Hyperchaotic Attractors Research Articles

The 4-dimensional hyperchaotic memristive circuit based on Chua’s circuit

Recently, there has been a growing interest in chaotic memristive circuits. However, four-dimensional (4D) memristive system often can only exhibit common chaos with only one positive Lyapunov exponent. By replacing the resistor of Chua’s circuit with a memristor, we propose a new simple 4D memristive circuit in this paper. A major difference between our proposed system and the known chaotic or hyperchaotic system is that our modified system has infinitely many stable and unstable equilibria. We show that the system can exhibit rich complex dynamic behaviors, such as limit cycles, chaos and hyperchaos. Further numerical study and circuit simulation verify the existence of a hyperchaotic attractor in the memristive circuit, which gives a positive answer about whether there exists hyperchaos in 4D memristive systems.

Theoretical Analysis and Circuit Verification for Fractional-Order Chaotic Behavior in a New Hyperchaotic System

A novel nonlinear four-dimensional hyperchaotic system and its fractional-order form are presented. Some dynamical behaviors of this system are further investigated, including Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations, and calculated Lyapunov exponents. A simple fourth-channel block circuit diagram is designed for generating strange attractors of this dynamical system. Specifically, a novel network module fractance is introduced to achieve fractional-order circuit diagram for hardware implementation of the fractional attractors of this nonlinear hyperchaotic system with order as low as 0.9. Observation results have been observed by using oscilloscope which demonstrate that the fractional-order nonlinear hyperchaotic attractors exist indeed in this new system.

Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points

Based on three-dimensional (3D) Lu chaotic system, we introduce a four-dimensional (4D) nonlinear system with infinitely many equilibrium points. The Lyapunov-exponent spectrum is obtained for the 4D chaotic system. A hyperchaotic attractor and a chaotic attractor are emerged in this 4D nonlinear system. Furthermore, to verify the existence of hyperchaos, the chaotic dynamics of this 4D nonlinear system is also studied by means of topological horseshoe theory and numerical computation.

Generating variable number of wings from a novel four-dimensional hyperchaotic system with one equilibrium

In this paper, a novel four-dimensional smooth system is presented. What particularly interests us is that the novel multi-wing chaotic system has only one equilibrium point at the origin. Furthermore, the strange phenomenon that different chaotic attractors such as the two-wing, three-wing chaotic attractors and four-wing hyperchaotic attractor could be generated by changing a single system parameter makes this system unique. By applying either analytical or numerical methods, basic properties of the system such as phase portraits, Poincaré mapping, bifurcation diagram and Lyapunov exponents are investigated to observe chaotic motions. The physical existences of the multi-wing chaotic attractors are verified by an electronic circuit.

A New Five Dimensional Hyperchaotic System and its Fractional Order Form

In this paper, a new five-dimensional hyperchaotic system by introducing two additional states feedback into a three-dimensional smooth chaotic system. With three nonlinearities, this system has more than one positive Lyapunov exponents. Based on the fractional derivative theory, the fractional-order form of this new hyperchaotic system has been investigated. Through predictor-corrector algorithm, the system is proved by numerical simulation analysis. Simulation results are provided to illustrate the performance of the fractional-order hyperchaotic attractors well.

On the transition to hyperchaos and the structure of hyperchaotic attractors

Using a recently proposed algorithmic scheme for correlation dimension analysis of hyperchaotic attractors, we study two well-known hyperchaotic flows and two standard time delayed hyperchaotic systems in detail numerically. We show that at the transition to hyperchaos, the nature of the scaling region changes suddenly and the attractor displays two scaling regions for embedding dimension M ≥ 4. We argue that it is an indication of a strong clustering tendency of the underlying attractor in the hyperchaotic phase. Because of this sudden qualitative change in the scaling region, the transition to hyperchaos can be easily identified using the discontinuous changes in the dimension (D2) at the transition point. We show this explicitely for the two time delayed systems. Further support for our results is provided by computing the spectrum of Lyapunov Exponents (LE) of the hyperchaotic attractor in all cases. Our numerical results imply that the structure of a hyperchaotic attractor is topologically different from that of a chaotic attractor with inherent dual scales, at least for the two general classes of hyperchaotic systems we have analysed here.

Dynamical Analysis of a Fractional Order Multi-Wing Hyper-Chaotic System

The dynamic behaviors of fractional-order systems have attracted increasing attentions recently. In this paper, a fractional-order four-wing hyper-chaotic system which has a rich variety of dynamic behaviors is proposed. We numerically study the dynamic behaviors of this fractional-order system with different conditions. Hyper-chaotic behaviors can be found in this system when the order is lower than 3 and four-wing hyper-chaotic attractors similar to integer order system can be generated. The lowest order for Hyper-chaos to exist in this system is 3.6 and the lowest order for chaos to exist in this system is 2.4.

Robust control for fractional-order four-wing hyperchaotic system using LMI

This paper reports a new fractional-order four-dimensional (4D) smooth autonomous system which is special since it can generate a four-wing hyperchaotic attractor but has only one zero equilibrium. The issue of robust control for the fractional-order hyperchaotic system with uncertainties and interference is considered. On the basis of the fractional Lyapunov stability theorem, a linear state feedback controller is derived to guarantee robust Mittag–Leffler stability of the fractional-order hyperchaotic system. The control strength matrix is obtained by solving the linear matrix inequality (LMI). The effectiveness and robustness of the proposed scheme is verified via numerical simulation.

Circuit Realization of Grid Multi-Wing Hyperchaotic System

This paper investigates a grid multi-wing hyperchaotic system firstly, then an improved module-based circuits are designed for generating the hyperchaotic attractor. Numerical simulation and circuit implementation have testified the feasibility of the proposed method.

A 5D HYPERCHAOTIC SYSTEM WITH THREE POSITIVE LYAPUNOV EXPONENTS COINED

This paper reports the finding of a five-dimensional (5D) new hyperchaotic system with three positive Lyapunov exponents, which is obtained by adding a nonlinear controller to the first equation of a 4D hyperchaotic system. The algebraical form of the hyperchaotic system is very similar to the 5D controlled Lorenz-like systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under unique equilibrium or three equilibria. To further analyze the new system, the corresponding hyperchaotic and chaotic attractor are firstly numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation, analysis of power spectrum and Poincaré projections. Moreover, some complex dynamical behaviors such as the stability of hyperbolic or nonhyperbolic equilibrium and two complete mathematical characterizations for 5D Hopf bifurcation are rigorously derived and studied.

Controlling hyperchaotic complex systems with unknown parameters based on adaptive passive method

The aim of this paper is to study the control of hyperchaotic complex nonlinear systems with unknown parameters using passive control theory. An approach is stated to design the passive controller and estimate the unknown parameters based on the property of the passive system. The feasibility and effectiveness of the proposed approach is demonstrated through its application to the hyperchaotic complex Lü system, as an example. The estimated values of the unknown parameters are calculated. The analytical form of the complex controller is derived and used in the numerical simulation to control the hyperchaotic attractors of this example. Block diagrams of this example using Matlab/Simulink are constructed after and before the control to ensure the validity of the analytical results. Other examples of hyperchaotic complex nonlinear systems can be similarly treated.

Grid Multiscroll Hyperchaotic Attractors Based on Colpitts Oscillator Mode with Controllable Grid Gradient and Scroll Numbers

Through introducing two piecewise-linear triangular wave functions in a three-dimensional spiral chaotic Colpittsoscillator model, a four-dimensional grid multiscroll hyperchaotic system is constructed. Interestingly, by adjusting abuild-in parameter in a variable of one triangle wave function, the control of the gradient of the multiscroll grid isachieved. Whereas by deploying the zero points of the two triangular wave functions to extend the saddle-focusequilibrium points with index-2 in phase space the scroll numbers do not only increase along with the number ofturning points, but they can also generate arbitrary multiples of products. The basic dynamical behaviors of theproposed four-dimensional multiscroll hyperchaotic system are analyzed. Finally, the hardware experimental circuit isdesigned and the interrelated circuit implementation is realized. The experimental results are in agreement with boththeoretical analyses and numerical simulations, which verify the feasibility of the design methods.

Explicit ultimate bound sets of a new hyperchaotic system and its application in estimating the Hausdorff dimension

Ultimate bound estimation of chaotic systems is a difficult yet interesting mathematical question. At present, explicit ultimate bound sets can be analytically obtained only for some special chaotic systems, and few results are known for hyper-chaotic ones. In this paper, through the Lagrange multiplier method and set operations, one derives two kinds of explicit ultimate bound sets for a novel hyperchaotic system. Based on the estimated result and optimization method, one further estimates the Hausdorff dimension of the hyperchaotic attractor. Numerical simulations show the effectiveness and correctness of the conclusions. The investigations enrich the related results for the hyperchaotic systems, and provide support for the assertion: hyperchaotic systems are ultimately bounded.

Complex complete synchronization of two nonidentical hyperchaotic complex nonlinear systems

In this paper, we introduce the definition of complex complete synchronization (CCS) of hyperchaotic complex nonlinear systems that have not been introduced recently in the literature. This type of synchronization can study only for complex nonlinear systems. On the basis of Lyapunov function, a scheme is designed to achieve the CCS of two nonidentical hyperchaotic attractors of these systems. The effectiveness of the obtained results is illustrated by a simulation example. Numerical results are plotted to show state variables, modules errors, and phases errors of these hyperchaotic attractors after synchronization to prove that CCS is achieved. Copyright © 2013 John Wiley & Sons, Ltd.

Hyperchaos Numerical Simulation and Control in a 4D Hyperchaotic System

A hyperchaotic system is introduced, and the complex dynamical behaviors of such system are investigated by means of numerical simulations. The bifurcation diagrams, Lyapunov exponents, hyperchaotic attractors, the power spectrums, and time charts are mapped out through the theory analysis and dynamic simulations. The chaotic and hyper-chaotic attractors exist and alter over a wide range of parameters according to the variety of Lyapunov exponents and bifurcation diagrams. Furthermore, linear feedback controllers are designed for stabilizing the hyperchaos to the unstable equilibrium points; thus, we achieve the goal of a second control which is more useful in application.

Dynamics of a New Hyperchaotic System with Only One Equilibrium Point

A new 4D hyperchaotic system is constructed based on the Lorenz system. The compound structure and forming mechanism of the new hyperchaotic attractor are studied via a controlled system with constant controllers. Furthermore, it is found that the Hopf bifurcation occurs in this hyperchaotic system when the bifurcation parameter exceeds a critical value. The direction of the Hopf bifurcation as well as the stability of bifurcating periodic solutions is presented in detail by virtue of the normal form theory. Numerical simulations are given to illustrate and verify the results.

Anti-lag synchronisation of hyperchaotic complex non-linear systems

In this paper, we present the anti-lag synchronisation ALS of two identical or non-identical hyperchaotic complex non-linear systems. Based on Lyapunov function a scheme is designed to achieve ALS of hyperchaotic attractors of these systems. The ALS of two identical complex Chen systems and two different hyperchaotic complex Lorenz and Lu systems are taken as two examples to verify the feasibility of the presented scheme. Numerical simulations are computed to demonstrate the effectiveness of the analytical expressions of the controller.

Dynamics and Feedback Control for a New Hyperchaotic System

In this paper, stability and Hopf bifurcation analysis of a novel hyperchaotic system are investigated. Four feedback control strategies, the linear feedback control method, enhancing feedback control method, speed feedback control method and delayed feedback control method, are used to control the hyperchaotic attractor to unstable equilibrium. Moreover numerical simulations are given to verify the theoretical results. Keywords—Feedback control, Hopf bifurcation, hyperchaotic system, stability.

HYPERCHAOS IN A MEMRISTOR-BASED MODIFIED CANONICAL CHUA'S CIRCUIT

After the successful solid state implementation of the memristor, memristor-based circuits have received a lot of attention. In this paper, a memristor with cubic nonlinear characteristics is employed in the modified canonical Chua's circuit. A systematic study of hyperchaotic behavior in this circuit is performed with the help of nonlinear tools such as Lyapunov exponents, phase portraits and bifurcation diagrams. In particular, an imitative memristor circuit is examined to reveal the construction of hyperchaotic attractors.

Design and Implementation of Grid Multiwing Hyperchaotic Lorenz System Family via Switching Control and Constructing Super-Heteroclinic Loops

This paper initiates a systematic methodology for generating various grid multiwing hyperchaotic attractors by switching control and constructing super-heteroclinic loops from the piecewise linear hyperchaotic Lorenz system family. By linearizing the three-dimensional generalized Lorenz system family at their two symmetric equilibria and then introducing the state feedback, two fundamental four-dimensional linear systems are obtained. Moreover, a super-heteroclinic loop is constructed to connect all equilibria of the above two fundamental four-dimensional linear systems via switching control. Under some suitable conditions, various grid multiwing hyperchaotic attractors from the real world applications can be generated. Furthermore, a module-based circuit design approach is developed for realizing the designed piecewise linear grid multiwing hyperchaotic Lorenz and Chen attractors. The experimental observations validate the proposed systematic methodology for grid multiwing hyperchaotic attractors generation. Our theoretical analysis, numerical simulations and circuit implementation together show the effectiveness and universality of the proposed systematic methodology.