One-dimensional Maps Research Articles

Loewner Evolution Driven by One-dimensional Chaotic Maps

We investigated the Loewner evolution driven by the diffusion processes determined by one-dimensional chaotic maps. Our model works primarily as a deterministic analog of the stochastic Loewner evo...

On the construction of one-dimensional discrete chaos theory based on the improved version of Marotto’s theorem

The purpose of this paper is to design one-dimensional chaotic maps with a non-zero intercept in the Y-axis and research the basic dynamic characteristics. First, by studying the common conditions that make one-dimensional discrete dynamical systems chaotic under the modulo operation, the general theory of the chaotic behavior of one-dimensional nonlinear functions with a non-zero origin is presented. It is proved that the discrete dynamic system derived from the proposed theory exhibits chaos phenomena in the sense of Li-Yorke, utilizing the improved version of Marotto’s theorem. Furthermore, the part of the discrete dynamical functions satisfying the proposed chaos theory is given. The function track, bifurcation diagram and Lyapunov exponential spectrum are also analyzed. Numerical simulations agree with the analytical framework for the suggested chaos theory. Finally, given the application of chaos in cryptographic communication, two significant corollaries of the original functions as general polynomials are put forward according to the theory of chaos structure.

Efficient symmetric image encryption by using a novel 2D chaotic system

The authors know that a common and effective way to protect digital images security are to encrypt these images into white noise image. In this study, the authors have designed a new two-dimensional (2D) chaotic system which is derived from two existing one-dimensional (1D) chaotic maps. The simulation results show that the new 2D chaotic system is able to produce many 2D chaotic maps by selecting different 1D chaotic maps, and which have the wider chaotic ranges and more complex chaotic behaviours compared with the existing 1D chaotic maps. In order to investigate its applications, using the proposed 2D chaotic maps, the authors design a novel image encryption algorithm. First of all, the original image is scrambled by using the chaotic sequences which are generated by new 2D chaotic maps, Arnold transform and Hilbert curve. Then the scrambled image is confused and diffused by chaotic sequences. Finally, the performance of the proposed encryption algorithm is simulated and the experimental results show that the validity and reliability of the proposed algorithm is validated.

A new one-dimensional cosine polynomial chaotic map and its use in image encryption

In this paper, we propose a new real one-dimensional cosine polynomial (1-DCP) chaotic map. The statistical analysis of the proposed map shows that it has a simple structure, a high chaotic behavior, and an infinite chaotic range. Therefore, the proposed map is a perfect candidate for the design of chaos-based cryptographic systems. Moreover, we propose an application of the 1-DCP map in the design of a new efficient image encryption scheme (1-DCPIE) to demonstrate the new map further good cryptographic proprieties. In the new scheme, we significantly reduce the encryption process time by raising the small processing unit from the pixels level to the rows/columns level and replacing the classical sequential permutation substitution architecture with a parallel permutation substitution one. We apply several simulation and security tests on the proposed scheme and compare its performances with some recently proposed encryption schemes. The simulation results prove that 1-DCPIE has a better security level and a higher encryption speed.

Self-Organizing Map Network-Based Soil and Water Conservation Partitioning for Small Watersheds: Case Study Conducted in Xiaoyang Watershed, China

Soil and water conservation partitioning (SWCP) considers complex environmental statutes and development demands and serves as a scientific basis for conducting soil erosion management and practice. However, few studies have researched partitioning in small watersheds (< 50 km2), and guidelines for enabling region-specific measures are lacking. In this study, the Xiaoyang watershed located in the red soil region of southern China was selected as a representative small watershed in which to conduct partitioning. The pressure–state–response (PSR) model was used as a framework for establishing an indicator system that included soil erosion sensitivity, the soil erosion condition, and ecosystem services. With three soil and water conservation variables as the input layer, a one-dimensional self-organizing map was applied to identify clusters in the small watershed. The silhouette width was evaluated to determine the optimal number of regions. Based on the associated results, the Xiaoyang watershed was divided into five regions accounting for 82%, 9%, 8%, 2%, and 1% of the total area, respectively. This study provides a framework on which region-specific soil erosion measures can be planned, and it also provides a partitioning method that can be employed in other areas.

Chaotic Color Image Encryption Scheme Using Deoxyribonucleic Acid (DNA) Coding Calculations and Arithmetic over the Galois Field

This paper proposes a chaotic color image encryption scheme based on DNA-coding calculations and arithmetic over the Galois field. Firstly, three modified one-dimensional (1D) chaotic maps with larger key space and better chaotic characteristics are presented. The experimental results show that their chaotic intervals are not only expanded to 0,15, but their average largest Lyapunov Exponent reaches 10. They are utilized as initial keys. Secondly, DNA coding and calculations are applied in order to add more permutation of the cryptosystem. Ultimately, the numeration over the Galois field ensures the effect for the diffusion of pixels. The simulation analysis shows that the encryption scheme proposed in this paper has good encryption effect, and the numerical results verify that it has higher security than some of the latest cryptosystems.

Image Encryption Algorithm Based on Tent Delay-Sine Cascade with Logistic Map

We propose a new chaotic map combined with delay and cascade, called tent delay-sine cascade with logistic map (TDSCL). Compared with the original one-dimensional simple map, the proposed map has increased initial value sensitivity and internal randomness and a larger chaotic parameter interval. The chaotic sequence generated by TDSCL has pseudo-randomness and is suitable for image encryption. Based on this chaotic map, we propose an image encryption algorithm with a symmetric structure, which can achieve confusion and diffusion at the same time. Simulation results show that after encryption using the proposed algorithm, the entropy of the cipher is extremely close to the ideal value of eight, and the correlation coefficients between the pixels are lower than 0.01, thus the algorithm can resist statistical attacks. Moreover, the number of pixel change rate (NPCR) and the unified average changing intensity (UACI) of the proposed algorithm are very close to the ideal value, which indicates that it can efficiently resist chosen-plain text attack.

Distinguished correlation properties of Chebyshev dynamical systems and their generalisations

We show that, among all smooth one-dimensional maps conjugated to an N-ary shift (a Bernoulli shift of N symbols), Chebyshev maps are distinguished in the sense that they have least higher-order correlations. We generalise our consideration and study a family of shifted Chebyshev maps, presenting analytic results for two-point and higher-order correlation functions. We also review results for the eigenvalues and eigenfunctions of the Perron-Frobenius operator of Nth order Chebyshev maps and their shifted generalisations. The spectrum is degenerate for odd N. Finally, we consider coupled map lattices (CMLs) of shifted Chebyshev maps and numerically investigate zeros of the temporal and spatial nearest-neighbour correlations, which are of interest in chaotically quantized field theories.

Performance enhancement of LR WPAN spread spectrum system using chaotic spreading sequences

Performance enhancement using chaotic spreading sequences on Low Rate Wireless Personal Area Network is presented. Several one-dimensional chaotic maps are used for generating optimized chaotic sequences for spread spectrum applications. The system parameter of one-dimensional chaotic maps is varied to generate sample-based chaotic signals. Correlation properties analysis is used to generate optimized chaotic sequences. Spreading sequences with a high value of Golay figure of merit are chosen as optimized spreading sequences. Chaotic sequences with high Golay figure of merit factor are expected to have better synchronization and bit error performance. The LR-WPAN model is programmed for USRP (Universal Software Radio Peripheral) software defined radio platform using LabVIEW. The performance of chaotic spreading sequences is evaluated against Gold spreading sequences using the proposed system model. The performance evaluation is based on comparing bit error rate and error vector magnitude in multiple access scenario. The results show that optimized chaotic sequences have better performance than Gold codes.

A mathematical model for neuronal differentiation in terms of an evolved dynamical system

We attempted to create a mathematical model for neuronal differentiation. The present study was performed within the framework of self-organization with constraints by looking for an optimized informational unit. We treated networks of individual dynamical system units with an external input, which was provided by coupled one-dimensional maps with possible forms of unidirectionally feed-forward network, random network, small-world network, and fully-connected network. We used a genetic algorithm to maximize the information transmission for each type of network. Optimized maps were obtained depending on the coupling strength and network structure. These maps can be classified into three types: passive, excitable, and oscillatory. In particular, the excitable and oscillatory types of dynamical systems possess characteristics that are quite similar to those of neurons, whereas the passive and oscillatory types of dynamical system may represent glial cells.

Dynamic Effects Arise Due to Consumers' Preferences Depending on Past Choices.

We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case considers only focusing on the action done by one player. This action reduces the game’s map to a one-dimensional map, which is the logistic map. Using analytical and numerical simulation, the stability of fixed points of this map is studied. In the second case, we focus on the actions applied by both players. The fixed points, in this case, are calculated, and their stability is discussed. The conditions of stability are provided in terms of the game’s parameters. Numerical simulation is carried out to give local and global investigations of the chaotic behavior of the game’s map. In addition, we use a statistical measure, such as entropy, to get more evidences on the regularity and predictability of time series associated with this case.

A New Image Encryption Algorithm Based on Composite Chaos and Hyperchaos Combined with DNA Coding.

In order to obtain chaos with a wider chaotic scope and better chaotic behavior, this paper combines the several existing one-dimensional chaos and forms a new one-dimensional chaotic map by using a modular operation which is named by LLS system and abbreviated as LLSS. To get a better encryption effect, a new image encryption method based on double chaos and DNA coding technology is proposed in this paper. A new one-dimensional chaotic map is combined with a hyperchaotic Qi system to encrypt by using DNA coding. The first stage involves three rounds of scrambling; a diffusion algorithm is applied to the plaintext image, and then the intermediate ciphertext image is partitioned. The final encrypted image is formed by using DNA operation. Experimental simulation and security analysis show that this algorithm increases the key space, has high sensitivity, and can resist several common attacks. At the same time, the algorithm in this paper can reduce the correlation between adjacent pixels, making it close to 0, and increase the information entropy, making it close to the ideal value and achieving a good encryption effect.

A new discrete-space chaotic map based on the multiplication of integer numbers and its application in S-box design

In this paper, a new one-dimensional discrete-space chaotic map based on the multiplication of integer numbers and circular shift is presented. Dynamical properties of the proposed map are analyzed, and it exhibits chaotic behavior. The proposed map has fixed points for certain settings, but it is easy to completely avoid them. This map preserves all desirable properties of previous discrete-space chaotic maps and has improved characteristics related to orbit length, computational complexity and memory requirements. These improvements can be particularly useful when implementation in digital devices, which have limited memory and computational resources, is needed. S-box design method based on this chaotic map is presented as an example of its application in cryptography. The results of performance tests show that S-boxes with good cryptographic properties can be generated on the basis of this discrete-space chaotic map.

Renormalization of almost commuting pairs

In this paper we give a new proof of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demonstrate that a two-dimensional map which lies in the stable set of the renormalization operator possesses attractor which is topologically a circle. Such a circle is critical: the dynamics on it is topologically, but not smoothly, conjugate to a rigid rotation.

Designing a 2D infinite collapse map for image encryption

Due to the unpredictability and complexity properties, chaotic maps are widely applied in security, communication, and system control. Existing one-dimensional (1D) chaotic maps can be easily predicted and high-dimensional (HD) ones have more complex structures and higher computation costs. In order to enhance the chaotic performance, this paper proposes a new two-dimensional infinite collapse map (2D-ICM). Compared with existing 2D chaotic maps, 2D-ICM has better ergodicity, hyperchaotic property, unpredictability, and a wider chaotic region. To investigate its application, we further propose an image encryption algorithm using 2D-ICM. Simulation demonstrates that the proposed image encryption algorithm has excellent performance for protecting various kinds of images.

Degenerate Period Adding Bifurcation Structure of One-Dimensional Bimodal Piecewise Linear Maps

Motivated by a problem in the management of ecological populations, we study the bifurcation structure known as period adding structure for a family of one-dimensional bimodal piecewise linear maps...

О КАЛИБРОВКЕ АВТОНОМНОЙ МОДЕЛИ ТУНДРОВОЙ БИОЛОГИЧЕСКОЙ ПОПУЛЯЦИИ ЛЕММИНГОВ

The paper presents an autonomous model of the biological population of lemmings of a phenomenological type, designed in complex studies of tundra communities. In the model, the dynamics of a population is described through a difference equation relating the population in two neighboring years that depends on three parameters of the biological and ecological genesis. The combination of parameter values included in the equation under consideration determines a class of one-dimensional unimodal mappings of a dynamical system in which the bifurcation properties, asymptotics, and stability of trajectories were analytically and numerically studied. In this paper the main focus is made on model identification criterion. The method of identification sets is proposed to be used for calibration of the model. The identification sets method is based on the approximation and visualization of small-dimensional projections of a multidimensional graph of the error function given in the space of three environmental and two population parameters. This paper describes a case study of model identification using data on the tundra lemming population on the Taimyr Peninsula. It is shown that in this case two biological and ecological parameters allow for stable location distribution.

Security-Enhanced 3D-CAP-PON Based on Two-Stage Spherical Constellation Masking

A security-enhanced three-dimensional carrier-less amplitude phase passive optical network (3D-CAP-PON) based on two-stage spherical constellation masking is proposed in this paper, where Chua's circuit map and one-dimensional Logistic map are jointly applied to implement rotation-based and scaling-based 3D-constellation masking. The rotation masking converts the constellation into two spherical layers with different radii, while the scaling masking leads to a hollow sphere with thick wall. The dimensional expansion of the constellation, combined with the joint adoption of two chaotic models, enable more flexibility and security for constellation encryption. And the key space of the proposed scheme is 1.2 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">73</sup> when taking only initial values and step lengths of chaotic models into consideration. A 4 Gb/s encrypted 3DCAP-16 signal transmission over 25 km standard single mode fiber (SSMF) is experimentally demonstrated. The experimental results indicate that the proposed scheme can effectively protect the system from any illegal optical network unit (ONU) attack, proving to be a promising candidate for future physically secure 3D-CAP-PON.

SYNTHESIS OF STRIP-BAND TELECOMMUNICATION SYSTEM WITH INCREASED CONFIDENTIALITY OF INFORMATION TRANSMISSION THROUGH USE OF PSSEVDOVIDOVYPOVY

Currently, special attention is paid to the creation of broadband telecommunications systems with a noise-like signal. Such systems have indisputable advantages in noise immunity, in the confidentiality of information transmission, electromagnetic compatibility with neighboring electronic devices. The formation of a broadband signal in most cases is carried out by the method of direct spreading of the spectrum by manipulating the carrier frequency signal by a pulse pseudo-random sequence (PVP) –m-sequence, Gold, Kasami, Walsh sequence. Sequences such as a well-known attacker can pick up in the receiver and they can not be considered reliable protection of information. The use of the phenomenon of dynamic chaos provides an opportunity to find a new class of PVP, the structure of which is almost impossible to reproduce, so their use provides increased confidentiality of information transmission. PVP generators are built on the basis of one-dimensional chaotic mappings - logistic, quadratic and cubic, which are chaotic systems. When creating a PVP based on chaos, three secret encryption keys are used: for example, for logistic separation, this is the initial value, the equation parameter, and the start of the sequence. To increase the cryptosecurity of telecommunication systems, it is proposed to create PVP based on 2 chaotic signals (by logistic and quadratic mappings), which provides an increase in secret keys from three to six. The paper proposes a scheme for building a telecommunications system with high confidentiality of information transmission through the use of such PVP.

Метод анализа экономических процессов в условиях нестабильности (на примере анализа динамики выручки российских компаний)

New methods are proposed that allow using the properties of nonlinear dynamics models to analyze the dynamics of real socio-economic processes. For this, a complex of models has been developed in the form of connected nonlinear one-dimensional and multidimensional map pings that meet the general laws of socio-economic processes - development in a rapidly changing environment under resource constraints. The proposed set of models allows you to analyze the dynamics of processes under the assumptions of both their independence and interconnection with each other. The definition of interrelated processes is introduced as processes developing under conditions of general constraints. Three criteria are proposed for identifying interrelated processes based on empirical data. In the case of interrelation of processes, the parameters of one-dimensional and multidimensional models are also interrelated, which allows a deeper analysis of the situations under consideration. The use of the proposed method is demonstrated by the example of analyzing the dynamics of revenue of Russian companies in the pe­riod from 2006 to 2015 (38 thousand companies in total). All companies were divided into four groups, depending on the rate of revenue growth from 2010 to 2015. The hypothesis of a possible relationship between the decline of one and the parallel strengthening of another group of Russian firms after the crisis of 2008-2009 is investigated and rejected. Innovative firms that boosted their revenue growth potential after the crisis were targeting markets other than con ser­vative firms whose revenue growth potential was exhausted after the crisis. The peculiarities of the dynamics of revenue of groups of companies are visually presented in the form of normalized values, when the median estimates of each of the groups in 2007 are taken as a unit.