Exponential synchronization of fractional-order complex chaotic systems and its application

In this paper, a generalized scheme for synchronizing a fractional order chaotic system with another one or with a linear combination of two other fractional order chaotic systems is presented. Static (time-independent) or dynamic (time-dependent) synchronization that could generate multiple scaled versions of the response is discussed for some fractional order continuous chaotic systems based on differential equations. Non-Standard finite difference method suitable for fractional order chaotic systems is used to solve each system and get the responses. Analysis in the generalized fractional order case is presented, as well as results of simulations that match the corresponding analysis for different values of fractional order parameters.

In order to figure out the dynamical behaviour of a fractional-order chaotic system and its relation to an integer-order chaotic system, in this paper we investigate the synchronization between a class of fractional-order chaotic systems and integer-order chaotic systems via sliding mode control method. Stability analysis is performed for the proposed method based on stability theorems in the fractional calculus. Moreover, three typical examples are carried out to show that the synchronization between fractional-order chaotic systems and integer-orders chaotic systems can be achieved. Our theoretical findings are supported by numerical simulation results. Finally, results from numerical computations and theoretical analysis are demonstrated to be a perfect bridge between fractional-order chaotic systems and integer-order chaotic systems.

Now the chaotic system and wavelet transform are more and more widely used in the watermarking technology. At the same time, the fractional order chaotic system has more complex dynamic characteristics than the integer order system. So a new image watermarking scheme based on the fractional order Chen chaotic system and discrete wavelet transform is proposed. Chaotic sequences generated by chaotic system are used to encrypt the watermark image, and the processed watermark information is embedded into the original image by the discrete wavelet transform. Finally, the security analysis of the proposed watermarking algorithm is presented. The experimental results show that the proposed watermarking scheme has high security, and it has stronger robustness and invisibility compared with the previous work.

The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.

Fractional calculus is a 300 years topic, which has been introduced to real physics systems modeling and engineering applications. In the last few decades, fractional-order nonlinear chaotic systems have been widely investigated. Firstly, the most used methods to solve fractional-order chaotic systems are reviewed. Characteristics and memory effect in those method are summarized. Then we discuss the memory effect in the fractional-order chaotic systems through the fractional-order calculus and numerical solution algorithms. It shows that the integer-order derivative has full memory effect, while the fractional-order derivative has nonideal memory effect due to the kernel function. Memory loss and short memory are discussed. Finally, applications of the fractional-order chaotic systems regarding the memory effects are investigated. The work summarized in this manuscript provides reference value for the applied scientists and engineering community of fractional-order nonlinear chaotic systems.

Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos.

A modified adaptive control method is developed in this article and the parameters identification method is then applied in fractional order systems with unknown parameters. The new modified control method based on Lyapunov stability theory is successfully applied to investigate the synchronization of pair of fractional order systems amongst Genesio–Tesi, Qi and Chen systems. By means of the Adams–Bosford–Moulton method, the numerical results show that the modified method is easy to implement and reliable for synchronizing the two different fractional order chaotic systems.

This paper focuses on the finite-time generalized synchronization problem of non-identical fractional order chaotic (or hyper-chaotic) systems by a designing adaptive sliding mode controller and its application to secure communication. The effects of both disturbances and model uncertainties are taken into account. A novel fractional order integral sliding mode surface is designed and its stability to the origin is proved in a given finite time. By the aid of the fractional Lyapunov stability theory, a robust controller with adaptive update laws is proposed and its finite-time stability for generalized synchronization between two non-identical fractional-order chaotic systems in the presence of model uncertainties and external disturbances is derived. Numerical simulations are provided to demonstrate the effectiveness and robustness of the presented approach. All simulation results obtained are in good agreement with the theoretical analysis. According to the proposed generalized finite-time synchronization criterion, a novel speech cryptosystem is proposed to send or share voice messages privately via secure channel. Security and performance analyses are given to show the practical effect of the proposed theories.

In this brief paper, projective synchronization (Projective Syn.) in coupled integer-order and fractional-order chaotic systems are both studied. For the case of driving coupled integer-order chaotic systems, theoretical analysis is carried out directly based on the Lyapunov stability theory. For the case of coupled fractional order chaotic system, we took a modified approximate method to obtain an equivalent integer order model for the fractional order chaotic system, which is much accurater than traditional approximation methods mentioned in literatures. Based on theoretical analysis, the mechanism of the occurrence of projective synchronization in coupled the fractional order chaotic systems is deduced in detailed. Also, a simple controller for adjusting the scaling factor of projective synchronization is designed, which can lead the states' evolution to desired value. This state error feedback controller is easily fulfilled in practice. Numerical experiments are also given to show the rightness of the theoretical analysis and the effectiveness of our proposed method by taking the newly proposed Liu-Chen system as an illustration.

In this article, the novel fractional order three-dimensional chaotic system has been constructed. Its dynamical properties have been thoroughly analyzed. For varying fractional order the properties have also been studied. This model can be made applicable in the field of control systems, secure communication,image encryption etc.

In this paper, a fast control scheme is presented for the problem of Q-S synchronization between fractional chaotic systems with different dimensions and orders. Using robust control law and Laplace transform, a synchronization approach is designed to achieve Q-S synchronization between n-D and m-D fractional-order chaotic systems in arbitrary dimension d. This paper provides further contribution to the topic of Q-S synchronization between fractional-order systems with different dimensions and introduces a general control scheme that can be applied to wide classes of fractional chaotic and hyperchaotic systems. Numerical example and simulations are used to show the effectiveness of the proposed approach.

In this paper, we introduced a novel approach to achieve the data encryption. The fractional order Lorenz chaotic system is used to generate the chaotic sequence. The article denotes the characteristic of the chaotic sequence. Two example about the plain text and plain image is shown which using the approach we introduced.

Parameter estimation for fractional-order chaotic systems by improved bird swarm optimization algorithm

In this paper, the projective synchronization problem of two fractional-order different chaotic (or hyperchaotic) systems with both uncertain dynamics and external disturbances is considered. More particularly, a fuzzy adaptive control system is investigated for achieving an appropriate projective synchronization of unknown fractional-order chaotic systems. The adaptive fuzzy logic systems are used to approximate some uncertain nonlinear functions appearing in the system model. These latter are augmented by a robust control term to compensate for the unavoidable fuzzy approximation errors and external disturbances as well as residual error due to the use of the so-called e-modification in the adaptive laws. A Lyapunov approach is adopted for the design of the parameter adaptation laws and the proof of the corresponding stability as well as the asymptotic convergence of the underlying synchronization errors towards zero. The effectiveness of the proposed synchronization system is illustrated through numerical experiment results.

In this article, an adaptive modified projective synchronization scheme of fractional-order chaotic systems with unknown parameters and disturbance is investigated. The synchronization can be completed with any desired accuracy. Based on the stability theory of fractional-order systems, the adaptive controllers and the parameters’ updated laws are designed to achieve modified projective synchronization between the fractional-order chaotic systems with disturbance. Furthermore, all unknown parameters of the response system are identified simultaneously. Numerical simulations on the fractional-order chaotic Lorenz system further demonstrate that the proposed method is effective and robust.