Discrete Chaotic Systems Research Articles

Design and Synchronization of Chaos-Based True Random Number Generators and Its FPGA Implementation

In this paper, a new chaos-based true random number generator (TRNG) and a sliding mode controller (SMC) to synchronize the proposed TRNGs are proposed. Firstly, the continuous chaotic system is transformed into a discrete system that preserves the original continuous system’s chaotic behavior and makes it easy to realize with the field-programmable gate array (FPGA) for synchronization control. Then, a discrete SMC is introduced to solve the synchronization problem of the master-slave discrete chaotic systems. Subsequently, a novel hybrid function integrated with the El-Gamal algorithm is proposed to complete the TRNG design. Finally, the FPGA-based realization of the synchronized master-slave TRNGs is implemented. The randomness quality of the proposed TRNGs has been ensured by using Shannon’s entropy and histogram analysis. Furthermore, the National Institute of Standards and Technology (NIST) test suite has also been introduced to evaluate the proposed TRNGs and compare with the existing results in the literature.

On the Construction of Chaotic Dynamical Systems on the Box Fractal

In this paper, our main aim is to obtain two different discrete chaotic dynamical systems on the Box fractal ($B$). For this goal, we first give two composition functions (which generate Box fractal and filled-square respectively via escape time algorithm) of expanding, folding and translation mappings. In order to examine the properties of these dynamical systems more easily, we use the intrinsic metric which is defined by the code representation of the points on $B$ and express these dynamical systems on the code sets of this fractal. We then obtain that they are chaotic in the sense of Devaney and give an algorithm to compute periodic points.

FPGA REALIZATION OF COMPLEX LOGISTIC MAP FRACTAL BEHAVIOR

This paper studies the capability of digital architecture to mimic fractal behavior. As chaotic attractors realized digitally had opened many tracks, digital designs mimicking fractals may ultimately achieve the same. This study is based on a complex single-dimensional discrete chaotic system known as the generalized positive logistic map. The fractals realized from this system are linked to the results of the mathematical analysis to understand the fractal behavior with different variations. A digital hardware architecture manifesting the fractal behavior is achieved on FPGA, showing a fractal entity experimentally. With this digital realization, it is hoped that fractals can follow the example of chaotic attractors digital applications.

Analytic Programming – a Novel Tool for Synthesis of Controller for Chaotic Lozi Map

In this paper, it is presented a utilization of a novel tool for symbolic regression, which is analytic programming, for the purpose of the synthesis of a new feedback control law. This new synthesized chaotic controller secures the fully stabilization of selected discrete chaotic systems, which is the two-dimensional Lozi map. The paper consists of the descriptions of analytic programming as well as selected chaotic system, used heuristic and cost function design. For experimentation, Self-Organizing Migrating Algorithm (SOMA) and Differential evolution (DE) were used. Two selected experiments are detailed described.

Observer-Based H ∞ Fuzzy Synchronization and Output Tracking Control of Time-Varying Delayed Chaotic Systems

Tracking control for the output using an observer-based H ∞ fuzzy synchronization of time-varying delayed discrete- and continuous-time chaotic systems is proposed in this paper. First, from a practical point of view, the chaotic systems here consider the influence of time-varying delays, disturbances, and immeasurable states. Then, to facilitate a uniform control design approach for both discrete- and continuous-time chaotic systems, the dynamic models along with time-varying delays and disturbances are reformulated using the T-S (Takagi–Sugeno) fuzzy representation. For control design considering immeasurable states, a fuzzy observer achieves master-slave synchronization. Third, combining both a fuzzy observer for state estimation and a controller (solved from generalized kinematic constraints) output tracking can be achieved. To make the design more practical, we also consider differences of antecedent variables between the plant, observer, and controller. Finally, using Lyapunov’s stability approach, the results are sufficient conditions represented as LMIs (linear matrix inequalities). The contributions of the method proposed are threefold: (i) systemic and unified problem formulation of master-slave synchronization and tracking control for both discrete and continuous chaotic systems; (ii) practical consideration of time-varying delay, immeasurable state, different antecedent variables (of plant, observer, and controller), and disturbance in the control problem; and (iii) sufficient conditions from Lyapunov’s stability analysis represented as LMIs which are numerically solvable observer and controller gains from LMIs. We carry out numerical simulations on a chaotic three-dimensional discrete-time system and continuous-time Chua’s circuit. Satisfactory numerical results further show the validity of the theoretical derivations.

Research on Pseudorandom Number Generator Based on Several New Types of Piecewise Chaotic Maps

This paper proposes three types of one-dimensional piecewise chaotic maps and two types of symmetrical piecewise chaotic maps and presents five theorems. Furthermore, some examples that satisfy the theorems are constructed, and an analysis and model of the dynamic properties are discussed. The construction methods proposed in this paper have a certain generality and provide a theoretical basis for constructing a new discrete chaotic system. In addition, this paper designs a pseudorandom number generator based on piecewise chaotic map and studies its application in cryptography. Performance evaluation shows that the generator can generate high quality random sequences efficiently.

A new pseudorandom bits generator based on a 2D-chaotic system and diffusion property

A remarkable correlation between chaotic systems and cryptography has been established with sensitivity to initial states, unpredictability, and complex behaviors. In one development, stages of a chaotic stream cipher are applied to a discrete chaotic dynamic system for the generation of pseudorandom bits. Some of these generators are based on 1D chaotic map and others on 2D ones. In the current study, a pseudorandom bit generator (PRBG) based on a new 2D chaotic logistic map is proposed that runs side-by-side and commences from random independent initial states. The structure of the proposed model consists of the three components of a mouse input device, the proposed 2D chaotic system, and an initial permutation (IP) table. Statistical tests of the generated sequence of bits are investigated by applying five evaluations as well as the ACF and NIST. The results of five standard tests of randomness have been illustrated and overcome a value of 0.160 in frequency test. While the run test presents the pass value t0=4.769 and t1=2.929. Likewise, poker test and serial test the outcomes was passed with 3.520 for poker test, and 4.720 for serial test. Finally, autocorrelation test passed in all shift numbers from 1 to 10.

A flexible image encryption algorithm based on 3D CTBCS and DNA computing

In this paper, a novel image encryption algorithm based on a new discrete chaotic system is presented. The chaotic characteristics of the new chaotic system are analyzed by phase diagram, Lyapunov exponent, bifurcation diagram and complexity analysis. The randomness of chaotic sequences is test by the NIST SP 800-22 test package. The calculation time with different length sequence is recorded. Based on the analyses results, a flexible encryption scheme is designed for 2D image and 3D image. The hash value which is calculated from the SHA-256 hash function is used to change the initial conditions of the discrete system firstly. And then, the chaotic sequences are used to scramble and spread the value for the images by Arnold matrix and DNA diffusion algorithm. The security analysis shows that the proposed encryption algorithm possesses higher security features to preserve the subject and resist conventional attack. The results of this paper offer a different realization scheme for protection of image files.

Construction of a Class of High-Dimensional Discrete Chaotic Systems

In this paper, a class of n-dimensional discrete chaotic systems with modular operations is studied. Sufficient conditions for transforming this kind of discrete mapping into a chaotic mapping are given, and they are proven by the Marotto theorem. Furthermore, several special systems satisfying the criterion are given, the basic dynamic properties of the solution, such as the trace diagram and Lyapunov exponent spectrum, are analyzed, and the correctness of the chaos criterion is verified by numerical simulations.

Chaos-Based Synchronized Dynamic Keys and Their Application to Image Encryption with an Improved AES Algorithm

This study aimed to design chaos-based synchronized dynamic keys and develop an improved chaos-based advanced encryption standard (AES) algorithm with the proposed synchronized random keys. First, based on sliding mode control (SMC) technology, a rippling control scheme was introduced to guarantee the synchronization between master–slave discrete chaotic systems. Under the synchronization, the same dynamic random chaos signals could be simultaneously obtained at the transmitter and receiver in communication systems. Then, a novel modified AES cryptosystem with dynamic random keys based on chaos synchronization was presented. In a traditional AES cryptosystem, a static key is used, and it must be exchanged in advance and confirmed to be safely kept. However, in the proposed design, by introducing the synchronization technology of chaotic systems, the static key becomes dynamic and random, and it does not need to be kept or transmitted in open channels. Consequently, the disadvantage of key storage could be eliminated and the security of encryption could be improved. Finally, the developed chaos-based AES (CAES) algorithm has been applied to construct a novel image encryption algorithm. The statistical analysis, histogram, information entropy, and correlation indexes have been calculated and analyzed through simulation experiments in order to demonstrate the capability and improvement of this presented CAES cryptosystem.

Visually meaningful image encryption based on universal embedding model

Visually meaningful image encryption (VMIE) means that a plain image can be encrypted into a visually meaningful cipher image (VMCI), which makes the secret more imperceptible than noise-like cipher images. Here, we first present a universal embedding model (UEM) and further present a new UEM-based VMIE algorithm. The plain image is pre-encrypted and then embedded into the integer wavelet subbands of the host image in a dynamic way. In order to avoid the overflow, a threshold limiting function is used to modify the range of pixel values of the host image. To adapt different types of wavelet transform subbands, a UEM is proposed to be used to embed the secret information domain into the host domain. Moreover, in the embedding process, a four-dimensional discrete chaotic system is used to ensure the security of embedding. The numbers of embedded bits in the four embedded domains can be adjusted flexibly. A traversal algorithm is designed to find the optimal numbers of embedded bits for different types of wavelet transform in order to achieve an optimal visual quality of cipher images. A matrix-bit-depth-conversion algorithm is designed where a large matrix with low bit depth can be transformed into a small matrix with high bit depth to meet the input requirements of UEM, which means that the size of the plain image is not limited to a quarter of the size of the host image. Simulation results and performance comparisons show that the proposed VMIE algorithm can achieve a better visual quality than existing VMIE algorithms.

A universal method of chaos cascade and its applications.

This paper proposes a universal method for a cascade chaotic system (CCS). CCSs may own better performances, including larger maximum Lyapunov exponents, extended full mapping range of chaos, and more coefficient variations. The chaos-cascade theorems had been proposed in our previous papers, which are more suitable for discrete chaotic systems with the same domain. In this paper, we further improve a universal method to normalize arbitrary discrete chaotic systems for constructing a series of CCSs. Besides, the inheritance and enhancement of robustness from the subsystem are first explored for CCSs. Finally, the designed CCS is implemented on field programmable gate array board. The generated chaotic sequences are obtained by an oscilloscope and tested by NIST software.

Eigenbranes in Jackiw-Teitelboim gravity

It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

A Method to Determine the Most Suitable Initial Conditions of Chaotic Map in Statistical Randomness Applications

The processes and systems in the real world actually contain order and symmetry. Understanding these order and symmetrical behavior has been a common effort of scientists. Chaos theory has been an interesting topic to understand these order and symmetrical behavior. One of the most important reasons for this interest is the rich dynamics of chaotic systems. A remarkable application of these systems is statistical randomness. Especially in random number generator designs based on discrete time chaotic systems, an important design parameter affecting the success of the generator (statistical randomness properties) is the initial conditions of chaotic maps. Obtaining different initial conditions that will meet the statistical requirements is important in terms of generating different random number sequences. In order to determine the initial conditions, an algorithm that updates the initial conditions depending on the number of successful statistical tests is proposed. Although the proposed design approach is similar to a back propagation neural network, it has a unique design approach. The NIST SP 800-22 test suite has been used to analyze the statistical properties of the proposed generator. It is known that the NIST SP 800-22 test suite is a hypothesis test. Therefore, in order to show the success of the proposed method in the best way, various additional analysis studies have been carried out proving that the generator outputs have a uniform distribution. Using the proposed method, six different initial conditions have been determined that provide statistical random properties for the discrete-time chaotic systems known as logistic map and tent map. 1,000,000 bits have been generated using the obtained initial conditions. These bit values are then converted to decimal values between 0-15. It is observed that the obtained numbers have a uniform distribution. These outputs are thought to be applicable in many areas such as games, simulation, modeling, determination of optimization parameters and cryptology. It is shown in a practical application for cryptographic substitution-box designs to examine the success of the outputs.

On Dynamics of a Fractional-Order Discrete System with Only One Nonlinear Term and without Fixed Points

Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called “self-excited attractors” has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called “hidden attractors”, which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system.

Fractional Grassi–Miller Map Based on the Caputo h-Difference Operator: Linear Methods for Chaos Control and Synchronization

Investigating dynamic properties of discrete chaotic systems with fractional order has been receiving much attention recently. This paper provides a contribution to the topic by presenting a novel version of the fractional Grassi–Miller map, along with improved schemes for controlling and synchronizing its dynamics. By exploiting the Caputo h-difference operator, at first, the chaotic dynamics of the map are analyzed via bifurcation diagrams and phase plots. Then, a novel theorem is proved in order to stabilize the dynamics of the map at the origin by linear control laws. Additionally, two chaotic fractional Grassi–Miller maps are synchronized via linear controllers by utilizing a novel theorem based on a suitable Lyapunov function. Finally, simulation results are reported to show the effectiveness of the approach developed herein.

Different dimensional fractional-order discrete chaotic systems based on the Caputo h-difference discrete operator: dynamics, control, and synchronization

The paper investigates control and synchronization of fractional-order maps described by the Caputo h-difference operator. At first, two new fractional maps are introduced, i.e., the Two-Dimensional Fractional-order Lorenz Discrete System (2D-FoLDS) and Three-Dimensional Fractional-order Wang Discrete System (3D-FoWDS). Then, some novel theorems based on the Lyapunov approach are proved, with the aim of controlling and synchronizing the map dynamics. In particular, a new hybrid scheme is proposed, which enables synchronization to be achieved between a master system based on a 2D-FoLDS and a slave system based on a 3D-FoWDS. Simulation results are reported to highlight the effectiveness of the conceived approach.

Mapping topological characteristics of dynamical systems into neural networks: A reservoir computing approach.

Significant advances have recently been made in modeling chaotic systems with the reservoir computing approach, especially for prediction. We find that although state prediction of the trained reservoir computer will gradually deviate from the actual trajectory of the original system, the associated geometric features remain invariant. Specifically, we show that the typical geometric metrics including the correlation dimension, the multiscale entropy, and the memory effect are nearly identical between the trained reservoir computer and its learned chaotic systems. We further demonstrate this fact on a broad range of chaotic systems ranging from discrete and continuous chaotic systems to hyperchaotic systems. Our findings suggest that the successfully reservoir computer may be topologically conjugate to an observed dynamical system.

Novel discrete chaotic system via fractal transformation and its DSP implementation

Designing a discrete chaotic system via fractal transformation has become a new method for engineering applications. This method generates new discrete chaotic system through external mechanisms, instead of the traditional way of internal mechanisms. The way of building novel discrete chaotic system is enriched by fractal and mathematical operation. Taking one-dimensional ICMIC map and two-dimensional Hénon map as the seed maps, dynamics of the generated chaotic map is analyzed by bifurcations, complexity and spectrum distribution characteristics. The results show that the new discrete chaotic map has the advantages in complexity and distribution in the parameter space. Finally, the digital circuit of fractal chaotic system is implemented based on DSP technique. The feasibility of the circuit is verified. Therefore, it has good application prospects in secure communication.

The finite sample behavior of the 0–1 test for chaos

In a recent paper, Webel (2012) provides evidence of chaotic structures in the stock returns of all DAX members, by using the so-called 0–1 test developed by Gottwald and Melbourne (2004). The main aim of this paper is to show, through a Monte Carlo experiment, that the 0–1 test is severely oversized against leptokurtic random processes, hence Webel (2012) conclusions may not be reliable. Moreover, noise filtering may affect the power of the test against noisy discrete chaotic systems. Therefore, the application of this procedure on high frequency financial data, and in general on noise-filtered data, should be performed with caution.