一类三次方对称离散混沌系统的分岔控制

通过分析对称性破缺分岔机制, 采用了一个直接的、有效的线性控制器, 精确控制了一类三次方对称离散混沌系统发生对称性破缺分岔和倍周期分岔时分岔点的位置. 进而分析了系统对初始值的敏感性和对称性, 选择合适的吸引域, 将对称性破缺分岔进行进一步控制, 从而使得对称性破缺分岔所缺解枝得以恢复. 数值结果表明了该控制器的有效性. 关键词： 离散混沌系统 对称性破缺 倍周期 分岔控制     

A Fractional-Order Sinusoidal Discrete Map

In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.     

Symmetry-increasing bifurcation of chaotic attractors

Bifurcation in symmetric is typically associated with spontaneous symmetry breaking. That is, bifurcation is associated with new solution having less symmetry.In this paper we show that symmetry-increasing bifurcation in the discrete dynamics of symmetric mappings is possible (and is perhaps generic). The reason for these bifurcations may be understood as follows. The existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry. Typically, in computer experiments, what we see is a sequence of symmetry-breaking bifurcations leading to the existence of conjugate chaotic attractors. As the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry.We prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m-gon. In the last section we discuss period-doubling in the presence of symmetry.     

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S₃-transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z₃-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions.     

Universality and self-similarity in the bifurcations of circle maps

The bifurcation structure in a two-parameter family of circle maps is considered. These maps have a (topological) degree that may be different from one. A generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described. Continuity arguments are used to establish the existence of periodic orbits. By plotting the locus of parameter values associated with superstable cycles, self-similar bifurcations are found. These bifurcations are a generalization of the familiar period-doubling cascade in maps with one extrema, to two-parameter maps with two extrema. Finally, a scheme for the global organization of bifurcation in these maps is proposed.     

耦合发电机系统的分岔和双参数特性

在综合分析系统基本动力学特性的基础上,通过数值计算Lyapunov指数谱、分岔图等,讨论了耦合发电机系统的混沌分岔行为和周期窗口的性态变化;计算和分析了系统在二维参数空间的双参数特性.结果显示系统在倍周期分岔中会出现缺边现象,在双参数空间系统出现复杂的分岔结构,两个控制参数对系统动力学行为的影响特性有所差别. 关键词： 耦合发电机系统 分岔 周期窗口 双参数特性     

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.     

振动筛系统的两类余维三分岔与非常规混沌演化

建立了振动筛系统的动力学模型,推导出了其周期运动的Poincaré 映射,基于Poincaré 映射方法着重研究了系统Flip-Hopf-Hopf余维三分岔、三次强共振条件下的Hopf-Hopf余维三分岔以及三种非常规的混沌演化过程.研究结果表明,此两类余维三分岔点附近的动力学行为变得更加复杂和新颖,在分岔点附近出现了三角形吸引子、3T²环面分岔以及“五角星型”、“轮胎型”概周期吸引子,揭示了环面爆破、环面倍化以及T²环面分岔向混沌演化的过程,这些结果对于振动筛系统的动力学优化设计提供了理论参考. 关键词： 余维三分岔 非常规混沌演化 T²环面分岔')" href="#">T²环面分岔     

CHAOTIC AND NON-LINEAR DYNAMIC ANALYSIS OF A TWO-AXIS RATE GYRO WITH FEEDBACK CONTROL MOUNTED ON A SPACE VEHICLE

An analysis is presented in this paper for a two-axis rate gyro subjected to linear feedback control mounted on a space vehicle, which is spinning with uncertain angular velocity ω_z(t) about its spin of the gyro. For the autonomous case in which ω_z(t) is steady, the stability analysis of the system is studied by Routh-Hurwitz theory. For the non-autonomous case in which ω_z(t) is sinusoidal function, this system is a strongly non-linear damped system subjected to parametric excitation. By varying the amplitude of sinusoidal motion, periodic and chaotic responses of this parametrically excited non-linear system are investigated using the numerical simulation. Some observations on symmetry-breaking bifurcations, period-doubling bifurcations, and chaotic behavior of the system are investigated by various numerical techniques such as phase portraits, Poincaré maps, average power spectra, and Lyapunov exponents. In addition, some discussions about chaotic motions of this system can be suppressed and changed into regular motions by a suitable constant motor torque are included.     

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.     

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     

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.     

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.     

不连续导电模式DC-DC变换器的倍周期分岔机理研究

根据一般迭代映射的倍周期分岔定理,从数学上论证了电压型不连续导电模式(DCM) Boost和Buck变换器中倍周期分岔现象产生条件,由此揭示了DC-DC变换器中倍周期分岔现象发生的机理,完善了该类变换器倍周期分岔分析的理论和方法. 关键词： 倍周期分岔 迭代映射 Lyapunov 指数 施瓦茨导数     

Jerk系统耦合的分岔和混沌行为

通过分析耦合的Jerk系统的平衡点及其稳定性,给出了参数空间中不同的分岔集,进而将参数空间划分为对应于各种动力学行为的不同区域.探讨了耦合系统随不同参数变化的动力学演化过程,重点分析了系统耦合强度变化对其动力学行为的影响.揭示了多种运动模式共存及倍周期分岔等各种非线性现象的产生机理.     

The differential diffusion effect of the intermediate species on the stability of premixed flames propagating in microchannels

The propagation of premixed flames in adiabatic and non-catalytic planar microchannels subject to an assisted or opposed Poiseuille flow is considered. The diffusive–thermal model and the well-known two-step chain-branching kinetics are used in order to investigate the role of the differential diffusion of the intermediate species on the spatial and temporal flame stability. This numerical study successfully compares steady-state and time-dependent computations to the linear stability analysis of the problem. Results show that for fuel Lewis numbers less than unity, Le_F < 1, and at sufficiently large values of the opposed Poiseuille flow rate, symmetry-breaking bifurcation arises. It is seen that small values of the radical Lewis number, Le_Z, stabilise the flame to symmetric shape solutions, but result in earlier flashback. For very lean flames, the effect of the radical on the flame stabilisation becomes less important due to the small radical concentration typically found in the reaction zone. Cellular flame structures were also identified in this regime. For Le_F > 1, flames propagating in adiabatic channels suffer from oscillatory instabilities. The Poiseuille flow stabilises the flame and the effect of Le_Z is opposite to that found for Le_F < 1. Small values of Le_Z further destabilise the flame to oscillating or pulsating instabilities.     

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     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

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.     

Synchronization of period-doubling oscillations in vascular coupled nephrons

The mechanisms by which the individual functional unit (nephron) of the kidney regulates the incoming blood flow give rise to a number of nonlinear dynamic phenomena, including period-doubling bifurcations and intra-nephron synchronization between two different oscillatory modes. Interaction between the nephrons produces complicated and time-dependent inter-nephron synchronization patterns. In order to understand the processes by which a pair of vascular coupled nephrons synchronize, the paper presents a detailed analysis of the bifurcations that occur at the threshold of synchronization. We show that, besides infinite cascades of saddle-node bifurcations, these transitions involve mutually connected cascades of torus and homoclinic bifurcations. To illustrate the broader range of occurrence of this bifurcation structure for coupled period-doubling systems, we show that a similar structure arises in a system of two coupled, non-identical Ro?ssler oscillators.