Discrete Map Research Articles

Deep learning delay coordinate dynamics for chaotic attractors from partial observable data.

A common problem in time-series analysis is to predict dynamics with only scalar or partial observations of the underlying dynamical system. For data on a smooth compact manifold, Takens' theorem proves a time-delayed embedding of the partial state is diffeomorphic to the attractor, although for chaotic and highly nonlinear systems, learning these delay coordinate mappings is challenging. We utilize deep artificial neural networks (ANNs) to learn discrete time maps and continuous time flows of the partial state. Given training data for the full state, we also learn a reconstruction map. Thus, predictions of a time series can be made from the current state and several previous observations with embedding parameters determined from time-series analysis. The state space for time evolution is of comparable dimension to reduced order manifold models. These are advantages over recurrent neural network models, which require a high-dimensional internal state or additional memory terms and hyperparameters. We demonstrate the capacity of deep ANNs to predict chaotic behavior from a scalar observation on a manifold of dimension three via the Lorenz system. We also consider multivariate observations on the Kuramoto-Sivashinsky equation, where the observation dimension required for accurately reproducing dynamics increases with the manifold dimension via the spatial extent of the system.

Influence of Parametric Symmetry on the Dynamics of 3D Sinusoidal Discrete Systems

The discrete system serves an important role in mimicking collective dynamics found in continuous dynamical systems, which are relevant to many realistic natural and artificial systems. To investigate the dynamical transition of a discrete system, we employ three-dimensional sinusoidal discrete maps with an additional self feedback factor. Specifically, we focus on dynamical transitions with respect to the bifurcation parameter, sine function amplitude, and intensity of self feedback factors. We demonstrate the presence of symmetry in relation to parametric variation using two parameter diagrams. The study is then expanded to the network of sine maps in the presence of self-feedback factor. We discover that negative feedback exhibits the transition from cluster state to synchronization while raising the coupling strength for small-world network interactions. Furthermore, increasing feedback from negative to positive causes the transition from synchronization to desynchronization via chimera state for various complex network connectivities.

A whole discretization method for milling stability prediction considering the discrete vibration velocities

The discretization methods have been widely used for chatter stability prediction and can be divided into two categories: semi-discretization methods (SDMs) by discretizing the time-delayed displacements, and full-discretization methods (FDMs) by discretizing both the present and time-delayed displacements. In this paper, a whole discretization method is proposed to achieve a faster and more accurate prediction for the milling stability. In addition to the discrete vibration displacements, the discrete vibration velocities are included in the solution by rebuilding the integrated matrix. The prediction error caused by the separate discretization of the cutting force coefficient matrix and vibration state variables can be avoided. The milling stability is determined with the discrete map of the state based on the Floquet theory. The proposed model is compared with the zeroth-order SDM and the first-order FDM which also approximate the solution with two neighboring discrete state variables. It is shown that the local discretization error based on the proposed method achieves o((Δt)5), which is three orders higher compared to the zeroth-order SDM and the first-order FDM. In addition, when the same prediction errors are required, the computational time of the proposed method is 11–14% of the zeroth-order SDM, and 24–27% of the first-order FDM. The results prove that considering vibration velocity can increase the efficiency and accuracy of the chatter stability prediction compared to existing SDM and FDM algorithms.

Low‐light image enhancement via span correction function and discrete mapping model

This paper proposes a new low-light image enhancement method, which we call the Local Discrete Mapping Method. The new method limits the processing range to small areas with a high information relevance, which can better coordinate the enhancement quality of each area. First, the discrete mapping relationship of pixels (called discrete mapping points) globally occupying a small part of the critical gray value was extracted and designated to keep the enhancement amplitude of each local area consistent. Then, other free mapping points were adjusted according to the local features to achieve the best visual effect in each local area. In addition, this paper also proposes a span correction function that takes the gray span between local pixels as the adjustment object. The function can preserve the gray difference between freely mapped pixels to the maximum and significantly reduce detailed damage in the local area. Finally, we used 1500 test images and eleven objective evaluation indicators in the public dataset to comprehensively test the seven methods. The experimental results showed that the proposed method has an excellent dark area quality enhancement, brightness detail protection, overall noise suppression, and processing speed. It is significantly better than similar methods in terms of visual quality and quantitative testing.

Exploring non-invertible symmetries in free theories

Symmetries corresponding to local transformations of the fundamental fields that leave the action invariant give rise to (invertible) topological defects, which obey group-like fusion rules. One can construct more general (codimension-one) topological defects by specifying a map between gauge-invariant operators from one side of the defect and such operators on the other side. In this work, we apply such construction to Maxwell theory in four dimensions and to the free compact scalar theory in two dimensions. In the case of Maxwell theory, we show that a topological defect that mixes the field strength F and its Hodge dual ⋆F can be at most an SO(2) rotation. For rational values of the bulk coupling and the θ-angle we find an explicit defect Lagrangian that realizes values of the SO(2) angle φ such that cos φ is also rational. We further determine the action of such defects on Wilson and ’t Hooft lines and show that they are in general non-invertible. We repeat the analysis for the free compact scalar ϕ in two dimensions. In this case we find only four discrete maps: the trivial one, a ℤ2 map dϕ → −dϕ, a \U0001d4af-duality-like map dϕ → i ⋆ dϕ, and the product of the last two.

Novel image cryptosystem based on new 2D hyperchaotic map and dynamical chaotic S-box

Chaotic systems are widely used in image encryption due to their sensitivity to initial values, ergodicity, and other properties; many image encryption algorithms based on chaotic systems have been studied in the past few years. To obtain a more secure encryption algorithm, this work firstly proposes a new two-dimensional discrete hyperchaotic map, which has a wider continuous chaotic interval, larger Lyapunov exponents and passed all NIST and part of TestU01 tests. Then, we apply the proposed map to generate S-boxes and combine them in pairs; finally, twelve S-boxes are obtained, and the elements of the plaintext image are grouped, each group of pixels is summed, and modular operations are used to specify specific S-boxes. Next, each set of elements is bitwise XOR with the corresponding S-box. Finally, the cipher image is obtained by scrambling using chaotic signal. Experiments show that compared with some other encryption algorithms, the proposed S-box-based encryption method has higher security, and it resists to common attacks.

Design, Hardware Implementation on FPGA and Performance Analysis of Three Chaos-Based Stream Ciphers

In this paper, we come up with three secure chaos-based stream ciphers, implemented on an FPGA board, for data confidentiality and integrity. To do so, first, we performed the statistical security and hardware metrics of certain discrete chaotic map models, such as the Logistic, Skew-Tent, PWLCM, 3D-Chebyshev map, and 32-bit LFSR, which are the main components of the proposed chaotic generators. Based on the performance analysis collected from the discrete chaotic maps, we then designed, implemented, and analyzed the performance of three proposed robust pseudo-random number generators of chaotic sequences (PRNGs-CS) and their corresponding stream ciphers. The proposed PRNGs-CS are based on the predefined coupling matrix M. The latter achieves a weak mixing of the chaotic maps and a chaotic multiplexing technique or XOR operator for the output function. Therefore, the randomness of the sequences generated is expanded as well as their lengths, and divide-and-conquer attacks on chaotic systems are avoided. In addition, the proposed PRNGs-CS contain polynomial mappings of at least degree 2 or 3 to make algebraic attacks very difficult. Various experimental results obtained and analysis of performance in opposition to different kinds of numerical and cryptographic attacks determine the high level of security and good hardware metrics achieved by the proposed chaos system. The proposed system outperformed the state-of-the-art works in terms of high-security level and a high throughput which can be considered an alternative to the standard methods.

Response to: Commentary: Multidimensional discrete chaotic maps

Response to: Commentary: Multidimensional discrete chaotic maps

Control Parameter Stability Domain of BBMC Speed Control System with Finite-Time Control

The buck-boost matrix converter (BBMC) asynchronous motor speed control system based on finite time control has a good speed control performance, but its system stability is closely related to the selection of its control parameters. Therefore, a method for determining the stability region of control parameters is proposed. The discrete iterative mapping model of the BBMC speed control system is established, and the control function of the BBMC speed control system based on finite time control is determined. The BBMC output voltage sampling distribution diagram is used to analyze the influence of each control parameter in the finite time control on the stability of the speed control system. On this basis, the study determined the value range of each control parameter to realize the stable operation of the system. Finally, the correctness of the above theoretical analysis was verified through the BBMC output voltage waveform diagram and experiments. The stability region of control parameters can effectively ensure the stable operation of BBMC speed control system based on finite time control.

Hidden multistability of fractional discrete non-equilibrium point memristor based map

At present, the multistability analysis in discrete nonlinear fractional-order systems is a subject that is receiving a lot of attention. In this article, a new discrete non-equilibrium point memristor-based map with γ − th Caputo fractional difference is introduced. In addition, in the context of the commensurate and non-commensurate instances, the nonlinear dynamics of the suggested discrete fractional map, such as its multistability, hidden chaotic attractor, and hidden hyperchaotic attractor, are investigated through several numerical techniques, including Lyapunov exponents, phase attractors, bifurcation diagrams, and the 0 − 1 test. These dynamic behaviors suggest that the fractional discrete memristive map has a hidden multistability. Finally, to validate the presence of chaos, a complexity analysis is carried out using approximation entropy (ApEn) and the C 0 measure. The findings show that the model has a high degree of complexity, which is affected by the system parameters and the fractional values.

Routes toward chaos in a memristor-based Shinriki circuit.

In this paper, the complex routes to chaos in a memristor-based Shinriki circuit are discussed semi-analytically via the discrete implicit mapping method. The bifurcation trees of period-m (m = 1, 2, 4 and 3, 6) motions with varying system parameters are accurately presented through discrete nodes. The corresponding critical values of bifurcation points are obtained by period-double bifurcation, saddle-node bifurcation, and Neimark bifurcation, which can be determined by the global view of eigenvalues analysis. Unstable periodic orbits are compared with the stable ones obtained by numerical methods that can reveal the process of convergence. The basins of attractors are also employed to analyze the coexistence of asymmetric stable periodic motions. Furthermore, hardware experiments are designed via Field Programmable Gate Array to verify the analysis model. As expected, an evolution of periodic motions is observed in this memristor-based Shinrik's circuit and the experimental results are consistent with that of the calculations through the discrete mapping method.

Periodic gaits and flip bifurcation of a biped robot walking on level ground with two feasible switching patterns of motion

In this article, a biped robot walking on horizontal ground with two feasible switching patterns of motion (two-phase gait and three-phase gait) is presented. By using the first-order Taylor approximate at the equilibrium point, a simplified linear continuous dynamic equation is obtained to discuss the walking dynamics of the biped robot. Conditions for the existence and stability of period-1 gaits ( P ( 1 , 2 ) , P ( 1 , 3 ) ) and period-2 gaits ( P ( 2 , 2 , 2 ) , P ( 2 , 2 , 3 ) , P ( 2 , 3 , 3 ) ) are obtained by using a discrete map. Among the periodic gaits, the P ( 2 , 2 , 3 ) type gait has never been reported in previous studies. Flip bifurcation of periodic gait is investigated. Numerical results for periodic gaits and bifurcation diagram are in good agreement with the theoretical analysis.

On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.

A novel simultaneous permutation–diffusion image encryption scheme based on a discrete space map

The constant increase of devices connected to the internet and the development and spread of technologies such as 5G leads to substantial growth in data-sharing over non-secure channels in real-time applications, augmenting the need for fast cryptography algorithms. Algorithms such as the Advanced Encryption Standard (AES) are not optimized for image encryption, and hence, as an alternative, chaotic encryption strategies have been explored by researchers. However, due to the use of floating-point arithmetic and inefficient memory access, the performance of these chaotic ciphers is insufficient for the encryption of high-resolution images. Thus, this work proposes generating the chaotic sequences based on fixed-point arithmetic to increase computational efficiency, and using a simultaneous permutation–diffusion operation and block encryption to deal with the inefficient memory access. The proposed scheme leads to a throughput 353 times higher than the AES, 75 times higher than the previously higher value in the literature and 11 times higher than the Ascon algorithm. The cipher security evaluation is performed based on statistical attacks, with metrics such as Pearson, histogram variance, Shannon’s entropy, correlation coefficients, and NIST tests; differential attacks, with the NPCR, UACI, and MSE metrics; chosen plain-text, with the SSIM metric; brute-force, with the key-space; noise and occlusion attacks, with the PSNR metric.

Building discrete maps with memristor and multiple nonlinear terms

Designing new discrete maps attracts numerous studies in the literature. This work introduces a way to build discrete maps by combining a memristor and multiple nonlinear terms. Different approaches for building memristive maps have been reviewed. We present a general model for constructing discrete maps with a memristor and two nonlinear terms. New memristive maps are reported. In addition, we have studied the MNFM1 map in more details to discovery its specific features. It is possible to extend our approach to obtain a memristive map with multiple nonlinear terms. Memristive maps can be used in civil aviation applications.

Recurrent Spectral Network (RSN): Shaping a discrete map to reach automated classification

A novel strategy to automated classification is introduced which exploits a fully trained dynamical system to steer items belonging to different categories towards distinct asymptotic target destinations. These latter are incorporated into the model by taking advantage of the spectral decomposition of the operator that rules the linear evolution across the processing network. Non-linear terms act for a transient and allow to disentangle the data supplied as initial condition to the discrete dynamical system. The system effectively aligns along assigned directions, which reflect the specificity of the provided input and that are encoded in the loss function via suitable spectral projections. The network can be equipped with several memory kernels which can be sequentially activated for serial datasets handling. Our novel approach to classification, that we here term Recurrent Spectral Network (RSN), is successfully challenged against a simple test-bed model, created for illustrative purposes, as well as a standard dataset for image processing training.

Fully fixed-point integrated digital circuit design of discrete memristive systems

In this paper, the fully fixed-point digital integrated circuits of discrete memristive systems are designed with very low circuit area cost, where the two-stage pipeline and multi-cycle architecture is employed. Firstly, the mathematical models of the discrete memristor and a memristor chaotic map are presented and dynamics are analyzed. Secondly, data flow design is carried out and the feasibility of the fixed-point model is discussed, which provides a foundation for the further circuit design. Thirdly, different architectures are designed and we get the optimal results. As a result, gate area of the discrete memristor is 40288 μm2 and the equivalent gate counts are 4588, while gate area of the discrete memristor map is 40424 μm2 and the equivalent gate counts are 4604. Moreover, a chaotic pseudo random sequence generator is designed and the NIST test is passed. It provides the foundation for the further applications of the discrete memristor and discrete memristor chaotic systems.

A DISCRETE MEMRISTOR COUPLED TWO-DIMENSIONAL GENERALIZED SQUARE HYPERCHAOTIC MAPS

In this paper, a new discrete chaotic map is constructed by introducing a discrete memristor in two-dimensional generalized square maps to enhance its chaotic performance. First, the fixed points of the new maps are analyzed, and the effects of different parameters on the system performance are investigated by bifurcation diagrams, Lyapunov exponential spectra and phase diagrams. Second, the fixed points of the new maps are analyzed, and the effects of different parameters on the system performance are investigated by bifurcation diagrams, Lyapunov exponential spectra and phase diagrams. The distinctive characteristic of a discrete system is the coexistence of various types of attractors, and there is coexistence of hyperchaos and cycles in the present maps. It is worth mentioning that symmetric chaotic attractors with different positive and negative parameters are found during the study. In addition, the phenomenon of state transition between chaos and cycles is also found. Finally, the discrete maps are designed and implemented using a DSP platform. The results of the study provide a reference for the application of discrete amnesic chaotic maps.

MULTI-IMAGE HYBRID ENCRYPTION ALGORITHM BASED ON PIXEL SUBSTITUTION AND GENE THEORY

A hybrid encryption scheme for multi-image is proposed in this paper, which can encrypt different types and different sizes of images simultaneously. First, the order of the discrete map is extended from integer order to unequal fractional order. By choosing suitable parameters, the map exhibits chaotic behavior, and using the map for the designed image encryption system can enlarge the key space of the cryptosystem. Then, the plain images are pixel substituted. The chaotic sequences produced from the unequal fractional-order discrete map are shaped and sorted; the index sequences generated by the sorting are used to replace plain image pixels. After plain image pixels are replaced, DNA encoding, selective diffusion, and gene exchange are performed. The statistical properties of the images are masked by the diffusion algorithm. Finally, simulation experiments and security test results show that the designed multi-image hybrid encryption algorithm is effective and secure.

Multistability near the boundary of noise-induced synchronization in ensembles of uncoupled chaotic systems

The aim of this work is to study the possibility of the existence of multistability near the boundary of noise-induced synchronization in chaotic continuous and discrete systems. Ensembles of uncoupled Lorenz systems and logistic maps being under influence of a common source of white noise have been chosen as an object under study. Methods. The noise-induced synchronization regime detection has been performed by means of direct comparison of the system states being under influence of the common noise source and by calculation of the synchronization error. To determine the presence of multistability near the boundary of this regime, the multistability measure has been calculated and its dependence on the noise intensity has been obtained. In addition, for fixed moments of time, the basins of attraction of the synchronous and asynchronous regimes have been received for one of the systems driven by noise for fixed initial conditions of the other system. The result of the work is a proof of the presence of multistability near the boundary of noise-induced synchronization. Conclusion. It is shown that the regime of intermittent noise-induced synchronization, as well as the regime of intermittent generalized synchronization, is characterized by multistability, which manifests itself in this case as the existence in the same time interval of the synchronous behavior in one pair of systems being under influence of a common noise source, whereas in the other pair the asynchronous behavior is observed. The found effect is typical for both flow systems and discrete maps being under influence of a common noise source. It can find an application in the information and telecommunication systems for improvement the methods for secure information transmission based on the phenomenon of chaotic synchronization.