Observer-Based Synchronization of Chaotic Systems Satisfying Incremental Quadratic Constraints and Its Application in Secure Communication

Chaotic optical communication at high bit rates has attracted great attention because of its potential application in secure communications. In a chaotic optical communication system, a message can participate in the dynamics of the nonlinear system and further affect the characteristics of the chaotic carrier waveform. Message encoding can also change the synchronization quality if an encoding scheme breaks the symmetry between a transmitter and a receiver. In this presentation, we experimentally investigate the influence of message encoding on the characteristics of chaotic dynamics, chaos synchronization, and chaotic communication, using three encoding schemes, namely chaos shift keying, chaos masking, and chaos modulation, with the setup of a chaotic optical communication system with optoelectronic feedback. The chaos shift keying and chaos modulation schemes are found to be able to increase the complexity of the chaotic carrier waveform due to the random nature of the encoded message. Meanwhile, the chaos shift keying and chaos masking schemes are found to deteriorate the synchronization quality when the amplitude of the message is increased. Consequently, only the chaos modulation scheme has the desired feature in increasing the complexity of the chaotic carrier waveform while maintaining the quality of chaos synchronization, which is an important feature in secure communication.

In this paper, we mainly propose a chaotic secure communication scheme which is based on the synchronization of double-layered and multiple complex dynamical networks. Compared with the previous chaotic secure communication schemes, in which only two chaotic systems or just a single-layer network composed of multiple chaotic systems is used, the introduction of a double-layered and multiple complex networks model composed of many encryption/encryption units can not only reflect the complex characteristics of different nodes, but also can improve the complexity and security of information encryption. By using a clustering method, nodes with the same characteristics belong to the same subnet, while the nodes with different characteristics belong to different ones. The subnets in the transmitter and receiver are one-to-one correspondence and form a pair of matching subnets, but the node size of each subnet can be inconsistent. Each subnet is only responsible for encrypting a certain part of information, and thus, the synchronization between each pair of matching subnets plays a crucial role on the correct recovery of information. Multiple encryption/decryption units operating in parallel way can speed up the encryption of information, and the key space can grow with the number of nodes in the transmitter. The proposed scheme utilizes the chaotic signals generated by many chaotic systems as the key sequences and adopts the one-time-one-cipher encryption method. Moreover, this scheme is not subject to the constraint that the amplitude of the encrypted signal should be much smaller than that of the chaotic signal, and it is particularly suitable for the big data encryption. Both theoretical analysis and numerical simulation demonstrate the feasibility and effectiveness of the proposed scheme.

In this paper, the finite-time stochastic combination synchronization of three different chaotic systems is investigated. Based on the adaptive technique and the properties of Weiner process, a novel sufficient condition is obtained to ensure combination synchronization under stochastic perturbations. Moreover, a secure communication scheme based on the adaptive combination synchronization of three different systems, i.e., the Lorenz system, Chen system, and Lü system, with uncertainties, unknown parameters, and stochastic perturbation is presented. The simulation results show the feasibility of the proposed method.

In recent years, a large amount of work on chaos-based cryptosystems has been published (Kocarev (2001); Millerioux et al. (2008)). A general methodology for designing chaotic and hyperchaotic cryptosystems has been developed using the control systems theory (Grassi et al. (1999); Liao et al. (1999); Yang et al. (1997a;b)). The chaotic communication system is closely related to the concept of chaos synchronization. An overview of chaotic secure communication systems can be found in (Yang (2004)). He classified the continuous-time chaotic secure communication systems into four generations. In the third generation, the combination of the classical cryptographic technique and chaotic synchronization is used to enhance the degree of security. Specifically, Yang et al. proposed a new chaos-based secure communication scheme in an attempt to thwart the attacks (Yang et al. (1997a;b)). They have combined both conventional cryptographic method and synchronization of chaotic systems. Their cryptographic method consists of an encryption function (the multi-shift cipher), a decryption function (the inverse of the encryption function), a chaotic encrypter that generates the key signal for the encryption function, and a decrypter that estimates the key signal. The approach has a limitation since the cryptosystem design may fail if different chaotic circuits are utilized. So far, this generation has the highest security in all the chaotic communication systems had been proposed and has not yet broken. From the control theoretic perspective, the transmitter and the receiver in the chaotic communication system can be considered as the nonlinear plant and its observer, respectively. Grassi et al. proposed a nonlinear-observer-based decrypter to reconstruct the state of the encrypter (Grassi et al. (1999); Liao et al. (1999)). They extended the Chua’s oscillator to the observer-based decrypter. The cryptosystem does not require initial conditions of the encrypter and the decrypter belonging to the same basin of attraction. If we can design a decrypter without the knowledge of the parameters of the encrypter, the chaos-based secure communication systems are not secure, because the parameters of the encrypter is selected as static secret keys in the cryptosystem. Parameter identification and adaptive synchronization methods may be effective for intruders in building reconstruction mechanisms, even when a synchronizing system is not available. Therefore, it is important for secure issues to investigate whether adaptive identifiers without the system information of encrypter can be constructed or not. We have recently designed an observer-based chaotic communication system combining the cryptosystems proposed by Grassi et al. (Grassi et al. (1999)) and by Liao et al. (Liao et al. (1999)) that allows us to assign the relative degree and the zeros of its encrypter system (Matsuo et al. (2004)). Specifically, we constructed three cryptosystems based on a Chua’s circuit by assigning its relative degree and zeros. The cryptosystem consists of 20

Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication.

A generalized projective synchronization method is proposed based on unidirectional coupled synchronization by only transmitting a single driving variable. The proposed method allows us to arbitrarily direct the scaling factor to a desired value. Since chaos synchronization can be achieved by transmitting a single variable from driving system to response system, this method is more practical. Theoretical deduction and computer simulation show the effectiveness of the proposed method. A scheme for secure communication, which can increase the security of a communication system, is presented based on generalized projective synchronization of unified chaotic system.

The main contribution of this paper is to propose a nonlinear robust controller to synchronize general chaotic systems, such that the controller does not need the information of the chaotic system’s model. Following this purpose, in this paper, two methods are proposed to synchronize general forms of chaotic systems with application in secure communication. The first method uses radial basis function neural network (RBFNN) as a controller. All the parameters of the RBFNN are derived and optimized via particle swarm optimization (PSO) algorithm and genetic algorithm (GA). In order to increase the robustness of the controller, in the second method, an integral term is added to the RBF neural network gives an integral RBFNN (IRBFNN). The coefficients of the integral term and the parameters of IRBFNN are also derived and optimized via PSO and GA. The proposed methods are applied to the famous Lorenz chaotic system for secure communication. The performance and control effort of the proposed methods are compared with the recently proposed PID controller optimized via GA. Simulation results show the superiority of the proposed methods in comparison to the recent one in improving synchronization while using smaller control effort.

This research introduces an innovative control method designed for the synchronization and management of chaotic systems through the application of advanced neural network techniques. Specifically, a neural network-based sliding mode control framework is employed to enhance system stability and precision in synchronization tasks. Chaotic systems, particularly when arranged in master–slave configurations, exhibit behaviors that are highly sensitive to initial conditions and parameter variations, making them ideal candidates for the proposed approach. The core of this methodology leverages neural networks to estimate unknown nonlinear functions and dynamically adjust the control coefficients, ensuring high accuracy and adaptability in real-time. One of the key contributions of this study lies in addressing the complex issues of parametric uncertainty, external disturbances, and unmodeled dynamics that typically challenge conventional sliding mode control methods. By incorporating neural networks, the controller is equipped to effectively mitigate these uncertainties, ensuring robust performance even in the face of significant system variability. The adaptive nature of the control system allows for continuous adjustment, resulting in improved synchronization accuracy and faster convergence times. The stability and robustness of the proposed control system are rigorously proven using Lyapunov-based methods. Simulations show that synchronization of nonlinear chaotic systems occurs within 10 s, even under varying conditions. This efficiency demonstrates its practical applications in secure communications, biological systems, and power grids, marking a significant advancement in chaotic system control with broad industrial potential.

Recently, many continuous-time chaotic systems were synthesized using microcontrollers and FPGAs. This requires applying mathematical discretization to convert integration into recursion. Depending on the approximation algorithms, the speed of the numerical processors, and the number of bits used to represent data, different accuracies and stabilities could be obtained. This article explores the conditions necessary to faithfully generate signals that reflect the true behavior of the chaotic systems, while maintaining the same values for their Lya-punov exponents. This is very important for chaos control and synchronization, especially for applications in secure communication that rely on digital cryptography. The Lorenz system and the Duffing oscillator are investigated to illustrate the effect of having autonomous versus non-autonomous structures. In addition, the Nosé-Hoover dynamical model is investigated to detect the relationship between agility and numerical accuracy. The obtained results prove that identical deterministic chaotic systems can behave differently, for the same set of initial conditions, depending on the discretization algorithm. This added sensitivity necessitates careful design of the mathematical models required for digital implementations of chaotic systems. The article concludes with useful recommendations for best practices in designing, synthesizing, and implementing digital chaotic systems, while commenting on the best compromise between mathematical complexity and numerical accuracy.

본 논문에서는 카오스 신호를 이용하여 음성신호의 보안전송을 위한 아날로그 비화통신 시스템을 제안하고 통신성능을 분석하였다. 기존의 카오스 동기화 및 카오스 변조통신 알고리즘을 개선하여 통신환경에서 발생하는 다양한 조건들을 적용하여 음성신호의 복원능력을 모의실험 하였다. 일반적인 PC(Pecora ＆ Carroll) 제어기법과 제안한 SFB(Self-FeedBack) 마스킹 기법을 사용하여 송신단에서 음성신호를 카오스 신호로 마스킹하여 변조하고, 통신채널에 잡음신호를 추가하여 전송하였다. 수신단에서는 카오스 응답시스템을 이용하여 음성신호를 복조하고, 복원성능을 계산하기 위하여 아날로그 복원 에러 신호의 평균전력을 제안하여 계산하였다. 실험결과 마스킹 정도, 파라미터들의 민감성, 채널잡음 등에 대하여 PC 제어기법보다 피드백 제어기법의 복원성능이 우수함을 정량적인 데이터로 확인할 수 있었다. 또한 로렌쯔 카오스 비화통신시스템에 사용할 경우 파라미터들의 조합으로 암호키를 구성해야 하므로 파라미터 변화율에 대응하는 복원에러율의 관계를 모의실험 값으로 구하였다. This paper presents analog secure communication system about safe speech transmission using chaotic signals. We applied various conditions that happen in actuality communication environment modifying chaotic synchronization and chaotic communication schemes and analyzed restoration performance of speech signal to computer simulation. In transmitter, we made the chaotic masking signal which is added voice signal to chaotic signal using PC(Pecora ＆ Carroll) and SFB(self-feedback) control techniques and transmitted encryption signal to noisy communication channel And in order to calculate the degree of restoration performance, we proposed the definition of analog average power of recovered error signals in receiver chaotic system. The simulation results show that feedback control techniques can certify that restoration performance is superior to quantitative data than PC method about masking degree, susceptibility of parameters and channel noise. We experimentally computed the table of relation of parameter fluxion to restoration error rate which is applied the encryption key values to the chaotic secure communication.

A bidirectional high-speed chaotic optical communication scheme with physical layer encryption is proposed and studied theoretically. The external cavity optical feedback method is analyzed. An erbium-doped fiber amplifier is used to represent the all-optical delay feedback loop of the traditional external cavity; it can overcome the shortcomings of the traditional method such as lower precision and high equipment volume requirements. The bidirectional communication scheme is discussed; it sets the encryption device at the transmitter, uses the correlation between the two encryption signals to decrypt, and cancels the decryption device at the receiver, which not only simplifies the experimental equipment but also solves the problem that the receiver cannot decrypt synchronously due to the channel damage in remote communication. Further, it is easily applied in production and life. Finally, the simulation results show that the bit error rate is <10 − 5, and the bidirectional transmission of 10-Gb / s information over an 85-km single-mode fiber is successfully realized.

Chaos science is also used in the fields of medicine, biology and finance besides in the engineering fields such as communications, electronics and computer. Up till today, many different chaotic systems have been introduced in the literature in order to be used in these fields. The majority of these systems have usually been examined numerically and only some of them have electronic circuit design. The chaotic systems must be able to be done electronically especially in the fields such as secure communication, encryption, etc. In 1994 were found 19 attractor systems that were simple-structured but producing complex chaotic signals and named A to S by Sprott. In this study electronic circuit design of Sprott H chaotic system with the positive (non-inverting) second generation current conveyor with high gain and high bandwidth were performed in Orcad-PSPICE program. Numerical analysis results of Sprott H chaotic system and the designed electronic circuit outputs were found to be the same. The system can be used secure communication and encryption applications of which electronic circuit were designed.

In this work, an improved nonlinear estimator based scheme is introduced to enhance the performance of a previously proposed chaotic digital communication scheme. The earlier scheme was improved in two ways. First the speed of the scheme is improved by a factor greater than ten, therefore is more suitable for real time communication where all delays (including processing time) must be kept extremely small in applications such as radar communication, telephony, and videoconferencing. Second, the new scheme's bit error rate performance is superior to that of the previous one. The key factor for the fast decoding of this new algorithm is that it is designed based on faster divergence of the estimator in case of chaotic system mismatch. Simulation results are provided and show the feasibility and superiority of our scheme over the earlier proposed one.

The variable-order fractional (VOF) chaotic systems offer a promising solution for applications in secure communication due to their unique properties. This paper addresses the synchronisation problem in secure communication for these systems, which have uncertainties and external disturbances with unknown bounds. According to the variable-order fractional type Mittag-Leffler stability theorem, a fractional-order derivative is applied to a sliding surface, and suitable adaptive laws are devised to address uncertainties and disturbances. A variable-order fractional control strategy and a new criterion are developed to ensure the synchronisation error systems achieve asymptotic stability in finite time, for which the upper limit can be obtained. Simulation outcomes demonstrate the efficacy of the proposed synchronisation strategy in secure communication scenarios.

In recent years, chaotic systems have begun to take a substantial place in literature due to increasing importance of secure communication. Chaotic synchronization which has emerged as a necessity for secure communication can now be performed with many different methods. This study purposes the Master-Slave synchronization via active controller and real time simulation of five different chaotic systems such as Lorenz, Sprott, Rucklidge, Moore-Spiegel, Rössler. Master slave synchronization was performed because of synchronization realized between the same type of chaotic systems with different initial parameters and also because of the systems were expected to behave similarly as a result of synchronization. Active control method was used to amplify the difference signal between master and slave systems which have different initial parameters and to return back synchronization information to the slave system. The real time simulation and synchronization of the master and slave systems performed successfully in LabVIEW environment. Furthermore, for the real time implementation, analogue outputs of NI-DAQ card used and real time results also were observed on an oscilloscope and secure communication application using sinusoidal signal and an image encryption application achieved successfully.