Rabinovich System Research Articles

A novel adaptive synchronization algorithm for a general class of fractional-order complex-valued systems with unknown parameters, and applications to circuit realization and color image encryption

This article proposes an adaptive synchronization (AS) algorithm to synchronize a general class of fractional-order complex-valued systems with completely unknown parameters, which may appear in physical and engineering problems. The analytical and theoretical concepts of the algorithm rely on the mathematical framework of the Mittag-Leffler global stability of fractional-order systems. A specific control system is established analytically based on the fractional-order adaptive laws of parameters, and the corresponding numerical results are executed to verify the accuracy of the AS algorithm. The proposed synchronization method is evaluated using the fractional-order complex Rabinovich system as an attractive example. The electronic circuits of the new system with different fractional orders are designed. By utilizing the Multisim electronic workbench software, various chaotic/hyperchaotic behaviors have been observed, and a good agreement is found between the numerical results and experimental simulation. In addition, the approximation of the transfer function for different fractional-order are presented. And the corresponding resistor and capacitor values in the chain ship model (CSM) are estimated, which can be utilized in designing electronic circuits for other fractional-order systems. Furthermore, two strategies for encrypting color images are proposed using the AS algorithm and fractional-order adaptive laws of parameters. In the first strategy, the color image is treated as a single package and divided into two vectors. The first vector is embedded into transmitter parameters, while the second vector is injected into the transmitter state signals. In the second strategy, the primary RGB channel components of the original color image are extracted and separated into two vectors, and the same process is followed as in the first strategy. These strategies complicate the decryption task for intruders. Different scales of white Gaussian noise are added to color images to examine the robustness of the proposed color images encryption strategies.

Dynamical analysis of a new 5D hyperchaotic system

This paper reports a new five-dimensional (5D) autonomous hyperchaotic system that is obtained by introducing two linear controllers to the Rabinovich system. The dynamical behaviors, including the boundedness, dissipativity and invariance, existence and stability of nonzero equilibrium points are studied and analyzed. The existences of the hyperchaotic and chaotic attractors are numerically verified through analyzing phase trajectories, Lyapunov exponent spectrum, bifurcations and Poincaré maps. The results indicate that the new 5D Rabinovich system can exhibit rich and complex dynamical behaviors. Finally, the existence of Hopf bifurcation, the stability and expression of the Hopf bifurcation are investigated by using the normal form theory and symbolic computations. Some cases are employed to test and verify the theoretical results.

Dual Function Matrix Projective Synchronization for Fractional-order Hyperchaotic Systems

Abstract It is well known that the variability and complexity of projection proportionality factors of Dual Projective Synchronization (DPS) can effectively enhance signal confidentiality. However, in most literatures, the proportionality factors are some simple fixed constants, which can't ensure high security of information. For two pairs of fractional-order hyperchaotic systems (FOHS), how to expand the projection proportionality factors to increase its complexity? Then, our work will propose a new synchronization type, i.e., Dual Function Matrix Projective Synchronization (DFMPS) and realize the DFMPS for FOHS for the first time. Firstly, based on the traditional DPS, we generalize the proportionality factors to a general function matrix depending on time t, present the error functions and define the DFMPS. Then, for FOHS, the active controller and synchronization condition are designed and proved. At the same time, when the system is affected by parameter disturbance, the proposed active controller can eliminate the influence of parameter disturbance to the system's DFMPS, which indicates that the proposed control strategy has strong robustness. Finally, the DFMPS of two pairs of fractional-order hyperchaotic Chen and Rabinovich systems are realized, and synchronizing analysis and system robustness analysis are verified by numerical simulation. Particularly, the DFMPS can be degenerated to dual anti-synchronization, dual complete synchronization, DPS, modified DPS and dual matrix projective synchronization. This work extends the synchronization types for FOHS and offers a useful method to explore DFMPS for other fractional-order systems.

Global dynamical analysis of the integer and fractional 4D hyperchaotic Rabinovich system

In this paper, the dynamical behavior of integer and fractional 4D hyperchaotic Rabinovich system is studied. We use the Lagrange coefficient method to solve an optimization problem analytically so that find an accurate ultimate bound set (UBS) for the 4D hyperchaotic Rabinovich system. Furthermore, bifurcation diagrams, Lyapunov exponents, global attractive sets (GASs), and positive invariant sets (PISs) of the fractional-order system are studied. Finally, using the Mittag-Leffler function and Lyapunov function method, the Mittag-Leffler GAS and Mittag-Leffler PIS of the proposed system are estimated. Our investigations indicate that there is a close relationship between changing the parameters and the dynamical behavior of the system, Hamilton energy consumption and changing the bound of the variables. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.

A hyperchaos generated from Rabinovich system

<abstract><p>In this paper, we present a 4D hyperchaotic Rabinovich system which obtained by adding a linear controller to 3D Rabinovich system. Based on theoretical analysis and numerical simulations, the rich dynamical phenomena such as boundedness, dissipativity and invariance, equilibria and their stability, chaos and hyperchaos are studied. In addition, the Hopf bifurcation at the zero equilibrium point of the 4D Rabinovich system is investigated. The numerical simulations, including phase diagrams, Lyapunov exponent spectrum, bifurcations, power spectrum and Poincaré maps, are carried out in order to analyze and verify the complex phenomena of the 4D Rabinovich system.</p></abstract>

On the chaotic nature of the Rabinovich system through Caputo and Atangana–Baleanu–Caputo fractional derivatives

The Rabinovich system can describe different physical interactions, including waves in plasmas, a convective fluid flow inside a rotating ellipsoid, and Kolmogorov’s flow interactions. This study considers the Rabinovich system through Caputo and Atangana–Baleanu fractional derivatives to detect its chaotic nature. First, the existence and uniqueness of the solutions of the fractional-order systems are proved using the combination of the Picard–Lindelöf theorem and the Banach contraction principle. Then, a numerical approximation of the fractional systems is developed. The fractional Rabinovich system is found to exhibit a chaotic behavior verified via Lyapunov exponents. However, the fractional-order models do not enter into chaotic behavior at the same fractional-derivative order. Bifurcation diagrams referring to variation of the fractional-order derivatives are provided. Chaotic attractors for both cases of the fractional-derivative representation of the system are depicted. The two fractional-order models of the system show sensitivity to initial conditions. A master–response synchronization was developed in the context of the Atangana–Baleanu fractional derivative. The master and the response systems showed a strong correlation, proving the system’s applicability in solving real problems, including secure communications.

Approximate Closed-Form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

Based on some geometrical properties (symmetries and global analytic first integrals) of the Rabinovich system the closed-form solutions of the equations have been established. The chaotic behaviors are excepted. Moreover, the Rabinovich system is reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions are built using the Optimal Auxiliary Functions Method (OAFM). The advantage of this method is to obtain accurate solutions for special cases, with only an analytic first integral. An important output is the existence of complex eigenvalues, depending on the initial conditions and physical parameters of the system. This approach was not still analytically emphasized from our knowledge. A good agreement between the analytical and corresponding numerical results has been performed. The accuracy of the obtained results emphasizes that this procedure could be successfully applied to more dynamic systems with these geometrical properties.

Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

Chaotic Dynamics in Generalized Rabinovich System

The article is devoted to the study of the global behavior of the generalized Rabinovich system describing the process of interaction between waves in plasma. For this generalized system, we obtain the positive invariant set (ultimate bound) and globally exponential attractive set using the approach where we transform the initial problem of finding the corresponding set to the conditional extremum problem and solve this problem. Furthermore, the rate of the trajectories going from the exterior of the attractive set to the interior of the attractive set is also obtained. Numerical localization of attractor is presented. Meanwhile, the volumes of the ultimate bound set and the global exponential attractive set are obtained, respectively. The main innovation of this article lays in considering the generalized form of the Rabinovich system with [Formula: see text] and obtaining the results on the ultimate bound set and globally exponential attractive set for this general case.

Synchronization of Hyperchaotic Rabinovich Systems Based on Adaptive Controller

Synchronization of Hyperchaotic Rabinovich Systems Based on Adaptive Controller

Ergodicity of stochastic Rabinovich systems driven by fractional Brownian motion

The current paper is devoted to dynamics of stochastic Rabinovich systems driven by fractional Brownian motion. By using Krylov–Bogoliubov criterion and constructing of Lyapunov function, the existence of invariant measure of stochastic Rabinovich system is established. The uniqueness of invariant measure is also obtained by the strong Feller property and topological irreducibility. Therefore the considered system possess exactly one invariant measure, which is also an unique adapted stationary solution.

Synchronization of Fractional Hyperchaotic Rabinovich Systems via Linear and Nonlinear Control with an Application to Secure Communications

This paper proposes two approaches for the synchronization of a fractional hyperchaotic Rabinovich master-slave pair. The first approach introduces a linear control law to synchronize the systems in their commensurate case, i.e. with a constant fractional-order. The convergence of the proposed scheme is proven by means of an appropriate Lyapunov functional. In the incommensurate scenario, a nonlinear controller is developed based on our choice of a suitable set of control parameters. Numerical results are presented to illustrate the validity and convergence of the proposed synchronization schemes. In addition, the incommensurate control law is applied to a signal-masking-based secure communication system, whose performance is assessed through simulations.

Dynamical analysis of a new fractional-order Rabinovich system and its fractional matrix projective synchronization

This paper constructs a new physical system, i.e., the fractional-order Rabinovich system, and investigates its stability, chaotic behaviors, chaotic control and matrix projective synchronization. Firstly, two Lemmas of the new system's stability at three equilibrium points are given and proved. Next, the largest Lyapunov exponent, the corresponding bifurcation diagram and the chaotic behaviors are studied. Then, the linear and nonlinear feedback controllers are designed to realize the system's local asymptotical stability and the global asymptotical stability, respectively. It's particularly significant that, the fractional matrix projective synchronization between two Rabinovich systems is achieved and two kinds of proofs are provided for Theorem 4.1. Especially, under certain degenerative conditions, the fractional matrix projective synchronization can be reduced to the complete synchronization, anti-synchronization, projective synchronization and modified projective synchronization of the fractional-order Rabinovich systems. Finally, all the theoretical analysis is verified by numerical simulation.

Dynamics analysis and Hamilton energy control of a generalized Lorenz system with hidden attractor

Hidden attractor can be found in some dynamic systems. More commonly, it can be excited by the stabilized equilibria, or be generated from the systems without equilibria. The generalized Lorenz system transformed from the Rabinovich system is researched by detecting the generating mechanism under different parameters and initial values, and then we have the good fortune to discover that the hidden attractor is coexisting with the states of stabilization, period, chaos, and even transient chaos. At the same time, the Hamilton energy function of the system is given to discuss the energy transform when the system undergoes a series of oscillations. The compositional principle can be used to design a new chaos control method, which is called Hamilton energy control. By numerical simulating, the feedback gain in the present control method is assigned and then controls the system with hidden attractor to expected states effectively. The feature of the control method can be indicated that the Hamilton energy can be detected during the oscillation control processes.

Fractional matrix and inverse matrix projective synchronization methods for synchronizing the disturbed fractional‐order hyperchaotic system

In this paper, for synchronizing two actual nonidentical fractional‐order hyperchaotic systems disturbed by model uncertainty and external disturbance, the fractional matrix and inverse matrix projective synchronization methods are presented and the methods' correctness and effectiveness are proved. Especially, under certain degenerative conditions, the methods are reduced to study the complete synchronization, antisynchronization, projective (or inverse projective) synchronization, modified (or modified inverse) projective synchronization, and stabilization problem for the disturbed (or undisturbed) fractional‐order hyperchaotic systems. In addition, as the fractional matrix and inverse matrix projective synchronization methods' applications, the fractional‐order hyperchaotic Chen and Rabinovich systems disturbed by model uncertainty and external disturbance are constructed, and the matrix and inverse matrix projective synchronizations between the two disturbed systems are achieved, respectively. This work constructs a basic theoretical framework of fractional matrix and inverse matrix projective synchronization methods and provides a general method for synchronizing the actual disturbed fractional‐order hyperchaotic systems that are related to science and engineering.

Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden attractor in the case of multistability as well as a classical self-excited attractor. The hidden attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of attractors. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden attractor and hidden transient chaotic set in the case of multistability are given.

Master-Slave Synchronization of 4D Hyperchaotic Rabinovich Systems

This paper is concerned with master-slave synchronization of 4D hyperchaotic Rabinovich systems. Compared with some existing papers, this paper has two contributions. The first contribution is that the nonlinear terms of error systems remained which inherit nonlinear features from master and slave 4D hyperchaotic Rabinovich systems, rather than discarding nonlinear features of original hyperchaotic Rabinovich systems and eliminating those nonlinear terms to derive linear error systems as the control methods in some existing papers. The second contribution is that the synchronization criteria of this paper are global rather than local synchronization results in some existing papers. In addition, those synchronization criteria and control methods for 4D hyperchaotic Rabinovich systems are extended to investigate the synchronization of 3D chaotic Rabinovich systems. The effectiveness of synchronization criteria is illustrated by three simulation examples.

Joint image encryption and compression scheme based on a new hyperchaotic system and curvelet transform

This paper proposes a joint image encryption and compression scheme based on a new hyperchaotic system and curvelet transform. A new five-dimensional hyperchaotic system based on the Rabinovich system is presented. By means of the proposed hyperchaotic system, a new pseudorandom key stream generator is constructed. The algorithm adopts diffusion and confusion structure to perform encryption, which is based on the key stream generator and the proposed hyperchaotic system. The key sequence used for image encryption is relation to plain text. By means of the second generation curvelet transform, run-length coding, and Huffman coding, the image data are compressed. The joint operation of compression and encryption in a single process is performed. The security test results indicate the proposed methods have high security and good compression effect.

Hidden Attractors on One Path: Glukhovsky–Dolzhansky, Lorenz, and Rabinovich Systems

In this report, by the numerical continuation method, we visualize and connect hidden chaotic sets in the Glukhovsky–Dolzhansky, Lorenz and Rabinovich systems using a certain path in the parameter space of a Lorenz-like system.

A new fractional order hyperchaotic Rabinovich system and its dynamical behaviors

In the paper, the dynamical behaviors of a new fractional order hyperchaotic Rabinovich system are investigated, which include its local stability, hyperchaos, chaotic control and synchronization. Firstly, a new fractional order hyperchaotic Rabinovich system with Caputo derivative is proposed. Then, the hyperchaotic attractors of the commensurate and incommensurate fractional order hyperchaotic Rabinovich system are found. After that, four linear feedback controllers are designed to stabilize this fractional order system Finally, by using the active control method the synchronization is studied between the fractional order hyperchaotic and chaos controlled Rabinovich system In addition, the theoretical predictions are confirmed by numerical simulations.