Discrete-time Chaotic Systems Research Articles

The Time Series Classification of Discrete-Time Chaotic Systems Using Deep Learning Approaches

Discrete-time chaotic systems exhibit nonlinear and unpredictable dynamic behavior, making them very difficult to classify. They have dynamic properties such as the stability of equilibrium points, symmetric behaviors, and a transition to chaos. This study aims to classify the time series images of discrete-time chaotic systems by integrating deep learning methods and classification algorithms. The most important innovation of this study is the use of a unique dataset created using the time series of discrete-time chaotic systems. In this context, a large and unique dataset representing various dynamic behaviors was created for nine discrete-time chaotic systems using different initial conditions, control parameters, and iteration numbers. The dataset was based on existing chaotic system solutions in the literature, but the classification of the images representing the different dynamic structures of these systems was much more complex than ordinary image datasets due to their nonlinear and unpredictable nature. Although there are studies in the literature on the classification of continuous-time chaotic systems, no studies have been found on the classification of discrete-time chaotic systems. The obtained time series images were classified with deep learning models such as DenseNet121, VGG16, VGG19, InceptionV3, MobileNetV2, and Xception. In addition, these models were integrated with classification algorithms such as XGBOOST, k-NN, SVM, and RF, providing a methodological innovation. As the best result, a 95.76% accuracy rate was obtained with the DenseNet121 model and XGBOOST algorithm. This study takes the use of deep learning methods with the graphical representations of chaotic time series to an advanced level and provides a powerful tool for the classification of these systems. In this respect, classifying the dynamic structures of chaotic systems offers an important innovation in adapting deep learning models to complex datasets. The findings are thought to provide new perspectives for future research and further advance deep learning and chaotic system studies.

Robust digital image watermarking scheme with a fractional-order discrete-time chaotic scheme and DWT-SVD transform

A robust digital image watermarking system with a fractional-order discrete-time chaotic system and discrete wavelet transform-singular value decomposition is presented. The inclusion method inserts an encrypted image into the dynamics of an integer-order discrete-time chaotic system and the resulting cipher serves as a host image watermark. A watermarking function (DWT-SVD transform) watermarks a host image before the watermark is extracted and decrypted at the receiver. As a contribution, our suggested approach introduces a new watermarking system based on discrete-time chaotic systems, as well as a hybridization of integer and fractional-order systems to ensure the watermarking scheme’s robustness. Our method achieved good results in terms of robustness, with normalized cross correlation values above 0.99 when subjected to a range of attacks. Results of the simulation underline practicality and robustness achieved by this approach. They also indicate that the proposed system resists various attacks.

A new synchronization result for fractional-order discrete-time chaotic systems via bandlimited channels

Recently, digital chaotic communication with limited bandwidth has seized the attention of several researchers. This interest stems from the fact that all physical communication channels are limited in bandwidth for data transmission. In furtherance of this, researchers and investigators in this area have put forward various new communication strategies based on synchronizing and controlling coupled chaotic systems evolving in discrete time. In this article, we present a new robust communication scheme based on the synchronization of two fractional-order discrete-time chaotic systems connected by bandlimited channels. The major contribution is to give a sufficient condition in term of the existing and updated parameters of the incorporated fractional-order system on one hand and the inserted filter coefficients, on the other hand. Once applied for data communications, the currently drawn up synchronization scheme pledges to enhance the transmission robustness. Indeed, the new considered parameters act like novel secret keys which widen the cipher key space. Primary and pertinent simulation curves and results performed on Matlab software portray and illustrate the effectiveness of our approach.

Fractional Cumulative Residual Inaccuracy Information Measure and Its Extensions with Application to Chaotic Maps

The purpose of this work is to introduce fractional cumulative residual inaccuracy (FCRI) information, Jensen-cumulative residual inaccuracy (JCRI), and Jensen-fractional cumulative residual inaccuracy (JFCRI) information measure. Further, we study the FCRI information for some well-known models used in reliability, economics and survival analysis. The associated results reveal some interesting connections between the FCRI information measure and cumulative residual entropy and Gini mean difference measures. Applications to two chaotic discrete-time dynamical systems (Chebyshev and Logistic) are presented to illustrate the behavior of the proposed information measures. FCRI and JFCRI measures allow to determine regions of discrepancy between systems, depending on their respective fractional and chaotic map parameters.

A new result for synchronizing nonlinear discrete-time chaotic systems connected by bandlimited channels

A new result for synchronizing nonlinear discrete-time chaotic systems connected by bandlimited channels

NEURAL NETWORK METHOD FOR PARAMETER ESTIMATION OF FRACTIONAL DISCRETE-TIME UNIFIED SYSTEMS

Data-driven learning of the fractional discrete-time unified system is studied in this paper. A neural network method is suggested in the parameter estimation of fractional discrete-time chaotic systems. An optimization problem is obtained and the famous Adam algorithm is employed to train the neural network’s weights and parameters. The parameter estimation result is compared with that of the stepwise response sensitivity approach (SRSA). This paper provides a high accuracy method for parameter inverse problems. The method also can be applied to data-driven learning of other fractional chaotic systems.

Noise-modulated nonlocal coupling: Chimera suppression and beyond

The spatio-temporal dynamics of networks of chaotic discrete-time systems is studied numerically in the presence of colored Gaussian noise-modulated nonlocal coupling between elements. As individual nodes of the networks, a modified Ricker map and a cubic map are used. Without noise, the networks demonstrate various types of chimera states, such as phase or double-well chimeras. The nonlocal coupling is modulated by either a common noise source or independent noise sources. The impact of noise-modulated coupling (coupling noise) on the dynamics of the networks is analyzed for a number of different random initial conditions and noise realizations. It is shown that the network dynamics and settled spatio-temporal structures can be controlled by varying the intensity and spectral characteristics of the coupling noise.

On secure communication scheme-based mixed discrete-time hyperchaotic systems

The proposed secure communication system is based on mixed discrete-time chaotic systems in this study. To increase signal transmission security over the public channel, this system is composed of two chaotic systems on the transmitter and receiver sides. Borne and Gentina criterion associated with the Benrejeb arrow form matrix is used for stability study and system description to ensure the synchronization between these chaotic systems. The case of electrocardiogram encryption and decryption is considered an application to illustrate the effectiveness of the proposed chaotic encryption approach. The performance analyses of the proposed secure communication under the Additive White Gaussian Noise (AWGN) channel, brute force attack and some statistical attacks have shown that the proposed encryption scheme exhibited a high level of security, could maintain the confidentiality and integrity of the transmitted information and the original signal can be accurately recovered even in the presence of noise.A comparitive study between different methods is given to show the advantages and the drawbacks of the proposed communication system.

Input-to-state stability for discrete-time switched systems by using Lyapunov functions with relaxed constraints

<abstract><p>In this paper, input-to-state stability (ISS) is investigated for discrete-time time-varying switched systems. For a switched system with a given switching signal, the less conservative assumptions for ISS are obtained by using the defined weak multiple ISS Lyapunov functions (WMISSLFs). The considered switched system may contain some or all subsystems which do not possess ISS. Besides, for an ISS subsystem the introduced Lyapunov function could be increasing along the trajectory of the subsystem without input at some moments. Then for a switched system under any switching signal, the relaxed sufficient constraints for ISS are attained by using the defined weak common ISS Lyapunov functions. For this case, each subsystem of the considered system must be ISS. The proposed function may be increasing along the trajectory of each ISS subsystem of the considered system without input at some instants. The relationship between WMISSLFs for a switched system and the defined weak multiple Lyapunov functions for this switched system without input is set up. Three numerical examples are investigated to display the usefulness of the principal outcomes. According to the main conclusions, an intermittent controller is applied to ensure ISS for a discrete-time disturbed Chua's chaotic system.</p></abstract>

A novel security and authentication method for infrared medical image with discrete time chaotic systems

Objective:Hand vein images have become important biometric signs used for identification systems. Also, dorsal hand vein images have noteworthy advantages in terms of reliability and contactless procedure. Surgically changing the vascular pattern under the skin is extremely difficult. Therefore, the use of such patterns in identity recognition applications is increasing day by day. In this context, many studies have been carried out in essential areas such as image acquisition for different textures, image preprocessing, data security, image feature extraction and recognition. Methods:In this article, a new dorsal hand vein identification application has been carried out with the chaos-based security mechanism to protect personal data and the Improved SURF method to reduce the error matches of traditional SURF method in the current application. There are two improvements in the presented system. The first one finds the features representing only the vein segments, not the remaining tissue or hairly parts. The second one eliminates the mismatching points to increase the accuracy. Results:Both encryption and data hiding techniques are preferred together to protect person data. In addition, dorsal hand vein images taken with a camera for matching and recorded in the database of the people have been improved with the newly developed I-SURF feature extraction method. Thereby, with the developed application, in addition to keeping the personal identity information securely in the database with chaos-based methods, higher accuracy rate has been achieved in matching the new image taken with the camera.

Lyapunov Exponent Enhancement in Chaotic Maps with Uniform Distribution Modulo One Transformation

Most of the chaotic maps are not suitable for chaos-based cryptosystems due to their narrow chaotic parameter range and lacking of strong unpredictability. This work presents a nonlinear transformation approach for Lyapunov exponent enhancement and robust chaotification in discrete-time chaotic systems for generating highly independent and uniformly distributed random chaotic sequences. The outcome of the new chaotic systems can directly be used in random number and random bit generators without any post-processing algorithms for various information technology applications. The proposed Lyapunov exponent enhancement based chaotic maps are analyzed with Lyapunov exponents, bifurcation diagrams, entropy, correlation and some other statistical tests. The results show that excellent random features can be accomplished even with one-dimensional chaotic maps with the proposed approach.

Global stability of local fractional Hénon-Lozi map using fixed point theory

<abstract><p>We present an innovative piecewise smooth mapping of the plane as a parametric discrete-time chaotic system that has robust chaos over a share of its significant organization parameters and includes the generalized Henon and Lozi schemes as two excesses and other arrangements as an evolution in between. To obtain the fractal Henon and Lozi system, the generalized Henon and Lozi system is defined by adopting the fractal idea (FHLS). The recommended system's dynamical performances are investigated from many angles, such as global stability in terms of the set of fixed points.</p></abstract>

3D-Chaotic discrete system of vector borne diseases using environment factor with deep analysis

<abstract><p>Vector-Borne Disease (VBD) is a disease that consequences as of an infection communicated to humans and other animals by blood-feeding anthropoids, like mosquitoes, fleas, and ticks. Instances of VBDs include Dengue infection, Lyme infection, West Nile virus, and malaria. In this effort, we formulate a parametric discrete-time chaotic system that involves an environmental factor causing VBD. Our suggestion is to study how the inclusion of the parasitic transmission media (PTM) in the system would impact the analysis results. We consider a chaotic formula of the PTM impact, separating two types of functions, the host and the parasite. The considered applications are typically characterized by chaotic dynamics, and thus chaotic systems are suitable for their modeling, with corresponding model parameters, that depend on control measures. Dynamical performances of the suggested system and its global stability are considered.</p></abstract>

A general method for projective-lag synchronization of heterogeneous chaotic maps with different dimensions

Projective-lag synchronization of complex systems has attracted much attention in the past two decades. However, the majority of previous studies concentrated on continuous-time chaotic systems or discrete-time chaotic systems with the same dimensions. In our present study, a general method for projective-lag synchronization of different discrete-time chaotic systems characterized with different dimensions is first demonstrated. On the basis of stability theory of discrete-time dynamical systems and Lyapunov stability theory, general controllers are designed by using the active control method. The method could achieve projective-lag synchronization in both cases: and . The effectiveness and feasibility of the proposed method is demonstrated by the projective-lag synchronization between two-dimensional Lorenz discrete-time system and three-dimensional Stefanski map, as well as between the three-dimensional generalized Hénon map and the two-dimensional quadratic map, respectively.

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.

DCS-ELM: a novel method for extreme learning machine for regression problems and a new approach for the SFRSCC.

Extreme learning machine (ELM) algorithm is widely used in regression and classification problems due to its advantages such as speed and high-performance rate. Different artificial intelligence-based optimization methods and chaotic systems have been proposed for the development of the ELM. However, a generalized solution method and success rate at the desired level could not be obtained. In this study, a new method is proposed as a result of developing the ELM algorithm used in regression problems with discrete-time chaotic systems. ELM algorithm has been improved by testing five different chaotic maps (Chebyshev, iterative, logistic, piecewise, tent) from chaotic systems. The proposed discrete-time chaotic systems based ELM (DCS-ELM) algorithm has been tested in steel fiber reinforced self-compacting concrete data sets and public four different datasets, and a result of its performance compared with the basic ELM algorithm, linear regression, support vector regression, kernel ELM algorithm and weighted ELM algorithm. It has been observed that it gives a better performance than other algorithms.

Difference synchronization in nonidentical discrete-time chaotic systems with different dimensions using three scaling matrices

In this paper, we consider the difference synchronization for different chaotic maps with different dimensions using three scaling matrices. The proposed method allows us to study difference synchronization of two master discrete-time chaotic systems with dimension and one response discrete-time chaotic system with dimension Based on the Lyapunov stability theory and stability property of linear discrete-time systems, controllers are derived to achieve the difference synchronization among three nonidentical discrete-time chaotic systems with different dimensions in two cases: and For the case 2D Fold map and 2D Hénon map are used as the master systems, and the 3D Hitzl-Zele map is chosen as the slave system to simulate the difference synchronization. For the case 3D Stefanski map and 3D Baier-Klein map are adopted as the master systems, and 2D Lorenz discrete-time chaotic system is used as the slave system during the simulation. The numerical simulations illustrate the effectiveness and feasibility of the proposed method.

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with “self-excited attractors.” This paper makes a contribution to the topic of fractional-order discrete-time systems with “hidden attractors” by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system.

New synchronization results for a class of nonlinear discrete-time chaotic systems based on synergetic observer and their implementation

This paper presents new synchronization results for a class of nonlinear discrete-time chaotic systems based on a synergetic observer design with a circuit implementation. The synchronization master–slave method adopted is based on the synergetic theory which is exploited to propose a synergetic observer. The system (master) adopted is that of the discrete-time Lur’e forms. One of the main contributions of this paper is to exploit this simple special form as master in order to design a synergetic observer (slave) which allows us to control the rate of convergence of the synchronization error using the synergetic design parameter matrix. In order to achieve the synchronization method, sufficient conditions for asymptotic synchronization are given. Two simulation examples are provided to show the effectiveness of the method in two different cases: the master system respects and does not respect the Lur’e form. Furthermore, experimental implementation results using Arduino boards are illustrated to validate the proposed approach.

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.