Complex Chaotic Systems Research Articles

Modified projective synchronization between fractional order complex chaotic systems

In this paper, the modified projective synchronization between two fractional order complex chaotic systems is deeply studied. The controller is designed on the basis of stability theory of fractional order complex chaotic systems to synchronize the considered systems. Finally, the numerical solution is investigated to support it.

New insights into a chaotic system with only a Lyapunov stable equilibrium

This paper gives some new insights into a chaotic system. The considered system has only one Lyapunov stable equilibrium and positive Lyapunov exponent in some certain parameter range, which means there coexists chaotic attractors and Lyapunov stable equibirium in system. First, from the perspective of analyzing the global structure of the system, based on the Poincaré compactification technique, the complete description of its dynamical behavior on the sphere at infinity is presented. The obtaining results show that its global dynamical behavior is very complex. Second, from a geometric perspective, the Jacobi stability of the system is investigated, including the unique equilibrium and a periodic orbit. Interestingly, we find that the unique Lyapunov stable equilibrium is always Jacobi unstable, a Lyapunov stable periodic orbit falls into both Jacobi stable and Jacobi unstable regions. It is shown that we might witness chaotic behavior of the system before the trajectories enter a neighborhood of the equilibrium point. It is hoped that the investigation of this paper can contribute to better understand and study the complex chaotic system with stable equilibria.

Combination Projective Synchronization in Fractional-Order Chaotic System with Disturbance and Uncertainty

In this manuscript, we study combination projective synchronization (CPS). In CPS, matrix projective combination synchronization (MPCS) and inverse matrix projective combination synchronization (IMPCS) between non-identical fractional-order complex chaotic systems subjected to uncertainty and external disturbance is investigated. Matrix projective synchronization (MPS) and inverse matrix projective synchronization is obtained when the scaling factor is a constant matrix, which gives the assurance of high security in secure communication and image encryption. Based on the Lyapunov stability theory and appropriate active control technique, the MPCS and IMPCS between two master systems and one slave system has been achieved. Based on the MPCS synchronization, a scheme of secure communication is presented, and the message signals are transmitted using the chaotic signal masking method. Finally, numerical simulations have been provided, which shows that our theoretical results are in complete agreement will the graphical one.

Stabilization of a Class of Complex Chaotic Systems by the Dynamic Feedback Control

The stabilization problem of the complex chaotic system is investigated in this paper. First, a systematic method is proposed, by which a given complex chaotic system can be transformed into its equivalent real chaotic system. Then, both simple and physical controller is designed for the corresponding real chaotic system by the dynamic feedback control method, thereby the controller for the original complex chaotic system is obtained. Especially, for some complex system, the controller is obtained by the linear feedback control method. Finally, two illustrative examples with numerical simulations are used to verify the validity and effectiveness of the theoretical results.

Module-phase synchronization of fractional-order complex chaotic systems based on RBF neural network and sliding mode control

Combined with RBF neural network and sliding mode control, the synchronization between drive system and response system was achieved in module space and phase space, respectively (module-phase synchronization). The RBF neural network is used to estimate the unknown nonlinear function in the system. The module-phase synchronization of two fractional-order complex chaotic systems is implemented by the Lyapunov stability theory of fractional-order systems. Numerical simulations are provided to show the effectiveness of the analytical results.

Synchronization of Non-Integer Complex Chaotic Systems with Uncertainty and Disturbances via Adaptive Sliding Mode Technique

In this paper we concern the adaptive sliding mode control technique for synchronization of fractional order chaotic systems with uncertainties and disturbances. This technique is used to design control law through suitable sliding surface and estimate the external disturbances. Computational results using MATLAB verified the effectiveness of the proposed scheme.

Mitigating Drawbacks of Logistic Map for Image Encryption Algorithms

This paper identifies and analyses some drawbacks of the logistic map which is still one of the most used chaotic maps in image encryption algorithms. As some of the disadvantages are caused by inappropriate implementations of the logistic map, this paper proposes a set of rules which should lead to enhancement of the desired chaotic behavior. Probably the most important rule introduces alternating value of parameter utilized by the logistic map. With careful choice of values and an adapted quantization technique, some of the issues should be fixed and theoretically also the values of numerical parameters should be improved. These assumptions are verified by applying the proposed set of rules on an algorithm from our prior work. Effects of the proposed rules on the used algorithm are investigated and all necessary modifications are thoroughly discussed. The paper also compares obtained values of commonly used numerical parameters and computational complexity with some other image encryption algorithms based on more complex chaotic systems.

Color Image Encryption Algorithm Based on Dynamic Chaos and Matrix Convolution

This paper proposes a color image encryption algorithm based on a cloud model Fibonacci chaotic system, as well as a matrix convolution operation that can protect image content effectively and safely. The algorithm combines the cloud model with the generalized Fibonacci, creating a new complex chaotic system that realizes the dynamic random variation of chaotic sequences. The chaotic sequence is used to scramble the pixel coordinates of the mosaic images of the R, G, and B components of the color image. Then, the chaotic sequence value is used as a matrix convolution cloud algorithm that alternately updates the input value of the matrix convolution operation and the pixel value to obtain the permutation transformation of the original pixel value. Finally, the pixel values of the replacement and cloud model Fibonacci chaotic sequence and the pixel values of the front (rear) adjacent pixel points are subjected to a two-way exclusive XOR operation. Realizing the change of the arbitrary pixel value causes a chain transformation of the pixel values of all of the pixel points, and sequentially generates an encrypted image. Experiments show that the histogram of the encrypted image is smoother and adjacent pixels of the image have low correlation. In addition, this algorithm can resist attack experiments such as differential attack, select plaintext attack and noise attack and provides high encryption security, high anti-interference, and strong robustness. The dynamic chaotic system is used to realize the color image encryption of the dynamic key, and the encryption algorithm has higher security and the validity of the algorithm.

Modeling and Inverse Complex Generalized Synchronization and Parameter Identification of Non-Identical Nonlinear Complex Systems Using Adaptive Integral Sliding Mode Control

This paper presents the Inverse Complex Generalized Synchronization (ICGS) of non-identical nonlinear complex systems with unknown parameters. Using the philosophy of adaptive integral sliding mode control, an adaptive controller and laws regarding parametric upgradation are designed to realize ICGS and parameter identification of two non-identical chaotic complex systems with respect to a given complex map vector. To employ the control, the error system is transformed into a unique structure containing a nominal part and some unknown terms, which are computed adaptively. Then, the error system is stabilized by using integral sliding mode control. The stabilizing controller for the error system is constructed, which consists of the fractional-order control plus some compensator control. To avoid the chattering phenomenon, smooth continuous compensator control is incorporated instead of traditional discontinuous control. The compensator controller and the adapted law are derived in such a way that the time derivative of a Lyapunov function becomes strictly negative. This scheme is applied to synchronize a Memristor-Based Hyperchaotic Complex (MBHC) Lu system and a Memristor-Based Chaotic Complex (MBCC) Lorenz system, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system with entirely unknown parameters. The effectiveness and feasibility of the proposed scheme is validated through computer simulation using MATLAB software package.

Specialized study to perform complex phase synchronization of two chaotic complex systems including a similar structure of direct terms with modifying in nonlinear terms

We ponder a different kind of complex synchronization. We name this sort “complex phase synchronization” (CPHS). This review concentrates on two chaotic complex structures, including a similar structure of direct terms with totally differing or incompletely in nonlinear terms. A novel plan is needed to complete the complex phase synchronization (CPHS) for the chaotic complex structures. In this plan, we have utilized the dependability hypothesis to decide diagnostically the complex control capacities that are expected to accomplish CPHS. From this plan, we found that CPHS holds or incorporates two sorts of synchronizations (phase and anti‐phase synchronizations) and the state factors of the main structure synchronize with alternate state factors of the slave structure. We connected this plan, for instance, to study the CPHS of two chaotic complex Lü structures with incompletely changing in nonlinear terms. These complex (or mind‐boggling) structures resemble in various imperative areas of mineral science and building. Numerical outcomes are sketched out to uncover the phases and modules errors of these chaotic attractors, subsequently communicating that CPHS is fulfilled. A specific application in secure communication is designed. Amongst this secure communication, CPHS between transmitter and collector is the key to the information signs are regained.

Simplified method and synchronization for a class of complex chaotic systems

AbstractThis paper is devoted to investigate a class of complex chaotic systems and a linear correlation between the real and imaginary component of complex variables in these systems is found. Based on this linear relationship, a simplified law is proposed. First, complex Lorenz system is given to show the linear correlation, then it is simplified. Second, a simplified law is proposed to determine whether the complex system can be simplified, and the complex Lü system and hyperchaotic complex Lü system are used to verify the simplified law. Finally, a new synchronization control is proposed to synchronize complex Lorenz system and real Lorenz system. The theoretical analysis and numerical simulation prove the feasibility and better performance of this method.

Synchronization on the adaptive sliding mode controller for fractional order complex chaotic systems with uncertainty and disturbances

The present paper purports to examine and analyse the concept of non identical complex chaotic systems of fractional order with external bounded disturbances and uncertainties. Hybrid projective synchronization has been achieved between fractional order complex Lu-system (drive system) and complex T-system (slave system). The adaptive sliding mode control technique has been used to design control law through suitable sliding surface and estimate the uncertainties and external disturbances in order to establish the stability of controlled system by using allied theorems. Also we have compared our results with prior published literature results to determine the supremacy of considered methodology. Computer simulations outcomes have established the efficacy and adeptness of the prospective scheme.

Synchronization of complex networks of identical and nonidentical chaotic systems via model-matching control.

In this work, a synchronization scheme for networks of complex systems is presented. The proposed synchronization scheme uses a control law obtained with some definitions from graph theory and solving the Model-Matching Problem for complex networks. In particular, Rössler, Chen, Lorenz and Lü chaotic systems are used as complex chaotic systems into complex networks. Particular cases with regular and irregular networks of six identical chaotic systems are implemented, with some well-known topologies as star and ring small-world, and tree topologies. Highlighting, the obtained control law is applied to synchronize an irregular network of six different chaotic systems in a tree topology. The usefulness and advantages of the proposed synchronization scheme are highlighted performing numerical simulations of the chaotic complex networks.

Synchronization of Fractional-Order Complex Chaotic Systems Based on Observers.

By designing a state observer, a new type of synchronization named complex modified projective synchronization is investigated in a class of nonlinear fractional-order complex chaotic systems. Combining stability results of the fractional-order systems and the pole placement method, this paper proves the stability of fractional-order error systems and realizes complex modified projective synchronization. This method is so effective that it can be applied in engineering. Additionally, the proposed synchronization strategy is suitable for all fractional-order chaotic systems, including fractional-order hyper-chaotic systems. Finally, two numerical examples are studied to show the correctness of this new synchronization strategy.

Determination of the Backward Predictability Limit and Its Relationship with the Forward Predictability Limit

In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space. The forward predictability mainly focuses on the forward evolution of initial errors superposed on the initial state over time, while the backward predictability is mainly concerned with when the given state can be predicted before this state happens. From the results, there is a negative correlation between the local forward and backward predictability limits. That is, the forward predictability limits are higher when the backward predictability limits are lower, and vice versa. We also find that the sum of forward and backward predictability limits of each state tends to fluctuate around the average value of sums of the forward and backward predictability limits of sufficient states. Furthermore, the average value is constant when the states are sufficient. For different chaotic systems, the average value is dependent on the chaotic systems and more complex chaotic systems get a lower average value. For a single chaotic system, the average value depends on the magnitude of initial perturbations. The average values decrease as the magnitudes of initial perturbations increase.

Finite‐time multi‐switching sliding mode synchronisation for multiple uncertain complex chaotic systems with network transmission mode

This study mainly investigates finite-time multi-switching synchronisation of multiple uncertain complex chaotic systems with network transmission mode. Both the unknown parameters and disturbances are considered. By establishing the desired multi-switching rules, the definition of finite-time hybrid synchronisation for multi-systems is first given. Then, sliding mode control approach is used to design the synchronisation controllers, which guarantee that any states of each drive system can synchronise the expected states of every response system in finite time. Simultaneously, the adaptive laws effectively estimate the unknown parameters and the unknown constants of disturbances. Finally, simulation example and analysis display the correctness and effectiveness of all innovations.

Adaptive impulsive synchronization for a class of fractional order complex chaotic systems

In this paper, a new lemma is proposed to study the stability of a fractional order complex chaotic system without dividing the complex number into real and imaginary parts. The proving process of the new lemma combines the fundamental properties of the complex field and the fractional order extension of the Lyapunov direct method. It extends the fractional order extension of the Lyapunov direct method from the real number field to the complex number field. Based on the new lemma, we propose a new impulsive synchronization scheme for fractional order complex chaotic systems. The numerical simulation results also show the validity of our conclusion.

Study on Saturation Estimation of Initial Perturbations in Ensemble Marine Forecasting based on BGM Method

Because the ocean is a highly complex nonlinear chaotic system, its numerical prediction results are strongly uncertain due to the influence of the initial values and the model itself. Ensemble forecasting is an effective way to reduce the uncertainty of forecasting. The initial perturbations quality, especially the saturation estimation, is of practical significance for improving ensemble forecasting techniques. In this paper, the factors affecting initial perturbation saturation of ensemble forecasting are analyzed by BGM based on ROMS model, and the perturbation saturation velocities of the two methods are compared. The results show that the saturation velocity of the initial perturbation is faster than that of the initial perturbation space structure. The larger the initial perturbation is, the higher the saturation degree and velocity are, and the saturation time of the same variable varies at different layers. The saturation of different variables is also different; the perturbation saturation speed of the regional adjust perturbation method is faster than that of the classical static perturbation method.

Synchronisation of fractional-order complex systems and its application

In this paper, a passive control scheme based on the fractional-order calculus is proposed. We study the modified complex projective synchronisation between two identical fractional-order complex chaotic systems, and its application in the secure communication. The fractional-order complex chaotic Lorenz system is employed to encrypt the emitted signal. In the transmitter module, the information signal is modulated into one parameter of the Lorenz system. It is assumed that the same parameter is unknown in the receiver module. In order to synchronise two systems with different initial conditions, the controllers and an appropriate parameter update rule are designed. Theoretical analysis and numerical simulations show that this method is feasible and robust to some extent in the presence of channel noise.

Dynamical Analysis of a Novel Complex Chaotic System and Application in Image Diffusion

In this paper, a novel complex chaotic system is constructed. The characteristics of new complex chaotic system are analyzed by symmetric, dissipative and stability, then dynamical performances studied using bifurcation diagram, Lyapunov exponent spectrum and complexity. On the basis of this, an image diffusion algorithm is proposed based on the model of law of gravity. Security performances of the proposed algorithm are researched through the key space, statistics, information entropy, noise attack. The experimental results illustrates that the novel complex chaotic system has abundant dynamical behaviors and large parameters range, the security analysis shows that the proposed image encryption algorithm possess higher security features. Therefore, the research will provide theoretical guidance and experimental basis for chaotic secure communication and information security.