Generalized synchronization involving a linear combination of fractional-order chaotic systems

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

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.

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.

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.

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.

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.

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.

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.

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.

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, 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.

The identification of fractional order systems by multiscale multivariate analysis

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

The research on a time delayed fractional order financial chaotic system is a hot issue. In this paper, synchronization of time delayed fractional order financial chaotic system is studied. Based on comparison principle of linear fractional equation with delay, by using a fractional order inequality, a sufficient condition is obtained to guarantee the synchronization of master-slave systems. An example is exploited to show the feasibility of the theoretical results.

This paper presents a colour image encryption algorithm based on cross-coupled chaotic map and fractional order chaotic systems. Firstly the algorithm employs the cross-coupled chaotic skew tent map to perform the shuffling operation, and then uses the fractional order versions of Lorenz's system and Chen's system to disturb image pixel intensity values. Fractional order extensions of the chaotic systems provides a much larger key-space than their original integer order versions. The encrypted image exhibits uniform histogram and a very high entropy for all the three colour channels. Also, the property of zero correlation is satisfied by the intensity values of adjacent pixels in each channel of the encrypted image. Moreover, the algorithm possesses a key-space which is large enough to resist all possible kinds of statistical attacks. Theoretical analysis and experimental results thus demonstrate that the proposed scheme has an excellent efficiency and satisfactory security attributes. However, solving fractional order differential equations is a computationally heavy process. Some efficient computing techniques are successfully employed to deal with this problem, so that the encryption-decryption process is executed reasonably fast.