Hyperchaotic Attractors Research Articles

Coexistence of Chaos with Hyperchaos, Period-3 Doubling Bifurcation, and Transient Chaos in the Hyperchaotic Oscillator with Gyrators

In this paper, the dynamics of the paradigmatic hyperchaotic oscillator with gyrators introduced by Tamasevicius and co-workers (referred to as the TCMNL oscillator hereafter) is considered. This well known hyperchaotic oscillator with active RC realization of inductors is suitable for integrated circuit implementation. Unlike previous literature based on piecewise-linear approximation methods, I derive a new (smooth) mathematical model based on the Shockley diode equation to explore the dynamics of the oscillator. Various tools for detecting chaos including bifurcation diagrams, Lyapunov exponents, frequency spectra, phase portraits and Poincaré sections are exploited to establish the connection between the system parameters and various complex dynamic regimes (e.g. hyperchaos, period-3 doubling bifurcation, coexistence of attractors, transient chaos) of the hyperchaotic oscillator. One of the most interesting and striking features of this oscillator discovered/revealed in this work is the coexistence of a hyperchaotic attractor with a chaotic one over a broad range of system parameters. This phenomenon was not reported previously and therefore represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. A close agreement is observed between theoretical and experimental analyses.

Complex transient dynamics of hidden attractors in a simple 4D system**Project supported by the National Natural Science Foundation of China (Grant No. 51277017), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK2012583), and the Fundamental Research Funds for the Central Universities of China (Grant No. NS2014038).

A simple four-dimensional system with only one control parameter is proposed in this paper. The novel system has a line or no equilibrium for the global control parameter and exhibits complex transient transition behaviors of hyperchaotic attractors, periodic orbits, and unstable sinks. Especially, for the nonzero-valued control parameter, there exists no equilibrium in the proposed system, leading to the formation of various hidden attractors with complex transient dynamics. The research results indicate that the dynamics of the system shows weak chaotic robustness and depends greatly on the initial states.

Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors

In this paper, a new five-dimensional autonomous hyper-chaotic attractor with complex dynamics is introduced. Nonlinear behavior of the proposed system is different with most well-known chaotic and hyper-chaotic attractors and exists interesting aspects in this system. There are eight parameters to control with only one equilibrium point, and the new dynamics can generate four-wing hyper-chaotic and chaotic attractors for some specific parameters and initial conditions. One remarkable feature of the new system is that it can generate double-wing and four-wing smooth chaotic attractors with special appearance. By using the symmetry properties about the coordinates, initial conditions and bifurcation diagrams, many schemes are presented to indicate multi-scroll or multi-wing systems. The phenomenon of multiple attractors is discussed, which means that several attractors created simultaneously from different initial values, and according to this phenomenon, many strange attractors are obtained. Detailed information on the dynamic of the new system is obtained numerically by means of phase portraits, sensitivity to initial conditions, Lyapunov exponents spectrum, bifurcation diagrams and Poincare maps. The stability analysis of the new attractors has been investigated to understand dynamical behaviors nearby equilibrium points.

Applications of modularized circuit designs in a new hyper-chaotic system circuit implementation**Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 61403395), the Natural Science Foundation of Tianjin, China (Grant No. 13JCYBJC39000), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China, the Fund from the Tianjin Key

Modularized circuit designs for chaotic systems are introduced in this paper. Especially, a typical improved modularized design strategy is proposed and applied to a new hyper-chaotic system circuit implementation. In this paper, the detailed design procedures are described. Multisim simulations and physical experiments are conducted, and the simulation results are compared with Matlab simulation results for different system parameter pairs. These results are consistent with each other and they verify the existence of the hyper-chaotic attractor for this new hyper-chaotic system.

The chaotic Dadras–Momeni system: control and hyperchaotification

In this paper a novel three-dimensional autonomous chaotic system, the so called Dadras-Momeni system, is considered and two different control techniques are employed to realize chaos control and chaos synchronization. Firstly, the optimal control of the chaotic system is discussed and an open loop feedback controller is proposed to stabilize the system states to one of the system equilibria, minimizing the cost function by virtue of the Pontryagin’s minimum principle. Then, an adaptive control law and an update rule for uncertain parameters, based on Lyapunov stability theory, are designed both to drive the system trajectories to an equilibrium or to realize a complete synchronization of the system in periodic and chaotic regimes. Numerical simulations are included to show the effectiveness of the designed controllers in realizing the stabilization and the synchronization of the chaotic system. Finally, a new hyperchaotic system is introduced by adding a controller to the second equation of the three-dimensional autonomous Dadras-Momeni chaotic system and the corresponding hyperchaotic attractor is shown via computer simulations.

On modified time delay hyperchaotic complex Lü system

Time delays are often considered as sources of complex behaviors in dynamical systems. Much progress has been made in the research of time delay systems with real variables. In this article, we will focus our study on time delay complex systems. This paper investigates a modified time delay hyperchaotic complex Lu system. This system is constructed by including the constant delay to one of its states. The behaviors of our time delay system are greatly different from those of the original system without delay. By setting the parameters, we discuss the effect of delay variation on system stability. Numerically, we calculate the range of system parameters at which chaotic and hyperchaotic attractors of different order exist. We found that our system has hyperchaotic attractors of order $$ 2,3,\ldots ,6$$ . However, the modified complex Lu system without delay has only hyperchaotic attractors of order 2. Different forms of modified time delay hyperchaotic complex Lu system are constructed by including the delay into different states of this system. Chaos synchronization in modified time delay hyperchaotic complex Lu system is investigated. The active control method based on Lyapunov–Krasovskii function is used to synchronize the hyperchaotic attractors. In particular, studying the time evolution of errors, we show that this technique is very effective for controlling the behavior of our system.

A Sinusoidally Driven Lorenz System and Circuit Implementation

Another approach is developed for generating two-wing hyperchaotic attractor, four-wing chaotic attractor, and high periodic orbits such as period-14 from a sinusoidally driven based canonical Lorenz system. A sinusoidal function controller is introduced into a 3D autonomous Lorenz system, so that the abovementioned various hyperchaotic attractors, chaotic attractors, and high periodic orbits can be obtained, respectively, by adjusting the frequency of the sine function. In addition, an analog circuit and a digital circuit are also designed and implemented, with experimental results demonstrated. Both numerical simulations and circuit implementation together show the effectiveness of the proposed systematic methodology.

A new Lorenz-type hyperchaotic system with a curve of equilibria

Little seems to be known about hyperchaotic systems with a curve of equilibria. Based on the classical Lorenz system, this paper proposes a new four-dimensional Lorenz-type hyperchaotic system which has a curve of equilibria. This new system can generate not only hyperchaotic attractors but also chaotic, quasi-periodic and periodic attractors, as well as singular degenerate heteroclinic cycles. Of particular interest is the observation that there are four types of coexisting attractors of this new hyperchaotic system: (i) chaotic attractor and quasi-periodic attractor, (ii) chaotic attractor and singular degenerate heteroclinic cycle, (iii) periodic attractor and singular degenerate heteroclinic cycle, and (iv) different periodic attractors. Furthermore, many singular degenerate heteroclinic cycles are found, which may lead to complex dynamics of hyperchaotic system with a curve of equilibria.

Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria

This paper studies a four-dimensional (4D) memristive system modified from the 3D chaotic system proposed by Lu and Chen. The new system keeps the symmetry and dissipativity of the original system and has an uncountable infinite number of stable and unstable equilibria. By varying the strength of the memristor, we find rich complex dynamics, such as limit cycles, torus, chaos, and hyperchaos, which can peacefully coexist with the stable equilibria. To explain such coexistence, we compute the unstable manifolds of the equilibria, find that the manifolds create a safe zone for the hyperchaotic attractor, and also find many heteroclinic orbits. To verify the existence of hyperchaos in the 4D memristive circuit, we carry out a computer-assisted proof via a topological horseshoe with two-directional expansions, as well as a circuit experiment on oscilloscope views.

Hidden Hyperchaotic Attractors in a Modified Lorenz–Stenflo System with Only One Stable Equilibrium

This paper reports the finding of a four-dimensional (4D) non-Sil'nikov autonomous system with three quadratic nonlinearities, which exhibits some behavior previously unobserved: hidden hyperchaotic attractors with only one stable equilibrium. The algebraical form of the non-Sil'nikov chaotic attractor is very similar to the hyperchaotic Lorenz–Stenflo system but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact this system has only one stable equilibrium, but can exhibit hidden hyperchaos, chaos, periodic orbit. Moreover, the coexistence of attracting sets can be obtained in the system for some parameter values and different initial conditions, such as hyperchaotic attractor and point, hyperchaotic attractor and period orbit. To further analyze the new system, the ultimate bound and positively invariant set for the modified hyperchaotic Lorenz–Stenflo system are also obtained. Moreover, the complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.

A novel hyperchaotic system and its circuit implementation

A four-dimensional hyperchaotic system with five parameters is proposed. Its dynamical properties such as dissipativity, equilibrium points, Lyapunov exponent, Lyapunov dimension, bifurcation diagrams and Poincare maps are analyzed theoretically and numerically. Theoretical analyses and simulation tests indicate that the new system's dynamics behavior can be periodic attractor, chaotic attractor and hyperchaotic attractor as the parameter varies. Finally, the circuit of this new hyperchaotic system is designed and realized by Multisim software. The simulation results confirm that the chaotic system is different from the existing chaotic systems and is a novel hyperchaotic system. The system is recommendable for many engineering applications such as information processing, cryptology, secure communications, etc.

Local bifurcation analysis and ultimate bound of a novel 4D hyper-chaotic system

This paper presents a new four-dimensional smooth quadratic autonomous hyper-chaotic system which can generate novel two double-wing periodic, quasi-periodic and hyper-chaotic attractors. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. It is shown that this system has a wide hyper-chaotic parameter. The pitchfork bifurcation and Hopf bifurcation are discussed using the center manifold theory. The ellipsoidal ultimate bound of the typical hyper-chaotic attractor is observed. Numerical simulations are given to demonstrate the evolution of the two bifurcations and show the ultimate boundary region.

Chaos–hyperchaos transition in a class of models governed by Sommerfeld effect

The present paper points out the scenario of the hyperchaotic attractor formation (an attractor with at least two positive Lyapunov exponents) in a class of models governed by the Sommerfeld effects. The Sommerfeld effect is the result of the conservation law of energy. When a dynamical system is coupled to a power source, it acts like a energy sink for which a part of the source energy is spend to deform the system rather than increasing the drive speed. The Sommerfeld effect involves riddling bifurcation which explains the creation of the hyperchaotic attractor.

A multiplierless hyperchaotic system using coupled Duffing oscillators

We present a 4-dimensional electrical model for realizing a multiplierless hyperchaotic system using two Duffing oscillators with a nonlinear resistive coupling. The corresponding hyperchaotic attractor is not only numerically verified through investigating phase portraits and Lyapunov exponents spectra, but also realized experimentally by a simple electronic circuit.

Generation and suppression of a new hyperchaotic nonlinear model with complex variables

In this paper, we introduce a new hyperchaotic complex Chen model. This hyperchaotic complex system is constructed by adding a complex nonlinear term to the third equation of the chaotic complex Chen system with consideration it’s all variables are complex. The new system is a 6-dimensional continuous real autonomous hyperchaotic system. The properties of this system including invariance, dissipation, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams and hyperchaotic behavior are studied. Different forms of hyperchaotic complex Chen systems are constructed. We suppress the hyperchaotic behavior of our system via passive control method by using one complex controller. The hyperchaotic attractors of the new system are converted to its unstable trivial fixed point and tracked to its unstable non trivial fixed points and periodic orbits. Block diagrams of our system are designed by using Matlab/Simulink after and before the suppression process to ensure the validity of the analytical results.

A family of hyperchaotic multi-scroll attractors in [formula omitted

In this work, we present a mechanism how to yield a family of hyperchaotic multi-scroll systems in Rn based on unstable dissipative systems. This class of systems is constructed by a switching control law changing the equilibrium point of an unstable dissipative system. For each equilibrium point presented in the system a scroll emerges. The switching control law that governs the position of the equilibrium point varies according to the number of scrolls displayed in the attractor. Thus, if two systems display different numbers of scrolls, they have different switching control laws. This paper also presents a generalized theory that explains different approaches such as hysteresis and step functions from a unified viewpoint, extending the concept of chaos in R3 to hyperchaotic multi-scroll systems in Rn,n⩾4. An illustrative example of synchronizing a communication system is given based on the developed theory.

Multiple regenerative effects in cutting process and nonlinear oscillations

A turning process paradigm is considered to study multiple regenerative effects in cutting operations. The workpiece is considered to be a spatially continuous element, while the cutter is modeled as a discrete-parameter element. The resulting system is described by a combined partial differential equation–ordinary differential equation (ODE) model with a surface function that is used for updating the workpiece. The time delay in this model is allowed to be any integer multiple of the tooth-pass period. Analysis of this system reveals that the loss of contact between the workpiece and the cutter results in two principal features, namely, a non-smooth cutting force and multiple regenerative effects. The model of the spatially continuous work piece is cast into a system of ODEs through the semi-discretization method. Subsequent analysis results in a high-dimensional, non-smooth discrete-time map. Iterations of this mapping show that the time delay can vary in a wide range, and due to the multiple regenerative effect, this delay can be as high as ten times the constant delay value. Through parametric studies, it is learned that the system can exhibit stable cutting behavior, as well as periodic, quasi-periodic, chaotic and hyperchaotic behavior. With the choice of the non-dimensional cutting coefficient as a control parameter, bifurcations of the system responses are examined. The system is observed to possess rich dynamics, including multiple solutions. Supported by computations of correlation dimension, Kaplan–Yorke dimension, and Lyapunov spectrum, limit cycle, torus, chaotic, and hyperchaotic attractors are observed to be present in the considered parameter range. The findings can help further our understanding of multiple regenerative effects in cutting and drilling operations.

A new finding of the existence of hidden hyperchaotic attractors with no equilibria

The paper presents a new four-dimensional hyperchaotic system developed by extension of the generalized diffusionless Lorenz equations. The model is shown to not be equivalent to any hyperchaotic system that the authors know of. In particular, the model does not display any equilibria, but can exhibit two-scroll hyperchaos as well as chaotic, quasiperiodic and periodic dynamics. For certain parameter values, coexisting attractors can be observed, e.g. hyperchaotic and periodic attractors. Investigation of the proposed system is performed through a combination of numerical simulation and mathematical analysis in order to obtain time plots, phase portraits, Lyapunov exponents, and Poincaré sections.

A novel four-dimensional multi-wing hyper-chaotic attractor and its application in image encryption

In this paper, a novel four-dimensional chaotic system for generating multi-wing chaotic attractors is proposed, and chaotic and hyper-chaotic attractors are generated in different parameters. Besides, basic dynamical properties of the chaotic system, such as equilibrium point Poincaré mapping, dissipativity, power spectrum, Lyapunov exponent spectrum, bifurcation diagram are studied numerically and theoretically. An analog oscillator circuit is designed for implementing the four-wing hyper-chaotic attractors, and the hardware circuit experimental results are shown to be in good agreement with the numerical simulation results. Finally, the four-wing hyper-chaotic system is used for hybrid image encryption of physical chaos encryption and advanced encryption standard encryption algorithm. Because physical chaos is adopted in this system, there does not exist a definitive relationship between plaintexts and ciphertexts. And the statistical characteristics of ciphertexts should be better than those of any other encryption system.

A Novel Four-Wing Hyperchaotic Complex System and Its Complex Modified Hybrid Projective Synchronization with Different Dimensions

We introduce a new Dadras system with complex variables which can exhibit both four-wing hyperchaotic and chaotic attractors. Some dynamic properties of the system have been described including Lyapunov exponents, fractal dimensions, and Poincaré maps. More importantly, we focus on a new type of synchronization method of modified hybrid project synchronization with complex transformation matrix (CMHPS) for different dimensional hyperchaotic and chaotic complex systems with complex parameters, where the drive and response systems can be asymptotically synchronized up to a desired complex transformation matrix, not a diagonal matrix. Furthermore, CMHPS between the novel hyperchaotic Dadras complex system and other two different dimensional complex chaotic systems is provided as an example to discuss increased order synchronization and reduced order synchronization, respectively. Numerical results verify the feasibility and effectiveness of the presented schemes.