Advances in fractional chaotic systems

The essence of parameter estimation in fractional chaotic systems is a multi-dimensional parameter optimization problem, which is essential for achieving fractional chaos control and synchronization. In this article, identification of parameters and orders of fractional-order chaotic systems is transformed into optimization of some special nonnegative functions mathematically equivalently. Chemical Reaction Optimization (CRO), a general-purpose metaheuristic to search the optimal solution of the objective function, is selected to solve this optimization problem. Based on the parameter identification of fractional Lorenz chaotic system, fractional chaoticsystem and fractional Volta chaotic system, simulation results show that the algorithm is effective, robust and powerful.

Today, chaotic systems have become one of the most important tools for encrypting and secure transmission of information. Other applications of these systems in economics, geography, sociology, and the like are not hidden from anyone. Despite the presentation of various chaotic systems, it is necessary to study and present new and more accurate chaotic systems. It is obvious that fractional models are more accurate and yield better results than integer order models. In this paper, the synchronization and anti-synchronization of an innovative fractional order chaotic system is investigated based on the nonlinear control method. In the proposed chaotic system, there is an exponential term that leads to behaviour very different from the integer order chaotic systems. Two different approaches have been proposed to achieve the synchronization and anti-synchronization goals between the proposed new fractional chaotic systems. A backstopping approach has been used to synchronize, and in addition to achieving this goal, it also ensures stability in Lyapunov's concept. Anti-synchronization between the two new fractional systems is also achieved by applying the active control method, and subsequently Lyapunov stability is shown under the proposed method. The simulation results in MATLAB environment show the synchronization and anti-synchronization effectiveness for the proposed innovative fractional order chaotic system.

In this paper, based on sliding mode control and adaptive control theory, the synchronization of two different fractional order chaotic systems is investigated. First, a fractional sliding surface with strong robustness is designed and a suitable adaptive sliding controller is constructed, then the error states of the systems are controlled to the sliding surface via the method to guarantee the synchronized behaviors between two fractional chaotic systems. Numerical simulations on the hyper Chen chaotic systems and Chen chaotic system are also carried out respectively. Simulation results show that the generalized errors tend to zero after a short time, and the effectiveness and feasibility of this method are well verified.

Stability about fractional chaotic system is studied and a theory about fractional chaotic system is proposed and proved under intermittent control in this paper. Based on the theorem, a controller is designed to realize the intermittent synchronizing fractional unified chaotic system. Numerical simulation demonstrates the effectiveness of the theorem.

To synchronize fractional chaotic systems with different orders, a method is proposed in which a fractional chaotic system with different orders is changed into a fractional chaotic system with the same order but different structures, according to the properties of fractional differential equation. This method is successfully used to synchronize fractional Lorenz chaotic systems. Numerical simulation demonstrates the effectiveness of the method.

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.

Synchronizing chaotic systems using control based on a special matrix structure and extending to fractional chaotic systems

Random number generator design is one of the practical applications of nonlinear systems. This study used random number generation and sound encryption application with a fractional chaotic system. Random numbers were generated with the Langford chaotic system, and a sound encryption application was carried out for the secure transmission of voice messages. Randomization performance of numbers was evaluated by employing NIST-800-22 statistical tests, which meet the highest international requirements. It was observed that the distributions of these generated random numbers reached the desired level of randomness after the examination. Unlike the integer-order random number generators widely used in the literature, the fractional-order Langford chaotic system was employed to generate and analyze random numbers and demonstrate their utilization in sound encryption. Random numbers generated from a fractional degree-based chaotic system developed in this study can be used in cryptology, secret writing, stamping, statistical sampling, computer simulations, dynamic information compression and coding.

In this paper the synchronization between complex fractional order chaotic system and integer order hyper chaotic system has been investigated. Due to increased complexity and presence of additional variables, it seems to be very interesting and can be associated with real life problems. Based on the idea of tracking control and non linear control, we have designed the controllers to obtain the synchronization between the chaotic systems. To establish the efficacy of the methods computations have been carried out. Excellent agreement between the analytical and computational studies has been observed. The achieved synchronization is illustrated in field of secure communication. The results have been compared with published literature.

This paper introduces a new three-dimensional chaos system that is different from any existing chaotic systems. The basic dynamic properties of the suggested system have been analyzed through theoretical and numerical studies and the rich chaotic behavior has been confirmed using phase portraits, Lyapunov analysis and bifurcation diagram. A simple model with a nonlinear quadratic term, unstable saddle equilibrium points and broadband chaotic behavior are some of the interesting features of the suggested system. Then, due to higher accuracy of fractional order models than integer order models, a novel fractional chaotic system has been extracted based on the suggested chaotic system and new nonlinear methods planned to achieve the synchronization and anti-synchronization goals for the system. A backstepping approach is designed to synchronize and ensure stability in Lyapunov’s concept. Also, the anti-synchronization between the two novel fractional systems is achieved by applying the active control technique, and subsequently Lyapunov stability is shown under the proposed scheme. The simulation results in MATLAB environment show the synchronization and anti-synchronization efficiency for the proposed innovative fractional order turbulent system.

<abstract><p>This paper presents a chaotic complex system with a fractional-order derivative. The dynamical behaviors of the proposed system such as phase portraits, bifurcation diagrams, and the Lyapunov exponents are investigated. The main contribution of this effort is an implementation of Mittag-Leffler boundedness. The global attractive sets (GASs) and positive invariant sets (PISs) for the fractional chaotic complex system are derived based on the Lyapunov stability theory and the Mittag-Leffler function. Furthermore, an effective control strategy is also designed to achieve the global synchronization of two fractional chaotic systems. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.</p></abstract>

This study is devoted to developing a chaotic secure communication system based on the fractional model by designing a novel fractional filter. The complexity of fractional-order systems is exploited to increase security. Channel noise and disturbance can impress or even disturb the received data. We chose this particular apparatus on account of the fact that the proposed filter can preciously estimate the fractional-order chaotic states with regard to the disturbance in a noisy environment, which results in receiving data precisely. This filter is applied to a fractional chaotic Lorenz system to illustrate the advantages of the proposed filter in state estimation of fractional chaotic systems and decrypting data from chaotic states.

This paper concerns with the fractional order unified chaotic control based on passivity. A hybrid control strategy combined with fractional state feedback and passive control is proposed, derived from the properties of fractional calculus and the concept of passivity. The fractional chaotic system with the hybrid controller proposed can be stabilized at its equilibrium under different initial conditions. Numerical simulation results present the verification on the effectiveness of the proposed control method.

Recently, generating visually secure cipher images by compressive sensing (CS) techniques has drawn much attention among researchers. However, most of these algorithms generate cipher images based on direct bit substitution and the underlying relationship between the hidden and modified data is not considered, which reduces the visual security of cipher images. In addition, performing CS on plain images directly is inefficient, and CS decryption quality is not high enough. Thus, we design a novel cryptosystem by introducing vector quantization (VQ) into CS-based encryption based on a 3D fractional Lorenz chaotic system. In our work, CS compresses only the sparser error matrix generated from the plain and VQ images in the secret generation phase, which improves CS compression performance and the quality of decrypted images. In addition, a smooth function is used in the embedding phase to find the underlying relationship and determine relatively suitable modifiable values for the carrier image. All the secret streams are produced by updating the initial values and control parameters from the fractional chaotic system, and then utilized in CS, diffusion, and embedding. Simulation results demonstrate the effectiveness of the proposed method.

Data-driven learning of the fractional discrete-time unified system is studied in this paper. A neural network method is suggested in the parameter estimation of fractional discrete-time chaotic systems. An optimization problem is obtained and the famous Adam algorithm is employed to train the neural network’s weights and parameters. The parameter estimation result is compared with that of the stepwise response sensitivity approach (SRSA). This paper provides a high accuracy method for parameter inverse problems. The method also can be applied to data-driven learning of other fractional chaotic systems.