Unstable manifolds for the hyperchaotic Rössler system

We analyse stability of the generalized four-variable Rössler oscillating system depending on selected control parameters, by using analytic and Hurwitz-Routh methods. In contrast to the usual three-dimensional Rössler and Lorenz systems, we show that always there exists at least one unstable direction, and the number of positive local Lyapunov exponents may be different for both fixed points. We have found two new types of Hopf bifurcation, in which the dimension of the unstable manifold can be increased or reduced by two. Hence there are many possibilities for hyperchaotic unstable manifolds of various dimensions. We have also calculated various ranges of the control parameters for which different unstable manifolds can be obtained. This allows a better characterization of stability of the attractors in the hyperchaotic regime.     

A novel hyperchaos in the quantum Zakharov system for plasmas

The nonlinear interaction of quantum Langmuir waves (QLWs) and quantum ion-acoustic waves (QIAWs) described by the one-dimensional quantum Zakharov equations (QZEs) is reinvestigated. A Galerkin type approximation is used to reduce the QZS to a simplified system (SS) of nonlinear ordinary differential equations which governs the temporal behaviors of the slowly varying envelope of the high-frequency electric field and the low frequency density fluctuation. This SS is then shown to establish the coexistence of novel hyperchaotic attractors, whose appearance is explained by means of the analysis of Lyapunov exponent spectra as well as the Kaplan-Yorke dimension. The system has an equilibrium point which depends parametrically on the nondimensional quantum parameter (H) proportional to quantum diffraction, the plasmon number (N) and the wave number of perturbation (α), and which can evolve into periodic, quasi-periodic, chaotic and hyperchaotic states in both semiclassical and quantum cases.     

General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters

This work is concerned with the general methods for modified projective synchronization of hyperchaotic systems. A systematic method of active control is developed to synchronize two hyperchaotic systems with known parameters. Moreover, by combining the adaptive control and linear feedback methods, general sufficient conditions for the modified projective synchronization of identical or different chaotic systems with fully unknown or partially unknown parameters are presented. Meanwhile, the speed of parameters identification can be regulated by adjusting adaptive gain matrix. Numerical simulations verify the effectiveness of the proposed methods.     

Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system

The stability conditions in fractional order hyperchaotic systems are derived. These conditions are applied to a novel fractional order hyperchaotic system. The proposed system is also shown to exhibit hyperchaos for orders less than 4. Based on the Routh-Hurwitz conditions, the conditions for controlling hyperchaos via feedback control are also obtained. A specific condition for controlling only fractional order hyperchaotic systems is achieved. Numerical simulations are used to verify the theoretical analysis.     

Adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters

This Letter investigates modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system using adaptive method. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the state of two hyperchaotic systems modified function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

Chaos Anti-synchronization between Two Novel Different Hyperchaotic Systems

We demonstrate that anti-synchronization can coexist in two different hyperchaotic systems of ratchets moving in different asymmetric potentials by active control method. By using rigorous mathematical theory, the sufficient condition is drawn for the stability of the error dynamics, where the controllers are designed by using the sum of the relevant variables in hyperchaotic systems. Numerical results are presented to justify the theoretical analysis strategy.     

一类具有四翼吸引子的超混沌系统

构造超混沌系统在目前并非难事,在Chua系统基础上构造具有多涡卷拓扑结构的吸引子也有很多系统方法,但较少有文献讨论基于Lorenz系统族的拥有多翼结构吸引子的光滑混沌系统,而同时具备超混沌特性和多翼拓扑结构吸引子的光滑混沌系统则更少报道.在现有超混沌系统的基础上,分析其共性,并利用坐标变化方法将其转为具有四个翼的超混沌吸引子.该方法在保留了系统原有超混沌特性的基础上,增加了吸引子的拓扑结构复杂性,使之更适合保密通信等领域的应用. 关键词： 多涡卷吸引子 多翼吸引子 超混沌     

Comparison of feedback control methods for a hyperchaotic Lorenz system

More and more attention has been payed to the hyperchaotic system for the huge potential applications of hyperchaotic system such as secure communication and more complex structure than chaotic system. So at present the controlling of the hyperchaotic system simply and effectively is a frontier topic of nonlinear science. In this Letter, for the latest hyperchaotic Lorenz system, four feedback control methods were studied with analytic solution and necessary numerical simulations. It is found that the enhancing feedback control approach is the best choice of the given four feedback control methods for its relatively simple external inputs and relatively small necessary feedback coefficient after comparison. The conclusion is a helpful for the choice of control methods of any other chaotic and hyperchaotic systems.     

Design and FPGA implementation of a memristor-based multi-scroll hyperchaotic system

A novel memristor-based multi-scroll hyperchaotic system is proposed. Based on a voltage-controlled memristor and a modulating sine nonlinear function, a novel method is proposed to generate the multi-scroll hyperchaotic attractors. Firstly, a multi-scroll chaotic system is constructed from a three-dimensional chaotic system by designing a modulating sine nonlinear function. Then, a voltage-controlled memristor is introduced into the above-designed multi-scroll chaotic system. Thus, a memristor-based multi-scroll hyperchaotic system is generated, and this hyperchaotic system can produce various coexisting hyperchaotic attractors with different topological structures. Moreover, different number of scrolls and different topological attractors can be obtained by varying the initial conditions of this system without changing the system parameters. The Lyapunov exponents, bifurcation diagrams and basins of attraction are given to analyze the dynamical characteristics of the multi-scroll hyperchaotic system. Besides, the field programmable gate array (FPGA) based digital implementation of the memristor-based multi-scroll hyperchaotic system is carried out. The experimental results of the FPGA-based digital circuit are displayed on the oscilloscope.     

四维系统中多涡卷混沌与超混沌吸引子的仿真研究

在双涡卷混沌吸引子的基础上，以变型蔡氏电路和四阶蔡氏电路为例，提出一种研究四维系统中多涡卷混沌与超混沌吸引子的新方法.根据这一方法，从数学上找到了一种能产生多涡卷的递推规律，其特点是只需给定三个初始值ma,mb和x1，由文中所导出的递推公式，可确定多涡卷吸引子中分段线性奇函数的各个转折点和平衡点的值，从而能在四维系统中产生多涡卷混沌与超混沌吸引子，并且这种方法可以推广到产生任意多个涡卷的情形，因此，它具有一般的规律性.理论分析、计算机模拟和电路仿真结果证明了该方法的可行性.     

Operational criterion for controlled dense coding with non-trivial tripartite entangled states

Hidden hyperchaotic attractors can be generated with three positive Lyapunov exponents in the proposed 5D hyperchaotic Burke–Shaw system with only one stable equilibrium. To the best of our knowledge, this feature has rarely been previously reported in any other higher-dimensional systems. Unidirectional linear error feedback coupling scheme is used to achieve hyperchaos synchronisation, which will be estimated by using two indicators: the normalised average root-mean squared synchronisation error and the maximum cross-correlation coefficient. The 5D hyperchaotic system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integration. In addition, fractional-order hidden hyperchaotic system will be considered from the following three aspects: stability, bifurcation analysis and FPGA implementation. Such implementations in real time represent hidden hyperchaotic attractors with important consequences for engineering applications.     

A new five-term simple chaotic attractor

A new chaotic attractor is presented with only five terms in three simple differential equations having fewer terms and simpler than those of existing seven-term or six-term chaotic attractors. Basic dynamical properties of the new attractor are demonstrated in terms of equilibria, Jacobian matrices, non-generalized Lorenz systems, Lyapunov exponents, a dissipative system, a chaotic waveform in time domain, a continuous frequency spectrum, Poincaré maps, bifurcations and forming mechanisms of its compound structures.     

Critical curves and coexisting attractors in a quasiperiodically forced delayed system

We study three critical curves in a quasiperiodically driven system with time delays, where occurrence of symmetry-breaking and symmetry-recovering phenomena can be observed. Typical dynamical tongues involving strange nonchaotic attractors (SNAs) can be distinguished. A striking phenomenon that can be discovered is multistability and coexisting attractors in some tongues surrounding by critical curves. The blowout bifurcation accompanying with on-off intermittency can also be observed. We show that collision of attractors at a symmetric invariant subspace can lead to the appearance of symmetry-breaking.     

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

The analysis of a novel 3-D autonomous system and circuit implementation

This Letter presents a new three-dimensional autonomous system with four quadratic terms. The system with five equilibrium points has complex chaotic dynamics behaviors. It can generate many different single chaotic attractors and double coexisting chaotic attractors over a large range of parameters. We observe that these chaotic attractors were rarely reported in previous work. The complex dynamical behaviors of the system are further investigated by means of phase portraits, Lyapunov exponents spectrum, Lyapunov dimension, dissipativeness of system, bifurcation diagram and Poincaré map. The physical circuit experimental results of the chaotic attractors show agreement with numerical simulations. More importantly, the analysis of frequency spectrum shows that the novel system has a broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Adaptive Functional Projective Lag Synchronization of a Hyperchaotic Rössler System

We explain the functional projective lag synchronization of a hyperchaotic Rössler system with four unknown parameters, where the output of the master system lags behind the output of the slave system proportionally. Based on Lyapunov stability theory, an active control method and adaptive control law are employed to make the states of two hyperchaotic Rössler systems asymptotically synchronized. Finally, some numerical examples are provided to show the effectiveness of our results.     

基于微控制器数字硬件实现的网格涡卷超混沌系统

基于微控制器(MCU)设计了一个通用的四维混沌系统数字硬件实验电路,由此实现了9×7网格涡卷的混沌和超混沌吸引子的生成.本文基于由Colpitts振荡器模型延伸出的四维多涡卷超混沌系统,通过引入单位锯齿波函数替换原系统中的三角波函数,构建了一个便于MCU数字硬件实现的新的网格涡卷超混沌系统,并对新系统网格涡卷吸引子的形成机理进行了分析和数值仿真.通过采用Euler算法对新系统进行离散化,在实验电路的有效动态范围内可以生成比原系统更多网格涡卷数量的吸引子.实验结果有效验证了本文基于MCU实现的网格涡卷超混沌系统的可行性.     

A Hyperchaotic Attractor Coined from Chaotic Lü System

We report a new hyperchaotic attractor coined from the chaotic Lü system by using a state feedback controller. Theoretical analyses and simulation experiments are conducted to investigate the dynamical behaviour of the proposed hyperchaotic system     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

A novel 3D autonomous system with different multilayer chaotic attractors

This Letter proposes a novel three-dimensional autonomous system which has complex chaotic dynamics behaviors and gives analysis of novel system. More importantly, the novel system can generate three-layer chaotic attractor, four-layer chaotic attractor, five-layer chaotic attractor, multilayer chaotic attractor by choosing different parameters and initial condition. We analyze the new system by means of phase portraits, Lyapunov exponent spectrum, fractional dimension, bifurcation diagram and Poincaré maps of the system. The three-dimensional autonomous system is totally different from the well-known systems in previous work. The new multilayer chaotic attractors are also worth causing attention.