Horseshoe and entropy in a fractional-order unified system

This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincar'e section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.     

NON-LINEAR DYNAMICS AND CONTROL OF CHAOS FOR A TACHOMETER

The dynamic behaviors of a rotational tachometer with vibrating support are studied in the paper. Both analytical and computational results are used to obtain the characteristics of the system. The Lyapunov direct method is applied to obtain the conditions of stability of the equilibrium position of the system. The center manifold theorem determines the conditions of stability for the system in a critical case. By applying various numerical analyses such as phase plane, Poincaré map and power spectrum analysis, a variety of periodic solutions and phenomena of the chaotic motion are observed. The effects of the changes of parameters in the system can be found in the bifurcation diagrams and parametric diagrams. By using Lyapunov exponents and Lyapunov dimensions, the periodic and chaotic behaviors are verified. Finally, various methods, such as the addition of a constant torque, the addition of a periodic torque, delayed feedback control, adaptive control, Bang-Bang control, optimal control and the addition of a periodic impulse are used to control chaos effectively.     

The analysis of complex behaviours of a novel three dimensional autonomous system

This paper introduces a new three dimensional autonomous system with five equilibrium points.It demonstrates complex chaotic behaviours within a wide range of parameters,which are described by phase portraits,Lyapunov exponents,frequency spectrum,etc.Analysis of the bifurcation and Poincar’e map is used to reveal mechanisms of generating these complicated phenomena.The corresponding electronic circuits are designed,exhibiting experimental chaotic attractors in accord with numerical simulations.Since frequency spectrum analysis shows a broad frequency bandwidth,this system has perspective of potential applications in such engineering fields as secure communication.     

Non-linear dynamics and control of chaos for a rotational machine with a hexagonal centrifugal governor with a spring

The dynamic behaviors of the rotational machine with a hexagonal centrifugal governor which is subjected to external disturbance are studied in the paper. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of system. By applying numerical results, phase diagrams, power spectrum, Poincaré maps, and Lyapunov exponents are presented to observe periodic and chaotic motions. The effect of the parameter changes in the system can be found in the bifurcation and parametric diagrams. Finally, eight methods are used to control chaos effectively.     

A new hyperchaos system and its circuit simulation by EWB

This paper reports a new hyperchaotic system evolved from the three-dimensional Lü chaotic system. The Lyapunov exponents spectrum and the bifurcation diagram of this new hyperchaotic system are obtained. Hyperchaotic attractor, periodic orbit and chaotic attractor are obtained by computer simulation. A circuit is designed to realize this new hyperchaotic system by electronic workbench.     

三自由度含间隙碰撞振动系统Neimark-Sacker分岔的反控制

分岔反控制作为传统分岔控制的逆问题, 其目的是在预先指定的系统参数点通过控制主动设计出具有所期望特性的分岔解. 以一类三自由度含间隙双面碰撞振动系统为研究对象, 在不改变原系统平衡解结构的前提下, 考虑到在碰撞振动系统反控制过程中由Poincaré映射的隐式特点和传统的映射Neimark-Sacker分岔临界准则带来的困难, 通过对原系统施加线性反馈控制器并利用不直接依赖于特征值计算的Neimark-Sacker分岔显式临界准则研究了此系统的分岔反控制问题. 首先对原系统施加线性反馈控制, 建立闭环控制系统的六维Poincaré映射. 由于六维映射的雅克比矩阵的特征值没有解析的表达式, 利用高维映射Neimark-Sacker分岔的显式临界准则, 获得了系统出现拟周期碰撞振动运动的控制参数区域. 然后采用中心流形-正则形方法分析了拟周期分岔解的稳定性. 数值仿真结果表明本文方法可以在指定的系统参数点通过控制设计出稳定的拟周期碰撞运动.     

A novel hyperchaos system only with one equilibrium

This Letter presents a new hyperchaotic system by introducing an additional state feedback into a three-dimensional quadratic chaotic system. The system only has one equilibrium, but it can evolve into periodic, quasi-periodic, chaotic and hyperchaotic dynamical behaviors. Basic bifurcation analysis of the new system is investigated by means of Lyapunov exponent spectrum and bifurcation diagrams. We find that the new hyperchaotic system possesses two big positive Lyapunov exponents within a large range of parameters. Therefore, the new hyperchaotic system may have good application prospects.     

二维正弦离散映射的分岔和吸引子

由一个正弦映射和一个三次方映射通过非线性耦合,构成一个新的二维正弦离散映射. 基于此二维正弦离散映射得到系统的不动点以及相应的特征值,分析了系统的稳定性,研究了系统的复杂非线性动力学行为及其吸引子的演变过程. 研究结果表明：此二维正弦离散映射中存在复杂的对称性破缺分岔、Hopf分岔、倍周期分岔和周期振荡快慢效应等非线性物理现象. 进一步根据控制变量变化时系统的分岔图、Lyapunov指数图和相轨迹图分析了系统的分岔模式共存、快慢周期振荡及其吸引子的演变过程,通过数值仿真验证了理论分析的正确性. 关键词： 正弦离散映射 对称性破缺分岔 Hopf分岔 吸引子     

A hyperchaos generated from Lorenz system

This paper presents a four-dimension hyperchaotic Lorenz system, obtained by adding a nonlinear controller to Lorenz chaotic system. The hyperchaotic Lorenz system is studied by bifurcation diagram, Lyapunov exponents spectrum and phase diagram. Numerical simulations show that the new system’s behavior can be convergent, divergent, periodic, chaotic and hyperchaotic when the parameter varies.     

基于周期性扩频的单相H桥逆变器非线性现象的研究

将周期性扩频技术用于变换器中来抑制其电磁干扰和噪声已经得到了论证并被广泛的应用, 但却忽略了其中的非线性现象. 在分析了周期性扩频技术和单相SPWM-H桥正弦逆变器的精确频闪映射模型的基础上, 研究了采用周期性扩频技术的单相H桥正弦逆变器的分岔和混沌现象, 建立了H桥正弦逆变器扩频后的离散模型, 通过时域图、折叠图、分岔图及李雅亚普诺夫指数谱对其非线性现象进行了分析. 研究表明, 扩频后的H桥正弦逆变器在进入非线性区域后更容易进入混沌区, 并且周期扩频的频率会对系统产生初始分岔点的位置产生影响. 关键词： H桥逆变器 周期性扩频 分岔 混沌     

The Plateau and Symmetrical Structure of Lyapunov Exponents in 2-Dimensional Homogeneous. Dissipative Map

The universal crossover behavior of Lyapunov exponents in transition from conservative limit to dissipative limit of dynamical system is studied. We discover numerically and prove analytically that for homogeneous dissipative two-dimensional maps, along the equal dissipation line in parameter space, two Lyapunov exponents λ₁ and λ₂ of periodic orbits possess a plateau structure, and around this exponent plateau value, there is a strict symmetrical relation between λ₁ and λ₂ no matter whether the orbit is periodic, quasiperiodic, or chaotic.The method calculating stable window and Lyapunov exponent plateau widths is given. For Hénon map and 2-dimensional circle map, the analytical and numerical results of plateau structure of Lyapunov exponents for period-1,2 and 3 orbits are presented.     

Liu混沌系统的混沌分析及电路实验的研究

研究了一种新型混沌系统——Liu混沌系统的基本动力学行为以及电路实现的问题，给出了相图、庞卡莱映射、功率谱以及李雅普诺夫指数，基于李雅普诺夫指数谱和分叉图分析了系统参数对Liu混沌系统的影响.最后设计硬件电路证实了Liu混沌系统以及Liu混沌系统随系统参数变化时的各种状态的存在.给出数值仿真和电路实验的结果. 关键词： Liu混沌系统 分岔 电路实验     

CHAOS, CHAOS CONTROL AND SYNCHRONIZATION OF ELECTRO-MECHANICAL GYROSTAT SYSTEM

The dynamic behavior of electro-mechanical gyrostat system subjected to external disturbance is studied in this paper. By applying numerical results, phase diagrams, power spectrum, Period-T maps, and Lyapunov exponents are presented to observe periodic and chaotic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. Several methods, the delayed feedback control, adaptive control algorithm (ACA) control are used to control chaos effectively. Anticontrol of chaos destroyed the periodic motions and replaced by chaotic motion effectively by adding constant motor torque and adding periodic motor torque. Finally, synchronization of chaos in the electro-mechanical gyrostat system is studied.     

CHAOS, CHAOS CONTROL AND SYNCHRONIZATION OF A GYROSTAT SYSTEM

The dynamic behavior of a gyrostat system subjected to external disturbance is studied in this paper. By applying numerical results, phase diagrams, power spectrum, period-T maps, and Lyapunov exponents are presented to observe periodic and choatic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. For global analysis, the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping (MICM) method. Several methods, the delayed feedback control, the addition of constant torque, the addition of periodic force, the addition of periodic impulse torque, injection of dither signal control, adaptive control algorithm (ACA) control and bang-bang control are used to control chaos effectively. Finally, synchronization of chaos in the gyrostat system is studied.     

A novel hyperchaos evolved from three dimensional modified Lorenz chaotic system

This paper reports a new four-dimensional continuous autonomous hyperchaos generated from the Lorenz chaotic system by introducing a nonlinear state feedback controller. Some basic properties of the system are investigated by means of Lyapunov exponent spectrum and bifurcation diagrams. By numerical simulating, this paper verifies that the four-dimensional system can evolve into periodic, quasi-periodic, chaotic and hyperchaotic behaviours. And the new dynamical system is hyperchaotic in a large region. In comparison with other known hyperchaos, the two positive Lyapunov exponents of the new system are relatively more larger. Thus it has more complex degree.[第一段]     

Multiple period-doubling bifurcation route to chaos in periodically pulsed Murali-Lakshmanan-Chua circuit-controlling and synchronization of chaos

We consider a simple nonautonomous dissipative nonlinear electronic circuit consisting of Chua's diode as the only nonlinear element, which exhibit a typical period doubling bifurcation route to chaotic oscillations. In this paper, we show that the effect of additional periodic pulses in this Murali-Lakshmanan-Chua (MLC) circuit results in novel multiple-period-doubling bifurcation behavior, prior to the onset of chaos, by using both numerical and some experimental simulations. In the chaotic regime, this circuit exhibits a rich variety of dynamical behavior including enlarged periodic windows, attractor crises, distinctly modified bifurcation structures, and so on. For certain types of periodic pulses, this circuit also admits transcritical bifurcations preceding the onset of multiple-period-doubling bifurcations. We have characterized our numerical simulation results by using Lyapunov exponents, correlation dimension, and power spectrum, which are found to be in good agreement with the experimental observations. Further controlling and synchronization of chaos in this periodically pulsed MLC circuit have been achieved by using suitable methods. We have also shown that the chaotic attractor becomes more complicated and their corresponding return maps are no longer simple for large n-periodic pulses. The above study also indicates that one can generate any desired n-period-doubling bifurcation behavior by applying n-periodic pulses to a chaotic system.     

无反馈作用下混沌系统的振幅死亡

考虑了一种无反馈的作用机理，通过分析定常强迫作用下R?ssler系统的分岔图和Lyapunov指数谱，发现定常强迫能够导致混沌系统产生振幅死亡.这种现象的产生类似于耦合的极限环系统和耦合的R?ssler系统，弱定常强迫作用下的R?ssler系统经由一个周期运动被驱动到系统自身的一个平衡点上.进一步对受迫系统时间序列的研究表明，当定常强迫强度超过一个临界值时，系统的状态不断在小振幅周期运动与静止之间交替出现. 关键词： 振幅死亡 定常强迫 R?ssler系统     

Bifurcation structure and Lyapunov exponents of a modulated logistic map

We have studied the bifurcation structure of the logistic map with a time dependant control parameter. By introducing a specific nonlinear variation for the parameter, we show that the bifurcation structure is modified qualitatively as well as quantitatively from the first bifurcation onwards. We have also computed the two Lyapunov exponents of the system and find that the modulated logistic map is less chaotic compared to the logistic map.     

Ergodicity breaking induced by external coupled spatial-time noise processes

We report on the dynamics in a parameter plane of a continuous-time damped system driven by a periodic forcing. The dynamics is characterized by considering the Lyapunov exponents spectrum and conventional bifurcation diagrams, to discriminate periodic, quasiperiodic, and chaotic behaviors for each point in this parameter plane, according two parameters are simultaneously varied. Periodic structures born in a quasiperiodic region and embedded in a chaotic region, the so-called Arnold tongues, are observed. We show that the Arnold tongues periodic distribution is highly organized in a mixed set of two period-adding sequences. Other three typical periodic structures born and embedded in a chaotic region were observed, also individually organized in a mixed set of two period-adding sequences.     

A new hyperchaotic system and its linear feedback control

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviours by varying parameter k. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.