Chen System Research Articles

The $$\psi $$-Hilfer fractional calculus of variable order and its applications

In this paper, we present the $$ \psi $$ -Hilfer fractional derivatives of variable order (FDVO) of 3 types I, II and III, versions A and B, as well as their combinations. In addition, we propose approximations and relations between both derivatives, i.e., $$ \psi $$ -Hilfer FDVO and $$ \psi $$ -Caputo FDVO. With regard to the $$ \psi $$ -Hilfer FDVO type II, we discuss the stability of the FVO nonlinear systems 7 solutions by means of one-parameter Mittag-Leffler functions of variable order. Examples 8 involving the FDVO Lu and Chen systems are also presented.

A New 4D Four-Wing Memristive Hyperchaotic System: Dynamical Analysis, Electronic Circuit Design, Shape Synchronization and Secure Communication

In this paper, a simple four-wing chaotic attractor is first proposed by replacing the constant parameters of the Chen system with a periodic piecewise function. Then, a new 4D four-wing memristive hyperchaotic system is presented by adding a flux-controlled memristor with linear memductance into the proposed four-wing Chen system. The memristor mathematical structure model is simple and easy to implement. Dynamical analysis and numerical simulation of the memristive hyperchaotic system are carried out. Then, the electronic circuit of the hyperchaotic system is designed and implemented. The results of numerical simulation are in good agreement with the electronic circuit experiment. In addition, shape synchronization control for the 4D four-wing memristive hyperchaotic system is realized, and a communication system is designed by using the shape synchronization method. Finally, secure signal masking application is implemented on Matlab platform. In the developed secure communication scheme, the information signal overlaps with the chaotic masking signal, which improves the security of the system.

Nonlinear observer design based on immersion and invariance method: an insight to chaotic systems

This paper presents some use of reduced order observers for two classes of chaotic systems. The first class includes rod-type plasma torch and Arneodo systems and the second class contains Lorenz and Chen systems. In this regard, a recently developed approach called immersion and invariance method is employed. During the estimation the first state of the system is assumed measured. Implementation of the proposed nonlinear observer is simple and does not need high gain to estimate unavailable state in comparison with conventional methods by rendering manifold observer design inspired form sliding mode theory. An important point of this design is to find appropriate mapping that give rise to exponential stability of observer dynamic. It is worth mentioning that the problem of designing a reduced order observer can be turned into the problem of finding a proper invariant manifold. Stability of the proposed scheme is proven via Lyapunov theorem whilst the simulation results confirm the significance of the proposed immersion and invariance observer.

Dynamic analysis, FPGA implementation, and cryptographic application of an autonomous 5D chaotic system with offset boosting

An autonomous five-dimensional (5D) system with offset boosting is constructed by modifying the well-known three-dimensional autonomous Liu and Chen system. Equilibrium points of the proposed autonomous 5D system are found and its stability is analyzed. The proposed system includes Hopf bifurcation, periodic attractors, quasi-periodic attractors, a one-scroll chaotic attractor, a double-scroll chaotic attractor, coexisting attractors, the bistability phenomenon, offset boosting with partial amplitude control, reverse period-doubling, and an intermittency route to chaos. Using a field programmable gate array (FPGA), the proposed autonomous 5D system is implemented and the phase portraits are presented to check the numerical simulation results. The chaotic attractors and coexistence of the attractors generated by the FPGA implementation of the proposed system have good qualitative agreement with those found during the numerical simulation. Finally, a sound data encryption and communication system based on the proposed autonomous 5D chaotic system is designed and illustrated through a numerical example.

Signal flow graph and control of realizable autonomous nonlinear Chen model with quaternion variables

In this work, we continue our study of autonomous nonlinear dynamical systems with quaternion variables. We present a novel chaotic Chen system with quaternion variables. This system in real form is high (nine) dimensional. We study the dynamics and basic properties of this model with quaternion variables. We also evaluate the stability of the trivial points and restrict the conditions under which the nonlinear novel quaternion Chen system has negative, zero, or positive Lyapunov exponents. A signal flow graph and an electronic circuit implementation are designed, analyzed, and constructed to realize the novel Chen system. The control problem of chaotic nonlinear systems with quaternion variables is investigated. We propose and design an approach to build a controller for these systems. The viability and usefulness of the suggested approach are illustrated by an application to the chaotic Chen model with quaternion variables. Using the proposed method, the attractors of the quaternion Chen model are transformed to trivial fixed point, quasi-periodic, or periodic (limit cycle) states. The mathematical results for the controller according to this method are confirmed numerically.

Jacobi analysis for an unusual 3D autonomous system

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

A novel chaotic system without equilibria, with parachute and thumb shapes of Poincare map and its projective synchronisation

In this paper, a three-dimensional novel chaotic system and its projective synchronisation are investigated. The proposed chaotic system has no equilibria. The topological structure of proposed chaotic system is different form Lorenz, Rossler and Chen systems. Different qualitative and quantitative tools such as time series, phase plane, Poincare section, bifurcation plot, Lyapunov exponents, Lyapunov spectrum, and Lyapunov dimension are used to evidence the chaotic behaviour of the proposed system. Further, the projective synchronisation between the proposed chaotic systems is achieved using nonlinear active control. Active control laws are designed, by using sum of the relevant variables of the proposed chaotic systems, to ensure the convergence of error dynamics. The required global asymptotic stability condition is derived using Lyapunov stability theory. Simulation is done in MATLAB environment to verify the theoretical approach. Simulation results reveal that the objectives of the paper are achieved successfully.

Numerical approximation of the system of fractional differential equations with delay and its applications

In this paper, the system of fractional differential equations with delay is studied. The computational method based on the hybrid functions of the general block-pulse functions and Legendre polynomials is presented to obtain the numerical results for the solution of our problem. In fact, the system under study is converted to the corresponding nonlinear algebraic system of equations, which is resolved by using the collocation technique. Also, the error estimate for the suggested computational method that verifies the convergence is presented. Finally, we examine two models arising in engineering and applied science, including the Chen and human immunodeficiency viruses systems, to illustrate the efficiency of the method.

Image encryption based on hyper-chaotic multi-attractors

Linear time-delay feedback method makes the stable system generate the infinite-dimensional hyper-chaos, which possesses more than one positive Lyapunov exponent, infinite-dimensional, a wider chaotic parameter range, and multi-attractors, including the single-scroll attractor, the double-scroll attractor, and the composite multi-scroll attractor. The infinite-dimensional hyper-chaotic multi-attractors Chen system generated by linear time-delay feedback (HCMACS) is used for image encryption. Firstly, the state time sequences of the infinite-dimensional HCMACS are preprocessed to achieve the ideal statistical property. Secondly, two random matrices generated by random number generators are used to expand the image size. Finally, the preprocessed chaotic sequences are used to confuse and diffuse the expanded digital image to obtain the encrypted image. The special feature of the proposed method is the theoretically infinite-dimensional secret key space. Results of analysis and computer simulation indicate that the encryption algorithm has good encryption performance, which can effectively resist various attacks.

Finite time synchronization of fractional chaotic systems with several slaves in an optimal manner

Optimal synchronization of chaotic fractional differential equations with one master and several slaves in finite time is the main aim of this paper. To achieve this goal, we convert the finite time synchronization problem to a fractional optimal control problem; then by solving it, we achieve the active control. In this way, we use Bernstein polynomials and prove that the corresponding minimization problem is a quadratic convex problem. Some examples using the famous Lorenz, Chen, Lu, and Liu systems are given to show the efficiency of the method.

One-Way Hash Function Based on Delay-Induced Hyperchaos

A scheme for constructing one-way Hash function based on hyperchaos induced by time delay and key-stream function iteration is proposed in this paper. In this scheme, the plaintext and secret key are used as the initial value in two hyperchaotic Chen systems; these values are evolved in a hyperchaotic way during a predefined period. The results of the evolution are quantified and iterated using key-stream function iteration to confuse and diffuse the plaintext and secret key. The cipher block chaining mode is used to generate a 128 bits Hash value for a plaintext of arbitrary length. Theoretical analysis and simulation results indicate that the proposed algorithm has satisfactory performance, such as value compression function, irreversibility, initial value sensitivity, forgery resistance and collision resistance.

Topological horseshoe in a single-scroll Chen system with time delay

Abstract The proof that an attractor is chaotic is not trivial. A single-scroll attractor in the Chen system with time delay is investigated through both theory and simulation. The Chen system with time delay is infinite dimensional parameterized by the time delay τ. The detailed procedure operations for finding topological horseshoe in the delay differential equation are different from that one in ordinary differential equation. We show the existence of chaos by using both the topological horseshoe theory and its corollary, and the Smale horseshoe construction if the geometry of the attractor on the 2D plane section satisfies certain conditions. This paper presents both methods for establishing the presence of the horseshoe in a Chen system with time delay. In the first method, we select two quadrilaterals in the 2D transversal section, and calculate the relationship of the quadrilaterals under the map. In the second method, we select quadrilaterals in the neighborhood of a short unstable periodic orbit in the section and obtain the approximate location of the quadrilaterals under the map, yielding the Smale horseshoe. By using the above two methods, the geometrical expansion of the quadrilaterals under the map satisfies the Topological Horseshoe Corollary and also the Smale horseshoe construction, thus showing that the time-delayed single-scroll attractor is indeed chaotic.

Adaptive combination synchronisation of unknown chaotic Lorenz, Lü, Rössler and Chen systems

This paper studies the adaptive combined synchronisation (ACS) of different unknown chaotic systems. In the proposed scheme, the state vectors of a response system synchronise with the corresponding state vectors of the combined dual drive systems. The Lyapunov stability theory proves the asymptotic stability of the closed loop system at the origin. This work provides parameter update laws that estimate the actual values of unknown parameters. Important numerical simulation is included to show the effectiveness of the propose synchronisation scheme.

Image Encryption Based on Hash Table Scrambling and DNA Substitution

This paper proposes an image encryption algorithm based on hash table structure scrambling and DNA substitution. The algorithm uses the classical `scrambling-diffusion' process, and the pseudo-random sequence used in each process is generated by the hyper-chaotic Chen system. Firstly, in the process of scrambling, two sequences with no repeated values are constructed by using the closed hash method in the hash table structure for chaotic sequences, and the plain image is scrambled twice according to the two sequences. Then, in the process of diffusion, DNA coding and decoding rules and DNA substitution rules are made according to the chaotic sequences. The image is dynamically encoded, substituted and decoded with DNA. In this way, while modifying the gray value of image pixels, it also destroys the strong correlation between adjacent pixels. Through a series of verifications, it is concluded that the algorithm not only has good encryption effect, high security and large key space, but also has the advantages of high key sensitivity and strong anti-attack capability.

一种基于新超混沌Chen系统的置乱替代相结合的加密算法

一种基于新超混沌Chen系统的置乱替代相结合的加密算法

An Efficient Nonstandard Finite Difference Scheme for Chaotic Fractional-Order Chen System

In this paper, an efficient nonstandard finite difference scheme for the numerical solution of chaotic fractional-order Chen system is developed. In the new method, an appropriate nonlocal framework in conjunction with the Grünwald-Letnikov approximation are applied for the discretization of fractional differential system. By constructing the discretization with the nonstandard finite difference scheme, high resolution of the system can be obtained, and the numerical instabilities of the nonlinear fractional-order Chen chaotic system can be also addressed to some extent. In addition, a new fractional derivative of the Caputo type is employed in the context of fractional-order Chen system to further decrease the computational complexity in the long-term treatment of fractional model. Numerical simulations demonstrate the applicability, accuracy and efficiency of the developed method.

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.

Multi-switching compound-compound anti-synchronization of non-identical hyperchaotic systems

ABSTRACT In this paper, a synchronization scheme for six non-identical hyperchaotic systems is investigated. The paper addresses a synchronization scheme of five leader (master) and one follower (slave) non-identical hyperchaotic systems. This synchronization scheme is first of its kind and different from existing schemes as instead of a scaling system a compound scaling signal is used. The presented scheme is termed as ‘multi-switching compound-compound anti-synchronization’ (MCCAS). Asymptotically stable synchronized states are established between leader and follower systems by employing nonlinear control method and Lyapunov stability criteria. Additional leader systems and hyperchaotic systems enhance the complexity of this synchronization scheme which is useful in secure communication. The information signal can be split and loaded with different signals of leader systems to increase the grade of security. To justify the theoretical results, an example is considered where Pang-Liu, Zheng, Chen, Liu and Newton-Leipnik hyperchaotic systems are considered as leader systems and Lorenz hyperchaotic system is considered as a follower system. Theoretical results are confirmed by graphical analysis.

Hyperchaos synchronization using univariate impulse control.

Rössler and Chen systems with time delay are shown to be hyperchaotic, which exhibits a more complex dynamics, including multiple positive Lyapunov exponents and infinite dimension. The hyperchaos has better application potential where hyperchaos synchronization is concerned. Univariate impulse control requires smaller perturbation to the response system, thus promising better performance. However, synchronization of two hyperchaotic systems using this control method is a challenging task due to the difficulty to guarantee synchronization stability using a minimum number of manipulated variables. In this paper, a univariate impulse control method is proposed for the synchronization of two hyperchaotic dynamics generated by time delay. A theorem is developed and proved to provide the sufficient conditions for the synchronization of time delay systems using the univariate impulse control. The upper bound of the impulse interval is proved to guarantee the asymptotic synchronization. Simulation and circuit experiment show the correctness of the analysis and the feasibility of the proposed method.

Chaos suppression in fractional order systems using state-dependent noise

Noise play a creative role in the evolution of periodic and complex systems which are essential for continuous performance of the system. The interaction of noise generated within one component of a chaotic system with other component in a linear or nonlinear interaction is crucial for system performance and stability. These types of noise are inherent, natural and insidious. This study investigates the effect of state-dependent noise on the bifurcation of two chaotic systems. Circuit realization of the systems were implemented. Numerical simulations were carried out to investigate the influence of state dependent noise on the bifurcation structure of the Chen and Arneodo-Coullet fractional order chaotic systems. Results obtained showed that state dependent noise inhibit the period doubling cascade bifurcation structure of the two systems. These results poses serious challenges to system reliability of chaotic systems in control design, secure communication and power systems.