One-dimensional Maps Research Articles

The dynamics of one-dimensional quasi-affine maps

We study the dynamics of the one-dimensional quasi-affine map x ↦ ⌊ λx + μ ⌋ , providing a complete description of the map's periodic points, and of the limit points of every x ∈ R under the map, for all real parameter values. Specifically, we establish the existence of regions of parameter values for which the map possesses n fixed points for all n ∈ N 0 ∪ { ∞ } , an explicit formula for the number of 2-cycles possessed by the map, and the ω-limit set of any x ∈ R under the map, which, depending on the parameter values, is either a singleton of a fixed point, a 2-cycle, { − ∞ , ∞ } , { ∞ } , or { − ∞ } .

A plaintext-related and ciphertext feedback mechanism for medical image encryption based on a new one-dimensional chaotic system

Abstract In recent years, Plaintext-Related Image Encryption (PRIE) algorithms have been introduced, demonstrating a commendable level of plaintext sensitivity to resist chosen plaintext attack (CPA). However, these approaches suffer from several drawbacks, including inability to fully reconstruct the original image, limited practical value and excessive computational demands etc.. Moreover, the exponential expansion of medical data necessitates the formulation of more secure and efficient encryption algorithms. In this paper, firstly, a novel one-dimensional chaotic map, designated as 1D-SAM, which strikes an excellent balance between structural complexity and chaotic performance is proposed. The 1D-SAM achieve a larger chaotic range and an elevated Lyapunov exponent, signifying enhanced dynamical complexity. Subsequently, we devise a lightweight medical image encryption system leveraging the 1D-SAM and an innovative diffusion architecture, termed the plaintext-related and ciphertext feedback mechanism(PRCFM). This encryption system is a symmetric-key cryptosystem, eliminating the need for transmitting supplementary data beyond the secret keys to the recipient. Notably, the encrypted image maintains identical dimensions to its original counterpart and is fully recoverable. Complete simulation experiments were conducted on a personal computer equipped with MATLAB R2021a, OS Windows 11, 2.60GHz CPU and 16GB RAM. The experimental results indicate that our encryption system, employing a single permutation-diffusion round, efficiently encrypts a 512×512 image in approximately 0.2854 seconds. Leveraging the advantages of the PRCFM, our approach demonstrates superior plaintext sensitivity, achieving an average number of pixels changing rate (NPCR) of 99.6051 and a unified average changed intensity (UACI) of 33.4452. In summary, our work addresses key limitations of contemporary encryption frameworks, exhibiting acceptable performance in both encryption speed and security strength.

From One-dimensional to Multidimensional Map Neural Networks

In the contemporary landscape, artificial neural networks transcend their traditional role of function approximation and have found diverse applications in fields such as image classification, machine translation, speech recognition, and natural language processing. In some datasets, traditional architectures exhibit low test and training accuracy, coupled with high loss and prolonged training times. This study aims to introduce innovative neural network architectures that outperform conventional models. The research presents a novel framework integrating one-dimensional and multidimensional map neural networks, consisting of three distinct architectures: 1D-Map, 2D-Map, and 3D-Map. A systematic performance comparison with traditional models is conducted following the implementation of these architectures. The evaluation spans four datasets, encompassing the domains of heat treatment of electroless Ni-P nano coatings, letter recognition, combined cycle power plants, and Seoul bike sharing demand. In the heat treatment dataset, the proposed 3D-Map architecture reached 0.055 higher average test accuracy than traditional MLP architecture. In the letter recognition dataset, the test accuracy of 3D-Map architecture was 0.0523 higher than the test accuracy of the LSTM architecture. In the combined cycle power plant dataset, the test accuracy of the 3D-Map architecture was 0.0612 more than the test accuracy of the MLP architecture. In the Seoul bike-sharing demand dataset, the test accuracy of the 2D-Map architecture was 0.0696 higher than the test accuracy of the LSTM architecture. The study's findings underscore the consistently superior performance of the proposed architectures compared to their traditional counterparts.

Mean field responses in disordered systems: an example from nonlinear MHD

Understanding the generation of large-scale magnetic fields and flows in magnetohydro-dynamical (MHD) turbulence remains one of the most challenging problems in astrophysical fluid dynamics. Although much work has been done on the kinematic generation of large-scale magnetic fields by turbulence, relatively little attention has been paid to the much more difficult problem in which fields and flows interact on an equal footing. The aim is to find conditions for long-wavelength instabilities of stationary MHD states. Here, we first revisit the formal exposition of the long-wavelength linear instability theory, showing how long-wavelength perturbations are governed by four mean field tensors; we then show how these tensors may be calculated explicitly under the ‘short-sudden’ approximation for the turbulence. For MHD states with relatively little disorder, the linear theory works well: average quantities can be readily calculated, and stability to long-wavelength perturbations determined. However, for disordered basic states, linear perturbations can grow without bound and the purely linear theory, as formulated, cannot be applied. We then address the question of whether there is a linear response for sufficiently weak mean fields and flows in a dynamical (nonlinear) evolution, where perturbations are guaranteed to be bounded. As a preliminary study, we first address the nature of the response in a series of one-dimensional maps. For the full MHD problem, we show that in certain circumstances, there is a clear linear response; however, in others, mean quantities – and hence the nature of the response – can be difficult to calculate.

Enhanced Chaotic Pseudorandom Number Generation Using Multiple Bernoulli Maps with Field Programmable Gate Array Optimizations

Certain methods for implementing chaotic maps can lead to dynamic degradation of the generated number sequences. To solve such a problem, we develop a method for generating pseudorandom number sequences based on multiple one-dimensional chaotic maps. In particular, we introduce a Bernoulli chaotic map that utilizes function transformations and constraints on its control parameter, covering complementary regions of the phase space. This approach allows the generation of chaotic number sequences with a wide coverage of phase space, thereby increasing the uncertainty in the number sequence generation process. Moreover, by incorporating a scaling factor and a sine function, we develop a robust chaotic map, called the Sine-Multiple Modified Bernoulli Chaotic Map (SM-MBCM), which ensures a high degree of randomness, validated through statistical mechanics analysis tools. Using the SM-MBCM, we propose a chaotic PRNG (CPRNG) and evaluate its quality through correlation coefficient analysis, key sensitivity tests, statistical and entropy analysis, key space evaluation, linear complexity analysis, and performance tests. Furthermore, we present an FPGA-based implementation scheme that leverages equivalent MBCM variants to optimize the electronic implementation process. Finally, we compare the proposed system with existing designs in terms of throughput and key space.

Extreme fractal dimension at periodicity cascades in parameter spaces.

In the parameter spaces of nonlinear dynamical systems, we investigate the boundaries between periodicity and chaos and unveil the existence of fractal sets characterized by a singular fractal dimension that deviates greatly from the fractal sets in their vicinity. This extreme fractal dimension stands out from the typical value previously considered universal for these parameter boundaries. We show that such singular fractal sets dwell along parameter curves, called extreme curves, that intersect periodicity cascades at their centers of stability across all scales of parameter spaces. The results reported here are generally demonstrated for the class of one-dimensional maps with at least two control parameters. Generalizations to other classes of systems are possible.

Semi-visual obfuscation image encryption algorithm based on [formula omitted]-type chaotic amplifier and self-hiding fuzzy

The level of privacy may vary across different parts of an image. This paper proposes a semi-visual obfuscation algorithm for images that takes into account the varying levels of privacy in different areas of the image. Firstly, we present a novel One-dimensional Uniform Chaotic Amplifier (1_DUCA) aimed at expanding the parameter range and enhancing the uniformity of the standard one-dimensional chaotic map. Next, we employ a detection algorithm or autonomous frame selection to identify the precise location of the area with strong privacy. Finally, we apply noise to blur the selected area and conceal vital bit information within the image. At this point, the image has certain visual effects, and only people with prior knowledge can recognize the image. Furthermore, in the last stage of image encryption, we employ a weight scrambling and high-low bit coupled diffusion technology to completely obscure the visual effects of the image. It is noteworthy that the experimental results and performance analysis have verified the practicality and security of the encryption algorithm. Moreover, they have also demonstrated the robust amplification effect of the employed amplifier.

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.

Mechanism of multistability in chaotic maps.

This research aims to investigate the mechanisms of multistability in chaotic maps. The study commences by examining the fundamental principles governing the development of homogeneous multistability using a basic one-dimensional chain-climbing map. Findings suggest that the phase space can be segmented into distinct uniform mediums where particles exhibit consistent movement. As critical parameter values are reached, channels emerge between these mediums, resulting in deterministic chaotic diffusion. Additionally, the study delves into the topic of introducing heterogeneous factors on the formation of heterogeneous multistability in the one-dimensional map. A thorough examination of phenomena such as multistate intermittency highlights the intimate connection between specific phase transition occurrences and channel formation. Finally, by analyzing two instances-a memristive chaotic map and a hyperchaotic map-the underlying factors contributing to the emergence of multistability are scrutinized. This study offers an alternative perspective for verifying the fundamental principles of homogenous and heterogeneous multistability in complex high-dimensional chaotic maps.

A secure fractal compression scheme based on irregular Latin square, Julia and 2D-FCICM

With the proliferation of digital image data, ensuring its secure and efficient storage and transmission has become important. One-dimensional (1D) chaotic maps come with their set of limitations, such as chaotic interval discontinuity and simple chaotic behavior. To increase the complexity of chaotic systems, this paper proposed a novel 2D fractal coupled model (2D-FCM) to generate chaotic systems and extend it to n-dimensional chaotic systems. Meanwhile, the newly generated 2D-FCICM is employed in secure fractal compression. Compared to existing chaotic systems, the pseudo-random sequences generated by the 2D-FCICM exhibit better randomness and complexity. To enhance the plaintext sensitivity of the algorithm, a localized magnified fractal image is produced using plaintext information. Control parameters for the chaotic system are then derived from both the fractal and the original images. Given the compressed data's I × 5 size, this study proposed a technique for generating irregular Latin squares and integrated it into the scrambling phase. The experimental findings indicate that this scheme boasts high security levels, strong plaintext sensitivity and key sensitivity, and is resilient against various statistical attacks.

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.

Pseudo-random number generation based on spatial chaotic map of Logistic type and its cryptographic application

The pseudo-random number generator (PRNG) based on chaos has been widely used in the fields of digital communication, cryptography and computer simulation. In this paper, we study a new PRNG based on spatial surface chaotic system (SSCS), which is constructed based on coupling mapping lattice (CML) and one-dimensional Logistic chaotic map. To verify the performance of this generator, we study its nonlinear dynamic properties, including the Lyapunov exponent, ergodicity and bifurcation phenomena. Moreover, we analyze the cryptographic properties such as key space, key sensitivity, correlation, histogram and information entropy, while performing NIST SP800-22 test and TestU01 test for the randomness of this system. Theoretical analysis and numerical simulation results show that the PRNG proposed in this paper has good complexity, ergodicity, sensitivity and randomness. Moreover, the key space can increase dynamically with the increase of encrypted data, especially suitable for the protection of big data such as multimedia, which is more advantageous than most existing chaotic mapping. The results of this paper will provide a new idea for the study of chaotic cryptography, and motivate the study of new discrete chaotic models with desirable statistical properties.

Color image encryption scheme for distributed architecture with SCFP chaotic map

Image protection mechanism in distributed cloud network is an essential component of information security field. In this paper, a novel one-dimensional sine-cosine fractional power chaotic map (SCFP) is proposed. Results of various dynamical system tests illustrate that SCFP exhibits superior chaotic behavior over its infinite positive real parameter range, whose complexity and unpredictability can guarantee the strength of image cryptosystem. Furthermore, a color image encryption scheme tailored for distributed architecture is devised. Firstly, a hybrid cryptographic mechanism is designed to perform diffusion and confusion encryption for image data and ECC public key encryption for intermediate keys. Secondly, the diffusion structure elevates processing units to row-column level, and the diffusion order is dictated by a pseudo-random sequence generated by SCFP. Thirdly, the confusion structure extends the unbiased and efficient Fisher-Yates algorithm into a 2D space, and adopts a design of dual plaintext-related key. Lastly, three techniques namely QOI lossless compression, DE information embedding and threshold secret sharing are integrated to resolve issues of data volume inflation, key synchronization difficulty and poor fault tolerance. Simulation experiments conducted on multiple color images demonstrate that the proposed scheme offers significant ciphertext randomness, sufficiently large key space and strong key sensitivity, which can ensure the integrity of image data and resist various typical cryptographic attacks, and outperforms existing schemes oriented to centralized architecture in terms of security and efficiency.

Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map

We study a family of one-dimensional quasi-periodically forced maps [Formula: see text], where [Formula: see text] is real, [Formula: see text] is an angle, and [Formula: see text] is an irrational frequency, such that [Formula: see text] is a real piecewise-linear map with respect to [Formula: see text] of certain kind. The family depends on two real parameters, [Formula: see text] and [Formula: see text]. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For [Formula: see text] and any [Formula: see text] there is only one continuous invariant curve. For [Formula: see text] there exists a smooth map [Formula: see text] such that: (a) For [Formula: see text], [Formula: see text] has two continuous attracting invariant curves and one continuous repelling curve; (b) For [Formula: see text] it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For [Formula: see text] it has one continuous attracting invariant curve. The case [Formula: see text] is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family [Formula: see text] for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when [Formula: see text].

Synchronization effects in multiplex networks of chaotic maps with memristive interlayer coupling

Models of two-layer multiplex networks of one-dimensional maps with memristive interlayer coupling are studied. In the absence of coupling, the layers exhibit different types of chimera states. Two types of maps are considered as the elements of networks: modified Ricker maps and cubic maps. The effects of complete and partial synchronization of the spatiotemporal dynamics of layers are observed in the case of a homogeneous and inhomogeneous network, respectively. The dependence of the synchronization threshold for the coupling parameter on the initial states of the memristors has been established. This dependence is partially preserved in the case of non-ideal memristors. Diagrams of the synchronization error are constructed on the plane of the initial state variable of memristors and the interlayer coupling coefficient. The effect of noise in memristive coupling elements on synchronization is considered.

Multi-dimension-precision chaotic encryption mechanism for Internet of Things

The massive amount of information processed in the Internet of Things(IoT) environment requires effective encryption schemes to ensure secure transmission. However, traditional encryption methods such as Elliptic Curve Cryptography (ECC) or Data Encryption Standard (DES) are not suitable for IoT environments where large amounts of data need to be quickly encrypted. In this paper, a multi-precision encryption mechanism based on multi-dimensional chaotic system is proposed. Firstly, the proposed scheme uses floating-point, 32-bit, 24-bit, and 16-bit data accuracy respectively, which can effectively reduce the complexity through fixed-point calculation. Secondly, aiming at the problem that traditional chaotic dynamic analysis methods such as the Lyapunov index cannot be applied, an analysis scheme based on complexity measure is proposed. Finally, a free dimension encryption mechanism based on Logistic, Tent, and PWLCM is proposed. By adding random disturbance to a one-dimensional chaotic map, the weak randomness of the single chaotic sequence is overcome effectively. Finally, the security analysis of the encryption scheme is carried out, and the result proves that the scheme can effectively resist the common attack means. In the 3D chaotic system, the encryption effect of 24-bit data accuracy is close to floating point accuracy, which is suitable for the IoT environment with large data volume, high encryption efficiency, and vulnerability.

Dynamics of One-Dimensional Maps and Gurtin–Maccamy’s Population Model. Part I. Asymptotically Constant Solutions

Dynamics of One-Dimensional Maps and Gurtin–Maccamy’s Population Model. Part I. Asymptotically Constant Solutions

Simplicity bias, algorithmic probability, and the random logistic map

Simplicity bias is an intriguing phenomenon prevalent in various input–output maps, characterized by a preference for simpler, more regular, or symmetric outputs. Notably, these maps typically feature high-probability outputs with simple patterns, whereas complex patterns are exponentially less probable. This bias has been extensively examined and attributed to principles derived from algorithmic information theory and algorithmic probability. In a significant advancement, it has been demonstrated that the renowned logistic map xk+1=μxk(1−xk), a staple in dynamical systems theory, and other one-dimensional maps exhibit simplicity bias when conceptualized as input–output systems. Building upon this work, our research delves into the manifestations of simplicity bias within the random logistic map, specifically focusing on scenarios involving additive noise. This investigation is driven by the overarching goal of formulating a comprehensive theory for the prediction and analysis of time series.Our primary contributions are multifaceted. We discover that simplicity bias is observable in the random logistic map for specific ranges of μ and noise magnitudes. Additionally, we find that this bias persists even with the introduction of small measurement noise, though it diminishes as noise levels increase. Our studies also revisit the phenomenon of noise-induced chaos, particularly when μ=3.83, revealing its characteristics through complexity-probability plots. Intriguingly, we employ the logistic map to illustrate a paradoxical aspect of data analysis: more data adhering to a consistent trend can occasionally lead to reduced confidence in extrapolation predictions, challenging conventional wisdom.We propose that adopting a probability-complexity perspective in analyzing dynamical systems could significantly enrich statistical learning theories related to series prediction and analysis. This approach not only facilitates a deeper understanding of simplicity bias and its implications but also paves the way for novel methodologies in forecasting complex systems behavior, especially in scenarios dominated by uncertainty and stochasticity.

Invariant measures for random piecewise convex maps

We establish the existence of Lebesgue-equivalent conservative and ergodic σ -finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighbourhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong recurrence to a fixed point.

Correlation between Hurst exponent and largest Lyapunov exponent on a coupled map lattice

Positive correlations have been made between the Hurst exponent and the largest Lyapunov exponent using various one-dimensional maps by other authors. We incorporate these maps into a Coupled Map Lattice (CML), and investigate whether the correlation still exists. With multiple Hurst exponents being calculated for each CML, differences between each have been shown, with the possibility of using one Hurst exponent to represent the system. Results also show that the Hurst exponent is a good indicator of chaos, and is in positive correlation with the largest Lyapunov exponent. We suggest that for a general class of CML, an increased Hurst exponent indicates an increased Lyapunov exponent 85% of the time. Given the Hurst exponent can frequently be calculated more quickly than the Lyapunov exponent we suggest it as a more time efficient measure. Through these connections, machine learning methods have been used to predict the largest Lyapunov exponent for different scenarios.