Complex Chaotic Systems Research Articles

Secure Communication of Fractional Complex Chaotic Systems Based on Fractional Difference Function Synchronization

Fractional complex chaotic systems have attracted great interest recently. However, most of scholars adopted integer real chaotic system and fractional real and integer complex chaotic systems to improve the security of communication. In this paper, the advantages of fractional complex chaotic synchronization (FCCS) in secure communication are firstly demonstrated. To begin with, we propose the definition of fractional difference function synchronization (FDFS) according to difference function synchronization (DFS) of integer complex chaotic systems. FDFS makes communication secure based on FCCS possible. Then we design corresponding controller and present a general communication scheme based on FDFS. Finally, we respectively accomplish simulations which transmit analog signal, digital signal, voice signal, and image signal. Especially for image signal, we give a novel image cryptosystem based on FDFS. The results demonstrate the superiority and good performances of FDFS in secure communication.

Modulus Synchronization of a Novel Hyperchaotic Real System and Its Corresponding Complex System

The synchronization of chaotic real systems or chaotic complex systems has been extensively discussed in previously conducted studies. However, much is not known regarding the synchronization between complex drive systems and real response systems. This study investigates a novel modulus synchronization between a hyperchaotic real system and its corresponding complex system. The systems are designed and their dynamical properties including invariance, dissipation, equilibria, Lyapunov exponents and the spectrum of them, as well as hyperchaotic attractors are analyzed in detail. Moreover, the adaptive controller is also developed based on Lyapunov stability theory. Furthermore, the numerical simulation experiments verify the effectiveness of our idea.

Dual phase and dual anti-phase synchronization of fractional order chaotic systems in real and complex variables with uncertainties

In the present article the nonlinear control method is used for dual phase and dual anti-phase synchronizations among fractional order chaotic systems with uncertainties. The control functions are designed to achieve synchronization with the help of nonlinear control technique. The nonlinear control method is found to be very effective and convenient to achieve dual phase and dual anti-phase synchronization with parametric uncertainties of the non-identical chaotic systems. The fractional order real and complex chaotic systems with parametric uncertainties are taken to illustrate dual phase and dual anti-phase synchronization process. Numerical simulation results show the possibility of dual phase and dual anti-phase synchronizations, which are carried out using Adams–Bashforth–Moulton method and graphical results are presented to display the effectiveness of the method. The striking feature of the article is the graphical demonstration of fast communication through signals between transmitter and receiver for the complex variable systems compared to the real variable systems.

Local Volterra multivariable chaotic time series multi-step prediction based on phase points clustering

To solve the multivariable multi-step prediction problem in chaotic complex systems, this paper proposes a local Volterra model based on phase points clustering. Firstly, reconstruct the phase space of the data and calculate the similarity of the evolution trajectories. According to the similarity, the initial clustering center of the observation point is calculated and the clustering is carried out by means of K mean. We find the cluster class nearest to the prediction phase, compare the predicted phase point with the evolutionary trajectory similarity of all the observed points in the cluster, select the optimal neighboring phase point, and the optimal neighboring phase point is used for training and multi-step prediction of the multivariable local Volterra model. The proposed model method can greatly reduce the time of multi-step prediction and improve the efficiency of prediction. Finally, by experimenting with the data of Beijing PM2.5 acquired from UCI machine learning database, the experimental results show that this model method has better predictive performance.

Sliding mode synchronization of fractional-order complex chaotic system with parametric and external disturbances

In this paper, the synchronization of two fractional-order complex chaotic systems with unknown parameters and external disturbances are studied. Based on the Lyapunov stability theory and fractional-order integral sliding surface, a novel active sliding mode controller is proposed to synchronize fractional-order complex chaotic systems. Moreover, the controller is robust to unknown parameters and external disturbances. Numerical simulations are implemented. Results indicate the proposed control strategy is robust and effective.

Parameter estimation of a complex chaotic system with unknown initial values

The parameter estimation of a chaotic system is an important issue in nonlinear science, and it has gained increased attention in recent years. However, the existing estimation methods must be based on the initial values known in the original system. Yet the initial values cannot be obtained in many cases, which is not conducive to the reconstruction and control of chaotic systems. In addition, these methods are unable to provide enough precision for high-dimensional complex chaotic systems. In this paper, a parameter estimation method with unknown initial values is developed, and a new algorithm called the chaos behaved particle swarm optimization algorithm is proposed. Three highlights, namely chaos initialization, special inertia weight and chaos search, are introduced into the algorithm. The simulation experiments are carried out for three complex chaotic systems, including two typical fractional-order hyperchaotic systems and a fractional-order multi-directional multi-scroll chaotic attractor system. Based on the results, the method proposed in this paper demonstrates more effectiveness and advantages than the other four existing algorithms.

Combination–combination phase synchronization among non-identical fractional order complex chaotic systems via nonlinear control

In the present article nonlinear control method is used for combination–combination phase synchronization among fractional order non-identical complex chaotic systems. The control functions are designed with the help of a new lemma and Lyapunov stability theory. The nonlinear control method is found to be very effective and convenient to achieve the said type of synchronization of the non-identical fractional order complex chaotic systems. Numerical simulations are carried out using Adams-Bashforth–Moulton method and the results are depicted through graphs for different particular cases.

New generalized chaos-geometric and neural networks approach to nonlinear dynamics of the complex systems

We present a new generalized approach to modeling nonlinear processes of chaotic systems based on the known concept of compact geometric attractors chaos theory methods in effective realization plus implemented neural networks simulation algorithm Using information on the phase space evolution of the nonlinear process in time and the neural networks simulation techniques can be considered as one of the fundamentally new approaches in the construction of global nonlinear prediction models for evolutionary dynamics of the complex chaotic systems and accurate description of the structure of the corresponding strange attractors

On synchronisation of a class of complex chaotic systems with complex unknown parameters via integral sliding mode control

Chaotic systems demonstrate complex behaviour in their state variables and their parameters, which generate some challenges and consequences. This paper presents a new synchronisation scheme based on integral sliding mode control (ISMC) method on a class of complex chaotic systems with complex unknown parameters. Synchronisation between corresponding states of a class of complex chaotic systems and also convergence of the errors of the system parameters to zero point are studied. The designed feedback control vector and complex unknown parameter vector are analytically achieved based on the Lyapunov stability theory. Moreover, the effectiveness of the proposed methodology is verified by synchronisation of the Chen complex system and the Lorenz complex systems as the leader and the follower chaotic systems, respectively. In conclusion, some numerical simulations related to the synchronisation methodology is given to illustrate the effectiveness of the theoretical discussions.

Synchronisation and Circuit Realisation of Chaotic Hartley System

Abstract Hartley chaotic system is topologically the simplest, but its dynamical behaviours are very rich and its synchronisation has not been seen in literature. This paper aims to introduce a simple chaotic system which can be used as alternative to classical chaotic systems in synchronisation fields. Time series, phase portraits, and bifurcation diagrams reveal the dynamics of the mentioned system. Chaotic Hartley model is also supported with electronic circuit model simulations. Its exponential dynamics are hard to realise on circuit model; this paper is the first in literature that handles such a complex modelling problem. Modelling, synchronisation, and circuit realisation of the Hartley system are implemented respectively in MATLAB-Simulink and ORCAD environments. The effectiveness of the applied synchronisation method is revealed via numerical methods, and the results are discussed. Retrieved results show that this complex chaotic system can be used in secure communication fields.

Synchronization of time delay systems with non-diagonal complex scaling functions

This paper deals with a kind of synchronization of dynamical systems with complex variables which is called generalized complex modified hybrid function projective synchronization (GCMHFPS) of time delay complex chaotic (hyperchaotic) systems. In other words, that the time delay complex systems can be synchronized up to a complex function transformation matrix. Moreover, the elements of the transformation matrix are complex functions of the states of drive system and time, where this matrix is not square. The idea of an active control method based on complex Krasovskii–Lyapunov functional is used to achieve GCMHFPS of time delay complex systems. Furthermore, based on the non-diagonal complex function transformation matrix, the modules and phases errors between one state of the complex response system and more than one of the states of the drive system are studied which have not been discussed before as far as we know. The analytical expression regarding the stability of this technique is derived and excellent agreement is found upon comparison with numerical calculations. In particular, we show through studying the time evolution of error, modulus and phase that the proposed scheme is effective for controlling time delay complex systems.

Synchronization and antisynchronization of N‐coupled fractional‐order complex chaotic systems with ring connection

This paper is devoted to investigate synchronization and antisynchronization of N‐coupled general fractional‐order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional‐order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional‐order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among N‐coupled fractional‐order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.

Secure communication with a chaotic system owning logic element

In recent years, various chaotic systems have been introduced in literature to be used in different branches of science. In the field of secure chaotic communication, more complex chaotic systems are proposed, such as high dimensional (4D, 5D), multi-scroll, surface equilibrium point, to enhance communication security level. In this study, a chaotic system owning logic element is used for increasing the level of security of chaotic communication system, which has not been previously implemented for this purpose in the literature. The used chaotic system includes signum functions, maximum function and absolute operation term. Also, there are analog devices and digital device in the electronic circuit of this chaotic system. These properties increase the complexity of the chaotic system and the security level. Sliding mode control method is preferred for the synchronization part of the secure communication. In this regard, numerical analysis and electronic circuit design of the secure chaotic communication system by using the chaotic system owning logic element have been performed successfully.

SPRING: a novel parallel chaos-based image encryption scheme

Due to the increasing demand on secure image transmission, image encryption has emerged as an active research field in recent years. Many of the proposed image encryption schemes are designed based on chaotic maps with permutation–diffusion architecture. While most of these schemes reported good statistical properties, they are slow in execution speed due to inherent data dependency of the proposed schemes. Some of these schemes are designed based on complex chaotic systems that require significant computational resources to obtain the keystream for encryption. In this paper, we propose SPRING, a novel image encryption scheme designed based on lightweight chaotic maps and simple logical and arithmetic operations, which is also highly optimized for massively parallel architecture (e.g. GPU). The extensive experimental results show that SPRING is not only secure but also able to achieve high encryption speed in single-core CPU, multi-core CPU and many-core GPU. Encrypting a 512 $$\times $$ 512 grayscale image in serial takes 0.9126 ms which is 220% faster than state-of-the-art ARX-based image encryption scheme proposed by Choi et al. SPRING can be implemented in parallel to encrypt the same image in 0.0862 ms by exploiting many-core GPU, which is 10 $$\times $$ faster than the serial version implemented using CPU.

Adaptive Dual Synchronization of Chaotic (Hyperchaotic) Complex Systems with Uncertain Parameters and Its Application in Image Encryption

Adaptive Dual Synchronization of Chaotic (Hyperchaotic) Complex Systems with Uncertain Parameters and Its Application in Image Encryption

Adaptive control of complex systems with unknown dynamics and input constraint: Applied to a chaotic elastic beam

SummaryOwing to the limitations of system identification and modeling techniques, there is usually some unknown dynamics in the mathematical models of the complex systems. In addition, external perturbations can affect the chaotic systems' responses and may destroy the desired control purpose. Consideration of such uncertain dynamics and external fluctuations in control applications is important in research and practice. On the other hand, because of the limited operation of control actuators, most of the practical implementations of control systems are forced with some input constraints. Therefore, this paper investigates the control problem of uncertain autonomous and/or nonautonomous complex chaotic systems in the presence of input saturation. The upper bounds of the unknown dynamics, modeling uncertainties, external perturbations, and the parameters of the saturation function are assumed to be unknown in advance. To make a fast control response, an adaptive nonsingular terminal variable structure controller is proposed to assure the finite‐time stability of the equilibrium states. Rigorous stability analysis is performed to prove the correct performance of the designed control algorithm. Numerical simulations on the unified system and a chaotic elastic beam model are developed to demonstrate the usefulness of the introduced adaptive control strategy. It is worth to notice that the derived adaptive nonsmooth sliding mode approach is general and it can be easily adopted for controlling of a wide class of uncertain MIMO nonlinear systems.

Complex modified function projective synchronization of complex chaotic systems with known and unknown complex parameters

Much progress has been made in the research of modified function projective synchronization (MFPS) for real (complex) chaotic systems with real parameters. The scaling functions are always chosen as real-valued ones in previous MFPS schemes for chaotic systems evolving in the same or inverse directions simultaneously. However, MFPS with different complex-valued scaling functions (CMFPS) has not been previously reported, where complex-variable chaotic (hyperchaotic) systems (CVCSs) evolve in different directions with a time-dependent intersection angle. Therefore, CMFPS is discussed for CVCSs with known and unknown complex parameters in this paper. By constructing appropriate Lyapunov functions defined on complex field, and employing adaptive control technique, a set of simple and practical sufficient conditions for achieving CMFPS are derived, and complex update laws for estimating unknown parameters are also given. The corresponding theoretical proofs and computer simulations are worked out to demonstrate the effectiveness and feasibility of the proposed schemes.

Hybrid projective synchronization of complex Duffing–Holmes oscillators with application to image encryption

Many works on hybrid projective synchronization (or simply ‘HPS’ for short) of nonlinear real dynamic systems have been performed, while the HPS of chaotic complex systems and its application have not been extensively studied. In this paper, the HPS of complex Duffing–Holmes oscillators with known and unknown parameters is separately investigated via nonlinear control. The adaptive control methods and explicit expressions are derived for controllers and parameters estimation law, which are respectively used to achieve HPS. These expressions on controllers are tested numerically, which are in excellent agreement with theory analysis. The proposed synchronization scheme is applied to image encryption with exclusive or (or simply ‘XOR’ for short). The related security analysis shows the high security of the encryption scheme. Concerning the complex Duffing–Holmes oscillator, we also discuss its chaotic properties via the maximum Lyapunov exponent. Copyright © 2017 John Wiley & Sons, Ltd.

On the dynamics, existence of chaos, control and synchronization of a novel complex chaotic system

In this paper, the authors have studied the dynamics of a novel complex chaotic system with fractional order derivative and found the existence of chaos. The novel complex system is simulated for integer as well as fractional orders which shows some unusual phenomena. The main contribution of this effort is an implementation of the Largest Lyapunov Exponent (LLE) criteria based on Wolf’s algorithm. The conditions for chaos control based on the fractional Routh–Hurwitz stability conditions and feedback control are given. Also synchronization between a fractional order novel chaotic system and a controlled fractional order novel system using the modified adaptive projective synchronization method for different scaling matrices has been obtained. Numerical simulation results are carried out using the Adams–Bashforth–Moulton method.

Synchronization between fractional order complex chaotic systems with uncertainty

In this article, the synchronization between fractional order complex chaotic systems with uncertainty is considered using Active control method. Synchronization of the fractional order chaotic systems with uncertainty have already been studied in real variable, but for the case of complex variable it is first of its kind. Due to complexity and the presence of additional variables, it is more interesting and important for the researchers working on the real life problem. The fractional order complex Lorenz system and complex T-system are chosen to address the theoretical result in the presence of uncertain parameters. Numerical simulation is carried out using Adams-Bashforth-Moulton method for different particular cases, which reveals that the method is reliable and effective, even for fractional order complex chaotic systems, and in the presence of uncertain terms.