques, the linear operator V, which relates the self-equilibrated internal solicitation to displacement-like nodal parameters, has been built through finite element discretization of displacement and strain.FindingsThe proposed work presented an elastic homogenization of the mechanical properties of an elementary cell with a geometry known in the literature, the isotropic truss. The matrix of elastic constants was calculated by subjecting the RVE to numerical load tests, simulated with a commercial FEM calculation code. This step showed the dependence of the isotropy properties, verified with Zener theory, on the density of the RVE. The isotropy condition of the material is only achieved for certain section ratios between body-centered cubic (BCC) and face-centered cubic (FCC), neglecting flexural effects at the nodes. The density that satisfies Zener’s conditions represents the isotropic geomatics of the isotropic truss.Originality/valueFor the isotropic case, the VFEM procedure was used to evaluate the isotropy of the limit domain and was compared with the Mises–Schleicher limit domain. The evaluation of residual ductility and dissipation energy allowed a measurement parameter for the limit anisotropy to be defined. The novelty of the proposal consisted in the formulation of both the linearized and the nonlinear limit locus of the material; hence, it furnished the starting point for further limit analysis of the structures whose elementary volume has been described through the proposed approach.

A new construction method of N-dimensional discrete sine hyperchaotic map

A new construction method of N-dimensional discrete sine hyperchaotic map

A 5D super-extreme-multistability hyperchaotic map based on parallel-cascaded memristors

A new 5-dimensional discrete memristors-based hyperchaotic map (5DMHM) is constructed by introducing four charge-controlled discrete memristors (DMs, coupled through parallel-cascade mixing operation) into the sine map. This 5DMHM has 4-dimensional space fixed points and its stability distributions are partitioned in coupling strength, memristor initial condition, and parameter planes. The dynamics of the map are investigated by employing 2D dynamical behavior and the first Lyapunov exponent (LE) distribution. The super extreme multistability is discovered through the basin of attraction, the initial-value-dependent LE spectra, bifurcation diagrams, the mean value of state variables, and trajectory plots. Moreover, a high level of complexity can be observed by calculating the spectral entropy complexity for various parameter planes. Subsequently, a microcontroller-based digital circuit is designed to implement the 5DMHM. The security analysis of the pseudorandom number generator (PRNG) designed by the 5DMHM is presented and the PRNG passes the NIST SP800-22 test. The comparison of the 5DMHM with the recent maps indicates a higher performance in secure communication and image encryption.

End-to-End Implicit Object Pose Estimation.

To accurately estimate the 6D pose of objects, most methods employ a two-stage algorithm. While such two-stage algorithms achieve high accuracy, they are often slow. Additionally, many approaches utilize encoding-decoding to obtain the 6D pose, with many employing bilinear sampling for decoding. However, bilinear sampling tends to sacrifice the accuracy of precise features. In our research, we propose a novel solution that utilizes implicit representation as a bridge between discrete feature maps and continuous feature maps. We represent the feature map as a coordinate field, where each coordinate pair corresponds to a feature value. These feature values are then used to estimate feature maps of arbitrary scales, replacing upsampling for decoding. We apply the proposed implicit module to a bidirectional fusion feature pyramid network. Based on this implicit module, we propose three network branches: a class estimation branch, a bounding box estimation branch, and the final pose estimation branch. For this pose estimation branch, we propose a miniature dual-stream network, which estimates object surface features and complements the relationship between 2D and 3D. We represent the rotation component using the SVD (Singular Value Decomposition) representation method, resulting in a more accurate object pose. We achieved satisfactory experimental results on the widely used 6D pose estimation benchmark dataset Linemod. This innovative approach provides a more convenient solution for 6D object pose estimation.

Design and analysis of discrete fractional-order chaotic map with offset-boosting behavior

Fractional calculus, as a more accurate tool for depicting the dynamics of complex systems, has been introduced into discrete chaotic maps. To further describe the offset-boosting behavior in discrete systems, a discrete fractional-order chaotic map (DFOCM) based on the Caputo difference operator is constructed. The mapping order of this fractional-order model controls the stability of the fixed point, thereby affecting the dynamic behavior of the map. The dynamics of DFOCM is studied using numerical simulation methods such as bifurcation diagrams and maximum Lyapunov exponents, revealing the presence of multistability. By comparing with integer-order map, it is found that DFOCM exhibit a larger chaotic region. Based on this, the difference between fractional order and integer order offset-boosting behavior is theoretically derived. Specifically, the offset-boosting behavior of fractional-order maps concerning mapping parameters is related to the initial state, which was further demonstrated through numerical simulations. SE complexity proves that the chaotic sequences generated by DFOCM have high unpredictability and pseudo-randomness. Finally, the proposed DFOCM is implemented on the DSP hardware platform, and the physical feasibility of numerical simulation is verified.

Design of a discrete memristive chaotic map: fractional-order memory, dynamics and application

In this paper, a discrete fracmemristor (DFM) model is derived based on the Caputo difference, and a new fractional-order chaotic map is designed. Dynamics of the proposed map is investigated in detail by means of Lyapunov exponent spectra, bifurcation diagrams, PE complexity and multistability analyses. Compared with the coupled discrete integer-order memristor (DIM), the map coupled with the DFM products richer dynamics, including larger attractor distribution, fewer numerically periodic windows, and higher complexity. Besides, the order becomes additional bifurcation parameter. Finally, the proposed map is implemented on Field-Programmable Gate Array (FPGA) platform, and applied in a pseudorandom number generator (PRNG), which further demonstrates its application value.

A new image encryption algorithm with feedback key mechanism using two-dimensional dual discrete quadratic chaotic map

A new image encryption algorithm with feedback key mechanism using two-dimensional dual discrete quadratic chaotic map

A High-Performance FPGA PRNG Based on Multiple Deep-Dynamic Transformations.

Pseudo-random number generators (PRNGs) are important cornerstones of many fields, such as statistical analysis and cryptography, and the need for PRNGs for information security (in fields such as blockchain, big data, and artificial intelligence) is becoming increasingly prominent, resulting in a steadily growing demand for high-speed, high-quality random number generators. To meet this demand, the multiple deep-dynamic transformation (MDDT) algorithm is innovatively developed. This algorithm is incorporated into the skewed tent map, endowing it with more complex dynamical properties. The improved one-dimensional discrete chaotic mapping method is effectively realized on a field-programmable gate array (FPGA), specifically the Xilinx xc7k325tffg900-2 model. The proposed pseudo-random number generator (PRNG) successfully passes all evaluations of the National Institute of Standards and Technology (NIST) SP800-22, diehard, and TestU01 test suites. Additional experimental results show that the PRNG, possessing high novelty performance, operates efficiently at a clock frequency of 150 MHz, achieving a maximum throughput of 14.4 Gbps. This performance not only surpasses that of most related studies but also makes it exceptionally suitable for embedded applications.

Simulink Modeling and Analysis of a Three-Dimensional Discrete Memristor Map

The memristor, a novel device, has been widely utilized due to its small size, low power consumption, and memory characteristics. In this paper, we propose a new three-dimensional discrete memristor map based on coupling a one-dimensional chaotic map amplifier with a memristor. Firstly, we analyzed the memristor model to understand its characteristics. Then, a Simulink model for this three-dimensional discrete memristor map was developed. Lastly, the complex dynamical characteristics of the system were analyzed via equilibrium points, bifurcation diagrams, Lyapunov exponent spectra, complexity, and multistability. This study revealed the phenomena of coexisting attractors and hyperchaotic attractors. Simulink modeling confirmed that the discrete memristors effectively enhanced the chaos complexity in the three-dimensional discrete memristor map. This approach addresses the shortcomings of randomness, the lack of ergodicity, and the small key space in a one-dimensional chaotic map, thereby enriching the theoretical analysis and circuit implementation of chaos.

Discrete model and non‐linear characteristics analysis of magnetic coupled resonant wireless power transfer system with constant power load

AbstractMagnetically coupled resonant wireless power transfer (MCR‐WPT) system with constant power load (CPL), finds extensive applications in military, industrial, and medical treatment. However, this system can easily exhibit complex non‐linear behaviors under certain parameter conditions due to the influences of constant power loads, switching devices, and feedback controllers. These behaviors limit the effectiveness of controllers and reduce the efficiency and stability of the system. It is necessary to study the non‐linear characteristics of the MCR‐WPT system with CPL. The stroboscopic discrete mapping of the MCR‐WPT system with CPL is established. Then the system's non‐linear dynamics are analyzed theoretically using a bifurcation diagram, the maximal Floquet multipliers, and the maximal Lyapunov exponent spectrum obtained from the proposed discrete mapping. The results show that the MCR‐WPT system with CPL will exhibit rich non‐linear dynamics with the variation of the power of CPL, such as cyclic fold bifurcation, Neimark–Scaker bifurcation, border collision bifurcations, chaos, etc. The excellent alignment of experimental and theoretical outcomes in corresponding states confirms the accuracy of the proposed discrete mapping and the nonlinear analysis of the system. The results of this study can provide a reference for selecting parameters for an actual MCR‐WPT system with CPL.

The hemodynamic response to co-occurring interictal epileptiform discharges and high-frequency oscillations localizes the seizure-onset zone.

To use intracranial electroencephalography (EEG) to characterize functional magnetic resonance imaging (fMRI) activation maps associated with high-frequency oscillations (HFOs) (80-250 Hz) and examine their proximity to HFO- and seizure-generating tissue. Forty-five patients implanted with intracranial depth electrodes underwent a simultaneous EEG-fMRI study at 3 T. HFOs were detected algorithmically from cleaned EEG and visually confirmed by an experienced electroencephalographer. HFOs that co-occurred with interictal epileptiform discharges (IEDs) were subsequently identified. fMRI activation maps associated with HFOs were generated that occurred either independently of IEDs or within ±200 ms of an IED. For all significant analyses, the Maximum, Second Maximum, and Closest activation clusters were identified, and distances were measured to both the electrodes where the HFOs were observed and the electrodes involved in seizure onset. We identified 108 distinct groups of HFOs from 45 patients. We found that HFOs with IEDs produced fMRI clusters that were closer to the local field potentials of the corresponding HFOs observed within the EEG than HFOs without IEDs. In addition to the fMRI clusters being closer to the location of the EEG correlate, HFOs with IEDs generated Maximum clusters with greater z-scores and larger volumes than HFOs without IEDs. We also observed that HFOs with IEDs resulted in more discrete activation maps. Intracranial EEG-fMRI can be used to probe the hemodynamic response to HFOs. The hemodynamic response associated with HFOs that co-occur with IEDs better identifies known epileptic tissue than HFOs that occur independently.

Controlling the Period-Bubbling Route to Chaos

We describe the period-bubbling attractors, their generation mechanism and chaos control strategy in nonlinear, unidimensional and two-parameter-dependent discrete maps. A theorem for the occurrence of sequential period-bubbling attractors leading to chaos, is presented. It delineates the role of a geometric parameter in the formation of anti-monotonic periodic sequences of each iterate of such discrete maps. The appearances of different period bubbles in two distinct Gaussian and cubic maps are demonstrated in accordance to the theorem. The delimited chaos through period-bubbling route is suppressed by employing linear density feedback with a proportional constant. The study discloses that the period-bubbling transitional route in unidimensional map cannot be suppressed by adopting constant feedback, which is in contrast to the control of period-doubling route to chaos. In Gaussian and cubic maps, the period-bubbling routes to chaos are gradually collapsed for linear density feedbacks, with the enhancements of the feedback strength parameter k. Interestingly, at [Formula: see text], the feedback-influenced Gaussian map experiences tangent bifurcation and chaos can re-enter into its dynamics through that scenario for [Formula: see text]. The tangent bifurcation in the feedback-affected Gaussian map is explained mathematically. However, no such transitional exchange has been perceived in cubic map, for augmented values of the feedback strength parameter.

Generating a Multi-Material Lattice Structure through a Modified Relative Density Mapping Algorithm

Multi-material lattice structures demonstrate superior mechanical advantages and lightweight potential, enabling it preferable in many engineering fields, especially with the rise of additive manufacturing. However, employing multi-scale topology optimization inherently involves computational complexities of designing the macromaterial distribution and microlattice geometric size simultaneously. In this paper, a modified relative density mapping (MRDM) method is proposed, which generates a multi-material lattice structure by projecting the continuum multi-material topology optimization results. The proposed method extends the standard relative density mapping (RDM) method to multi-material problem and improves the performance. To achieve such purpose, a new mapping relation considering multi-material density and stress information is introduced to generate mapping weight of materials, which is the most novel contribution. Second, to avoid intersection of different materials, two types of material mapping strategies, namely continuous mapping strategy, and discrete mapping strategy, are introduced to generate multi-material lattice structure using the mapping weight of materials. Several numerical examples are exhibited, which demonstrate that the proposed method is capable of generating a multi-material lattice structure with clear material interface and good performance.

A Two-Dimensional Discrete Memristor Map: Analysis and Implementation

In this paper, we present a novel two-dimensional discrete memristor map that is based on a discrete memristor model and a sine–arcsine one-dimensional map. First, an analysis is conducted on the memristor model to understand its characteristics. Then, the model is coupled with the sine–arcsine one-dimensional map to achieve the two-dimensional discrete memristor map. Our investigation reveals the presence of coexisting attractors and hyperchaotic attractors as the bifurcation parameters vary. Numerical simulations show that the discrete memristors effectively enhance the complexity of chaos in the sine–arcsine map. Furthermore, a digital circuit is designed to experimentally verify the new chaotic system. The research results can enrich the theoretical analysis and circuit implementation of chaos.

Symmetric Pseudo-Multi-Scroll Attractor and Its Application in Mobile Robot Path Planning

The symmetric multi-scroll strange attractor has shown great potential in chaos-based applications due to its high complexity in phase space. Here, the approach of symmetrization is employed for attractor doubling to generate pseudo-multi-scroll attractors in a discrete map, where a carefully selected offset constant is the key to organizing coexisting attractors. By choosing the Hénon map to generate the pseudo-multi-scroll attractor and implementing the digital circuit on a microcontroller, this study fills a significant gap in the research on discrete chaotic systems. The complexity performance is further validated using a pseudo-random number generator, demonstrating substantial academic contributions to the field of chaos theory. Additionally, a pseudo-multi-scroll attractor-based squirrel search algorithm is first developed, showcasing its practical application in mobile robot path planning. This work not only advances the theoretical understanding of chaotic systems but also provides practical methods for implementation in digital systems, offering valuable insights for policy-making in advanced robotic systems and intelligent manufacturing.

A visual security multi-key selection image encryption algorithm based on a new four-dimensional chaos and compressed sensing

In this article, a visual security image encryption algorithm based on compressed sensing is proposed. The algorithm consists of two stages: the compression and encryption stage and the embedding stage. The key streams in the compression and encryption stage are generated by a newly constructed four-dimensional discrete chaotic map, while the Gaussian measurement matrix is generated by a Chebyshev map, and both of their generations are related to the feature code of the carrier image, which enhances the security of the ciphertext. In the compression and encryption stage, a scrambling-cyclic shift-diffusion encryption structure is adopted for the compressed image in which the shift number in the cyclic shift stage and the diffusion key streams are dynamically changed according to each pixel value, so the algorithm can resist chosen plaintext attack. In the embedding stage, the carrier image is first subjected to integer wavelet transform to obtain the high-frequency and low-frequency components of the image, and then the intermediate ciphertext information is embedded into its high-frequency components. Finally, the carrier image is subjected to inverse integer wavelet transform to obtain a visually secure ciphertext image. The experimental results and security analysis indicate that the encryption scheme has a large key space, high decryption key sensitivity, similar histogram distribution between the carrier image and the visual security ciphertext image, and good robustness to noise attacks.