Discrete-time Chaotic Systems Research Articles

Design and ARM-Embedded Implementation of a Chaotic Map-Based Real-Time Secure Video Communication System

A systematic methodology is proposed for a chaotic map-based real-time video encryption and decryption system with advanced Reduced Instruction Set Computer machine (ARM)-embedded hardware implementation. According to the anticontrol principle of dynamical systems, first, an 8-D discrete-time chaotic map-based system is constructed, which possesses the required property of 1-1 surjection in the integer range $[{0,\,N-1}]$ , where $N$ is the number of frame pixels, suitable for position scrambling of each video frame. Then, an 8-D discrete-time hyperchaotic system is designed for encryption–decryption of red, green, and blue (RGB) tricolor pixel values. Using the ARM-embedded platform super4412 model with Cortex-A9 processor, together with the standard QT cross-platform, an integrated chaotic map-based real-time secure video communication system is designed, implemented, and evaluated. In addition, the security performance of the designed system is tested using criteria from the National Institute of Standards and Technology statistical test suite. The main feature of this method is that, both scrambling–antiscrambling of RGB tricolor pixel positions and encryption–decryption of pixel values are realized simultaneously for enhancing the security. As is well known, compared with numerical simulations, hardware implementation for such a secure video communication system is very difficult to achieve, but we successfully implemented and tested in a real-world network environment. Both theoretical analysis and experimental results validate the feasibility and real-time performance of the new secure video communication system.

Analysis and implementation of a novel robust transmission scheme for private digital communications using Arduino Uno board

In this paper, we present a new scheme for the secured transmission of discrete information based on hyperchaotic discrete dynamics. The system is a modified-Henon hyperchaotic discrete-time oscillator considered as transmitter and a delayed step-by-step observer used as receiver. The transmitter parameters play the role of secret keys of the transmission scheme. To increase the robustness of the secure data transmission against known plain-text attacks, the message to be sent is encrypted by additional secret keys and inserted by inclusion method in the chaotic discrete-time system dynamics. By this way, the parameters used as secret keys cannot be identified with usual techniques. Simulation results are presented to highlight the performances of the proposed method. One of the main contributions of this paper is to demonstrate the feasibility of discrete realization of a chaotic observer-based secured transmission scheme. Indeed, experimental implementation results using Arduino Uno board validate the proposed approach, since it exhibits good performances of throughput and cost in terms of resources used.

A discrete-time chaotic oscillator based on the logistic map: A secure communication scheme and a simple experiment using Arduino

This paper presents a modified discrete-time chaotic system obtained from the standard logistic map model. Then, a secure communication system is given and numerical experiments are carried out using the conceived discrete-time chaotic oscillator. Moreover, an experiment of our chaotic model is realized using the Arduino-UNO board.

A study on global stabilization of periodic orbits in discrete-time chaotic systems by using symbolic dynamics

A study on global stabilization of periodic orbits in discrete-time chaotic systems by using symbolic dynamics

Policy iteration optimal tracking control for chaotic systems by using an adaptive dynamic programming approach**Project supported by the National Natural Science Foundation of China (Grant Nos. 61034002, 61233001, 61273140, 61304086, and 61374105) and the Beijing Natural Science Foundation, China (Grant No. 4132078).

A policy iteration algorithm of adaptive dynamic programming (ADP) is developed to solve the optimal tracking control for a class of discrete-time chaotic systems. By system transformations, the optimal tracking problem is transformed into an optimal regulation one. The policy iteration algorithm for discrete-time chaotic systems is first described. Then, the convergence and admissibility properties of the developed policy iteration algorithm are presented, which show that the transformed chaotic system can be stabilized under an arbitrary iterative control law and the iterative performance index function simultaneously converges to the optimum. By implementing the policy iteration algorithm via neural networks, the developed optimal tracking control scheme for chaotic systems is verified by a simulation.

H ∞ synchronization of two different discrete-time chaotic systems via a unified model

This paper presents some novel synchronization methods for two discrete-time chaotic systems with different time delays, which are transformed into two unified models. First, the H∞ performance of the synchronization error dynamical system between the drive unified model and the response one is analyzed using the linear matrix inequality (LMI) approach. Second, the novel state feedback controllers are established to guarantee H∞ performance for the overall system. The parameters of these controllers are determined by solving the eigenvalue problem (EVP). Most discrete-time chaotic systems with or without time delays can be converted into this unified model, and H∞ synchronization controllers are designed in a unified way. The effectiveness of the proposed design methods are demonstrated by three numerical examples.

Control of Discrete Time Chaotic Systems via Combination of Linear and Nonlinear Dynamic Programming

In this article by introducing and subsequently applying the Min–Max method, chaos has been suppressed in discrete time systems. By using this nonlinear technique, the chaotic behavior of Behrens–Feichtinger model is stabilized on its first and second-order unstable fixed points (UFP) in presence and absence of noise signal. In this step, a comparison has also been carried out among the proposed Min–Max controller and the Pyragas delayed feedback control method. Next, to reduce the computation required for controller design, the clustering method has been introduced as a quantization method in the Min–Max control approach. To improve the performance of the acquired controller through clustering method obtained with the Min–Max method, a linear optimal controller is also introduced and combined with the previously discussed nonlinear control law. The resultant combined controller has been applied on the Henon map and through comparison with both Pyragas controller, and the linear optimal controller alone, its advantages are discussed.

Dynamic delayed feedback control for stabilizing the giant swing motions of an underactuated three-link gymnastic robot

This paper investigates the dynamics of the giant swing motions of an underactuated three-link gymnastic robot moving in a vertical plane by means of dynamic delayed feedback control (DDFC). DDFC, being one of useful methods to overcome the so-called odd number limitation in controlling a chaotic discrete-time system, is extended to control a continuous-time system such as a 3-link gymnastic robot with passive joint. Meanwhile, a way to calculate the error transfer matrix and the input matrix which are necessary for discretization is proposed, based on a Poincare section which is defined to regard the target system as a discrete-time system. Moreover, the stability of the closed-loop system by the proposed control strategy is discussed. Furthermore, some numerical simulations are presented to show the effectiveness in controlling a chaotic motion of the 3-link gymnastic robot to a periodic giant swing motion.

Adaptive fuzzy optimal control using direct heuristic dynamic programming for chaotic discrete-time system

In this paper, we aim to solve the optimal tracking control problem for the Henon Mapping chaotic system using the direct heuristic dynamic programming (DHDP) setting with filtered tracking error. The fuzzy logic system is used to approximate the long-term utility function. Compared with the results for chaotic discrete-time system, the cost of the controller is reduced. The Lyapunov analysis approach is utilized to prove the stability of the chaotic system. It is shown that the tracking error, the adaptation law and the control input retain the property of uniformly ultimate boundedness. A simulation example is given to demonstrate the effectiveness of the proposed approach.

An LPV framework for chaos synchronization in communication

This paper proposes a unified framework to achieve chaos synchronization of both classes of chaotic discrete-time systems, namely maps involving polynomial nonlinearities and piecewise linear maps. It is shown that all of those chaotic systems can be rewritten as a polytopic Linear Parameter Varying (LPV) system. A unified approach to tackle chaos synchronization problems encountered in communication is derived.

Adaptive near optimal neural control for a class of discrete-time chaotic system

In this paper, an adaptive critic neural network controller is designed for a class of discrete-time chaotic system. The critic neural network is used to approximate the long-term function. In contrast with the existing results for discrete-time chaotic systems, in this paper, a near optimal control input can be generated when the long-term function is minimized. It is proven that the tracking error, the adaptation laws and the control input are uniformly bounded. A simulation example is employed to illustrate the effectiveness of the proposed algorithm.

Design and ARM Platform-Based Realization of Digital Color Image Encryption and Decryption via Single State Variable Feedback Control

In this paper, discrete-time chaotic systems under single state variable feedback control are studied, and drive-response synchronization is investigated. To that end, ARM platform-based digital color image encryption and decryption using a chaotic stream cipher is designed and implemented, with experimental results demonstrated.

Design and Circuit Implementation of Discrete-Time Chaotic Systems with Modulus of Triangular Wave Functions

This paper investigates n-dimensional discrete-time systems with modulus of triangular wave functions. First, a proof is given to the existence of chaos in the sense of Li–Yorke. Then, taking the three-dimensional case as a typical example, a circuit is designed under the condition of a limited region, with experimental results demonstrated.

A Systematic Methodology for Multi-Images Encryption and Decryption Based on Single Chaotic System and FPGA Embedded Implementation

A systematic methodology is developed for multi-images encryption and decryption and field programmable gate array (FPGA) embedded implementation by using single discrete time chaotic system. To overcome the traditional limitations that a chaotic system can only encrypt or decrypt one image, this paper initiates a new approach to designn-dimensional (n-D) discrete time chaotic controlled systems via some variables anticontrol, which can achieve multipath drive-response synchronization. To that end, the designedn-dimensional discrete time chaotic controlled systems are used for multi-images encryption and decryption. A generalized design principle and the corresponding implementation steps are also given. Based on the FPGA embedded hardware system working platform with XUP Virtex-II type, a chaotic secure communication system for three digital color images encryption and decryption by using a 7D discrete time chaotic system is designed, and the related system design and hardware implementation results are demonstrated, with the related mathematical problems analyzed.

Sampled-data nonlinear observer design for chaos synchronization: A Lyapunov-based approach

This paper considers sampled-data based chaos synchronization using observers in the presence of measurement noise for a large class of chaotic systems. We study discretized model of chaotic systems which are perturbed by white noise and employ Lyapunov-like theorems to come up with a simple yet effective observer design. For the choice of observer gain, a suboptimal criterion is obtained in terms of LMI. We present semiglobal as well as global results. The proposed scheme can also be extended for discrete-time chaotic systems. Numerical simulations have been carried out to verify the effectiveness of theoretical results.

Stabilization of discrete-time chaotic systems via improved periodic delayed feedback control based on polynomial matrix right coprime factorization

Delayed feedback control, proposed by Pyragas,is one of useful control methods of stabilizing unstable periodic orbits of chaotic systems without their exact information. This paper discusses an improved periodic delayed feedback control for the stabilization of the discrete-time chaotic systems. A technique to solve the generalized Sylvester matrix equation, based on polynomial matrix right coprime factorization, is introduced in order to give an explicit solution to the periodic gains of the closed-loop system. Moreover, some examples are demonstrated to show the effectiveness of the proposed method.

Approximation-error-ADP-based optimal tracking control for chaotic systems with convergence proof

In this paper, an optimal tracking control scheme is proposed for a class of discrete-time chaotic systems using the approximation-error-based adaptive dynamic programming (ADP) algorithm. Via the system transformation, the optimal tracking problem is transformed into an optimal regulation problem, and then the novel optimal tracking control method is proposed. It is shown that for the iterative ADP algorithm with finite approximation error, the iterative performance index functions can converge to a finite neighborhood of the greatest lower bound of all performance index functions under some convergence conditions. Two examples are given to demonstrate the validity of the proposed optimal tracking control scheme for chaotic systems.

The synchronization between two discrete-time chaotic systems using active robust model predictive control

In this paper, we consider the synchronization of two discrete-time chaotic systems. A novel active robust model predictive strategy is proposed to guarantee the synchronization of two discrete-time systems in the presence of model uncertainty. The proposed approach reduces the synchronization to a convex optimization involving linear matrix inequalities. The numerical simulations illustrate the effectiveness of the proposed method.

Further stability results for a generalization of delayed feedback control

In this paper, we consider the stabilization of unstable periodic orbits for one-dimensional and discrete time chaotic systems. Various control schemes for this problem are available and we consider a recent generalization of delayed control scheme. We prove that if a certain condition, which depends only on the period number, is satisfied then the stabilization is always possible. We will also present some simulation results.

${\rm H}_{\infty}$ State Estimation for Discrete-Time Chaotic Systems Based on a Unified Model

This paper is concerned with the problem of state estimation for a class of discrete-time chaotic systems with or without time delays. A unified model consisting of a linear dynamic system and a bounded static nonlinear operator is employed to describe these systems, such as chaotic neural networks, Chua's circuits, Hénon map, etc. Based on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance analysis of this unified model using the linear matrix inequality approach, H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> state estimator are designed for this model with sensors to guarantee the asymptotic stability of the estimation error dynamic systems and to reduce the influence of noise on the estimation error. The parameters of these filters are obtained by solving the eigenvalue problem. As most discrete-time chaotic systems with or without time delays can be described with this unified model, H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> state estimator design for these systems can be done in a unified way. Three numerical examples are exploited to illustrate the effectiveness of the proposed estimator design schemes.