Bifurcation analysis of the logistic map via two periodic impulsive forces

The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincare′ map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.     

Dynamical behaviors of a chaotic system with no equilibria

Based on Sprott D system, a simple three-dimensional autonomous system with no equilibria is reported. The remarkable particularity of the system is that there exists a constant controller, which can adjust the type of chaotic attractors. It is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping and period-doubling route to chaos are analyzed with careful numerical simulations.     

耦合电路中的复杂振荡行为分析

讨论了两个非线性电路适当连接后的耦合系统随耦合强度变化的演化过程.给出了两子系统各自的分岔行为及通向混沌的过程,指出原子系统均为周期运动时,耦合系统依然会由倍周期分岔进入混沌,同时在混沌区域中存在有周期急剧增加及周期增加分岔等现象.而当周期运动和混沌振荡相互作用时,在弱耦合条件下,受混沌子系统的影响,原周期子系统会在其原先的轨道邻域内作微幅振荡,其振荡幅值随耦合强度的增加而增大,混沌的特征越加明显,相反,周期子系统不仅可以导致混沌子系统的失稳,也会引起混沌吸引子结构的变化. 关键词： 非线性电路 耦合强度 分岔 混沌     

Non-Smooth Bifurcation and Chaos in a DC-DC Buck Converter

A direct-current-direct-current （DC-DC） buck converter with integrated load current feedback is studied with three kinds of Poincaré maps. The external corner-collision bifurcation occurs when the crossing number per period varies, and the internal corner-collision bifurcations occur along with period-doubling and period-tripling bifurcations in this model. The multi-band chaos roots in external corner-collision bifurcation and often grows into 1-band chaos. A new kind of chaotic sliding orbits, which is more complex for non-smooth systems, is also found in this model.     

周期脉冲作用下Logistic映射的复杂动力学行为及其分岔分析

研究了两种周期脉冲作用下Logistic映射的复杂动力学行为. 随着参数的变化, 该系统产生平衡解、周期解、混沌等现象, 且该系统可经级联倍周期分岔到达混沌. 通过构造Poincaré 映射, 对周期脉冲作用下Logistic映射进行了分岔分析. 最后基于Floquet理论揭示了该系统周期解的分岔机理. 关键词： Logistic映射 脉冲 周期解 分岔机理     

分段线性电路切换系统的复杂行为及非光滑分岔机理

分析了分段线性电路系统在周期切换下的复杂动力学行为及其产生的机理. 基于平衡点分析, 给出了两子系统Fold分岔和Hopf分岔条件. 考虑了在不同稳定态时两子系统周期切换的分岔特性, 产生了不同的周期振荡, 并揭示了其产生的机理. 在不同的周期振荡中, 切换点的数量随参数变化产生倍化, 导致切换系统由倍周期分岔进入混沌. 关键词： 分段线性电路 切换系统 非光滑分岔     

神经元电活动不同节律模式的几种变化过程

利用神经元Chay模型,对实验中观察到的三种放电节律模式序列进行数值模拟,并应用余维1极限环分岔分析研究了其产生机理.首先考虑的是周期性放电模式的变化过程;其次,具有不同表象的一种超临界和一种亚临界倍周期簇放电序列产生并导致混沌现象的出现,然后以不同的方式转迁到逆超临界倍周期峰放电序列;最后研究无混沌的加周期簇放电序列,得出加周期分岔仅是一种与倍周期分岔密切相关的分岔现象.     

周期切换下Chen系统的振荡行为与非光滑分岔分析

研究了不同参数Chen系统之间进行周期切换时的分岔和混沌行为.基于平衡态分析,考虑Chen系统在不同稳态解时通过周期切换连接生成的复合系统的分岔特性,得到系统的不同周期振荡行为.在演化过程中,由于切换导致的非光滑性,复合系统不仅仅表现为两子系统动力特性的简单连接,而且会产生各种分岔,导致诸如混沌等复杂振荡行为.通过Poincaré映射方法,讨论了如何求周期切换系统的不动点和Floquet特征乘子.基于Floquet理论,判定系统的周期解是渐近稳定的.同时得到,随着参数变化,系统既可以由倍周期分岔序列进入混沌,也可以由周期解经过鞍结分岔直接到达混沌.研究结果揭示了周期切换系统的非光滑分岔机理.     

脉冲作用下Chen系统的非光滑分岔分析

研究了脉冲作用下Chen系统的复杂动力学行为. 对脉冲作用下的Chen系统进行了非光滑分岔分析. 该系统可经级联倍周期分岔到达混沌, 也可由周期解经鞍结分岔直接到达混沌. 最后通过Floquet理论揭示了该系统周期解的非光滑分岔机理.     

Dynamics of fractional-order sinusoidally forced simplified Lorenz system and its synchronization

In this paper, dynamical behaviors of the fractional-order sinusoidally forced simplified Lorenz are investigated by employing the time-domain solution algorithm of fractional-order calculus. The system parameters and the fractional derivative orders q are treated as bifurcation parameters. The range of the bifurcation parameters in which the system generates chaos is determined by bifurcation, phase portrait, and Poincaré section, and different bifurcation motions are visualized by virtue of a systematic numerical analysis. We find that the lowest order of this system to yield chaos is 3.903. Based on fractional-order stability theory, synchronization is achieved by using nonlinear feedback control method. Simulation results show the scheme is effective and a chaotic secure communication scheme is present based on this synchronization.     

非线性电路通向混沌的演化过程

给出了四阶非线性电路通向复杂性的两种演化模式,指出这两种模式与三个共存的平衡点有关.在第一种模式中,不稳定的平衡点由Hopf分岔导致了稳定的周期运动,经过倍周期分岔通向混沌,其所有的吸引子都保持对称结构;而在第二种模式中,另两个平衡点由Hopf分岔产生相互对称的极限环,并分别导致了两个混沌吸引子,其分岔过程步调一致,而且所有的吸引子都相互对称.随着参数的变化,这两个混沌吸引子相互作用形成一个扩大的混沌吸引子,导致与第一种分岔模式中定性一致的混沌运动.     

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.     

周期切换下Rayleigh振子的振荡行为及机理

本文研究了自治与非自治电路系统在周期切换连接下的动力学行为及机理.基于自治子系统平衡点和极限环的相应稳定性分析和切换系统李雅普诺夫指数的理论推导及数值计算.讨论了两子系统在不同参数下的稳态解在周期切换连接下的复合系统的各种周期振荡行为,进而给出了切换系统随参数变化下的最大李雅普诺夫指数图及相应的分岔图,得到了切换系统在不同参数下呈现出周期振荡,概周期振荡和混沌振荡相互交替出现的复杂动力学行为并分析了其振荡机理.给出了切换系统通过倍周期分岔,鞍结分岔以及环面分岔到达混沌的不同动力学演化过程.     

Complex chaos and bifurcations of semiconductor lasers subjected to optical injection

This paper presents the nonlinear dynamics and bifurcations of optically injection semiconductor lasers in the frame of relative weak injection strength. We consider the new modified rate equations model established recently and the behavior of the system is explored by means of bifurcation diagrams. However, the exact nature of the involved dynamics is well described by a detailed study of the changes of dynamics as a function of the effective gain coefficient. As results, we notice symmetry spectra of intensity, the sudden transition between chaos and stable limit cycle, double scroll attractors together with the phenomenon of a sequence of period-doubling route of chaos, strict crisis between the two basins attraction and the boundary crisis as well as the effects of frequency detuning and linewidth enhancement factor on the nonlinear behaviors.     

Symmetry chaotic attractors and bursting dynamics of semiconductor lasers subjected to optical injection

This paper presents the nonlinear dynamics and bifurcations of optically injected semiconductor lasers in the frame of relative high injection strength. The behavior of the system is explored by means of bifurcation diagrams; however, the exact nature of the involved dynamics is well described by a detailed study of the dynamics evolutions as a function of the effective gain coefficient. As results, we notice the different types of symmetry chaotic attractors with the riddled basins, supercritical pitchfork and Hopf bifurcations, crisis of attractors, instability of chaos, symmetry breaking and restoring bifurcations, and the phenomena of the bursting behavior as well as two connected parts of the same chaotic attractor which merge in a periodic orbit.     

单参数Lorenz混沌系统的电路设计与实现

为了研究混沌系统的性质及其应用,采用分立元件设计并实现了单参数Lorenz混沌系统,系统参数与电路元件参数一一对应.通过调节电路中的可变电阻,观察到了该单参数系统的极限环、叉式分岔、倍周期分岔和混沌等动力学现象,以及该系统由倍周期分岔进入混沌的过程.研究了分数阶单参数Lorenz系统存在混沌的必要条件,找出了分数阶单参数Lorenz系统出现混沌的最低阶数以及最低阶数随系统参数变化的一般规律.电路仿真与电路实现研究表明,单参数Lorenz系统具有物理可实现性、丰富的动力学特性以及理论分析与实验结果的一致性.     

Bifurcations near homoclinic orbits with symmetry

The effect of symmetry on bifurcations associated with homoclinic orbits to saddle-foci is analysed. With symmetry each homoclinic bifurcation contributes three periodic orbits to the global bifurcation picture as opposed to a single orbit in the general case. Bifurcations on these orbits are studied: there are sequences of saddle-node and period-doubling bifurcations, chaos and more complicated homoclinic orbits.     

Lyapunov exponent diagrams of a 4-dimensional Chua system

We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.     

Stability of periodic motion, bifurcations and chaos of a two-degree-of-freedom vibratory system with symmertical rigid stops

A two-degree-of-freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Such models play an important role in the studies of mechanical systems with clearances or gaps. The period-one double-impact symmetrical motion and its Poincaré map are derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motion are analyzed by the equation of Poincaré map. The routes from period-one double-impact symmetrical motion to chaos, via pitchfork bifurcations and period-doubling bifurcation, are studied by numerical simulation. Some non-typical routes to chaos, caused by grazing the stops and Hopf bifurcation of period two four-impact motion, are analyzed. Hopf bifurcations of period-one double-impact symmetrical and antisymmetrical motions are shown to exist in the two-degree-of-freedom vibratory system with two-sided stops. Interesting feature like the period-one four-impact symmetrical motion is also found, and its route to chaos is analyzed. It is of special interest to acquire an overall picture of the system dynamics for some extreme values of parameters, especially those which relate to the degenerated case of a single-degree-of-freedom system, and these analyses are presented here.     

非线性切换系统的动力学行为分析

建立了周期切换下的非线性电路模型,基于子系统平衡点及其稳定性分析,分别给出了其相应的fold分岔和Hopf分岔条件,讨论了子系统在不同平衡态下由周期切换导致的各种复杂行为,指出切换系统的周期解随参数的变化存在着倍周期分岔和鞍结分岔两种失稳情形,并相应地导致不同的混沌振荡,进而结合系统轨迹及其相应的分岔分析,揭示了各种振荡模式的动力学机理. 关键词： 周期切换 倍周期分岔 鞍结分岔 混沌