A Feigenbaum sequence of bifurcations in the Lorenz model

For some high values of the Rayleigh numberr, the Lorenz model exhibits laminar behavior due to the presence of a stable periodic orbit. A detailed numerical study shows that, forr decreasing, the turbulent behavior is reached via an infinite sequence of bifurcations, whereas forr increasing, this is due to a collapse of the stable orbit to a hyperbolic one. The infinite sequence of bifurcations is found to be compatible with Feigenbaum's conjecture.     

Quasiperiodicity involving twin oscillations in the complex Lorenz equations describing a detuned laser

Quasiperiodical motion in the complex Lorenz equations describing a detuned laser is shown to consist of twin oscillations: the first oscillation originates from Hopf bifurcation and the second is a parastic oscillation of the first one. Equations for the twin asymptotic oscillations are analytically derived in the center manifold, showing explicitly the parastic property of the second oscillation: its frequency is proportional to the square of the amplitude of the first one. The phase of the second oscillation shows also certainanholonomy which is very similar to the characteristics of Berry's phase. Numerical results show further that the first oscillation follows the sequence of bifurcations from simple periodic through period-doubling to chaos, as one continuously increases the control parameter, whereas the frequency of the parastic oscillation does not change qualitatively during the bifurcation process.     

The phase-modulated logistic map

We study the logistic mapping with the nonlinearity parameter varied through a delayed feedback mechanism. This history dependent modulation through a phaselike variable offers an enhanced possibility for stabilization of periodic dynamics. Study of the system as a function of nonlinearity and modulation parameters reveals new phenomena: In addition to period-doubling and tangent bifurcations, there can be bifurcations where the period increases by unity. These are extensions of crises that arise in nonlinear dynamical systems. Periodic orbits in this system can be systematized via the kneading theory, which in the present case extends the analysis of Metropolis, Stein, and Stein for unimodal maps.     

分数阶Lorenz系统的分析及电路实现

频域传递函数近似方法不仅是常用的 分数阶混沌系统相轨迹的数值分析方法之一, 而且也是设计分数阶混沌系统电路的主要方法. 应用该方法首先研究了分数阶Lorenz系统的混沌特性, 通过对Lyapunov指数图、分岔图和数值仿真分析, 发现了其较为丰富的动态特性, 即当分数阶次从0.7到0.9以步长0.1变化时, 该分数阶Lorenz系统既存在混沌特性, 又存在周期特性, 从数值分析上说明了在更低维的Lorenz系统中存在着混沌现象. 然后又基于该方法和整数阶混沌电路的设计方法, 设计了一个模拟电路实现了该分数阶Lorenz系统, 电路中的电阻和电容等数值是由系统参数和频域传递函数近似确定的. 通过示波器观测到了该分数阶Lorenz系统的混沌吸引子和周期吸引子的相轨迹图, 这些电路实验结果与数值仿真分析是一致的, 进一步从物理实现上说明了其混沌特性. 关键词： 分数阶系统 Lorenz系统 分岔分析 电路实现     

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.     

Lorenz混沌系统的近似有限时间稳定控制

针对Lorenz混沌系统,研究其有限时间稳定控制问题.考虑系统存在不确定非线性,提出一种可使受控Lorenz系统实现近似有限时间稳定的控制方法.改进并设计一种扩张状态观测器,解决了受控Lorenz系统中不确定非线性未知问题.通过引入奇异扰动性理论,分析了闭环系统的近似有限时间稳定性.仿真实验结果验证了该控制方法及扩张状态观测器的有效性. 关键词： Lorenz混沌系统 近似有限时间稳定 扩张状态观测器 奇异扰动     

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     

Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.     

基于符号序列描述的一类分段光滑系统中分岔现象与混沌分析

从拓扑序列出发，提出了描述DC/DC变换器一类分段光滑系统中的分岔现象和混沌行为的符号序列方法，根据最大子序列的性态判别分岔的类型，以及检测边界碰撞分岔的发生.例如,当发生倍周期分岔时，最大子序列保持不变；当发生边界碰撞分岔时，最大子序列发生变化；混沌态则没有最大子序列.研究表明，占空比是表征DC/DC变换器一类分段光滑系统动力学行为的一个最本质的量，“饱和非线性”是引起边界碰撞分岔产生的根本原因. 关键词： 符号序列 分岔 混沌 分段光滑系统     

Circuit implementation of a new hyperchaos in fractional-order system

This paper introduces a new four-dimensional （4D） hyperchaotic system, which has only two quadratic nonlinearity parameters but with a complex topological structure. Some complicated dynamical properties are then investigated in detail by using bifurcations, Poincare mapping, LE spectra. Furthermore, a simple fourth-order electronic circuit is designed for hardware implementation of the 4D hyperchaotic attractors. In particular, a remarkable fractional-order circuit diagram is designed for physically verifying the hyperchaotic attractors existing not only in the integer-order system but also in the fractional-order system with an order as low as 3.6.     

Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations

Two infinite sequences of orbits leading to turbulence in a five-mode truncation of the Navier-Stokes equations for a 2-dimensional incompressible fluid on a torus are studied in detail. Their compatibility with Feigenbaum's theory of universality in certain infinite sequences of bifurcations is verified and some considerations on their asymptotic behavior are inferred. An analysis of the Poincaré map is performed, showing how the turbulent behavior is approached gradually when, with increasing Reynolds number, no stable fixed point or periodic orbit is present and all the unstable ones become more and more unstable, in close analogy with the Lorenz model.     

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.     

Homoclinic bifurcation in Chua’s circuit

We report our experimental observations of the Shil’nikov-type homoclinic chaos in asymmetry-induced Chua’s oscillator. The asymmetry plays a crucial role in the related homoclinic bifurcations. The asymmetry is introduced in the circuit by forcing a DC voltage. For a selected asymmetry, when a system parameter is controlled, we observed transition from large amplitude limit cycle to homoclinic chaos via a sequence of periodic mixed-mode oscillations interspersed by chaotic states. Moreover, we observed two intermediate bursting regimes. Experimental evidences of homoclinic chaos are verified with PSPICE simulations.     

Memcapacitor model and its application in a chaotic oscillator

A memcapacitor is a new type of memory capacitor. Before the advent of practical memcapacitor, the prospective studies on its models and potential applications are of importance. For this purpose, we establish a mathematical memcapacitor model and a corresponding circuit model. As a potential application, based on the model, a memcapacitor oscillator is designed, with its basic dynamic characteristics analyzed theoretically and experimentally. Some circuit variables such as charge, flux, and integral of charge, which are difficult to measure, are observed and measured via simulations and experiments. Analysis results show that besides the typical period-doubling bifurcations and period-3 windows, sustained chaos with constant Lyapunov exponents occurs. Moreover, this oscillator also exhibits abrupt chaos and some novel bifurcations.In addition, based on the digital signal processing(DSP) technology, a scheme of digitally realizing this memcapacitor oscillator is provided. Then the statistical properties of the chaotic sequences generated from the oscillator are tested by using the test suit of the National Institute of Standards and Technology(NIST). The tested randomness definitely reaches the standards of NIST, and is better than that of the well-known Lorenz system.     

Heteroclinic dynamics in a model of Faraday waves in a square container

We study periodic orbits associated with heteroclinic bifurcations in a model of the Faraday system for containers with square cross-section and single-frequency forcing. These periodic orbits correspond to quasiperiodic surface waves in the physical system. The heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable Hamiltonian limit, some of which persist in the presence of small damping. The dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate Poincaré maps, with predictions that agree with numerical computations. The results suggest a great richness of possible dynamics of Faraday waves even in simple geometries and with single-frequency forcing.     

一个新四维非自治超混沌系统的分析与电路实现

在以Lorenz系统为基础的一个新超混沌系统上,加入一个驱动信号,组成了一个四维非自治超混沌系统.该系统基本动力学特性的变化,主要是通过控制外界输入信号的频率实现的.在不同的频率参数下,该系统体现出了周期轨、二维环面、混沌以及超混沌不同的动力学特性,其中周期轨特性较为显著.这个现象在数值仿真以及分形分析上得到了很好地验证.最后设计一个模拟电路,通过实验结果进一步验证了与数值仿真的一致性.     

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Systems such as fluid flows in channels and pipes or the complex Ginzburg–Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto–Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.     

Dependence of magnetic field generation by thermal convection on the rotation rate: A case study

Dependence of magnetic field generation on the rotation rate is explored by direct numerical simulation of magnetohydrodynamic convective attractors in a plane layer of conducting fluid with square periodicity cells for the Taylor number varied from zero to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We observe 5 types of hydrodynamic (amagnetic) attractors: two families of two-dimensional (i.e. depending on two spatial variables) rolls parallel to sides of periodicity boxes of different widths and parallel to the diagonal, travelling waves and three-dimensional “wavy” rolls. All types of attractors, except for one family of rolls, are capable of kinematic magnetic field generation. We have found 21 distinct nonlinear convective MHD attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge. In addition, we have observed a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The system is highly symmetric. We have found two novel global bifurcations reminiscent of the SNIC bifurcation, which are only possible in the presence of symmetries. The universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid on continuing the increase), remains unaltered, but we demonstrate that this “large-scale” picture lacks many significant details.     

Complex dynamic behaviors in hyperbolic-type memristor-based cellular neural network

This paper presents a new hyperbolic-type memristor model,whose frequency-dependent pinched hysteresis loops and equivalent circuit are tested by numerical simulations and analog integrated operational amplifier circuits.Based on the hyperbolic-type memristor model,we design a cellular neural network(CNN)with 3-neurons,whose characteristics are analyzed by bifurcations,basins of attraction,complexity analysis,and circuit simulations.We find that the memristive CNN can exhibit some complex dynamic behaviors,including multi-equilibrium points,state-dependent bifurcations,various coexisting chaotic and periodic attractors,and offset of the positions of attractors.By calculating the complexity of the memristor-based CNN system through the spectral entropy(SE)analysis,it can be seen that the complexity curve is consistent with the Lyapunov exponent spectrum,i.e.,when the system is in the chaotic state,its SE complexity is higher,while when the system is in the periodic state,its SE complexity is lower.Finally,the realizability and chaotic characteristics of the memristive CNN system are verified by an analog circuit simulation experiment.     

脉冲序列控制电流断续模式Buck变换器的动力学建模与边界碰撞分岔

通过建立一个开关周期内输出电容电荷变化量对应的输出电压变化量, 建立了工作于电感电流断续模式(discontinuous conduction mode, DCM)的脉冲序列(pulse train, PT)控制Buck变换器的近似离散时间模型, 研究了负载电阻及输入电压变化时PT控制DCM Buck变换器的边界碰撞分岔行为. 通过构造相应的迭代映射曲线, 分别分析了不同负载电阻时PT控制DCM Buck变换器的周期1、周期2和周期3运行轨迹的不动点稳定性, 揭示了PT控制DCM Buck变换器在不同周期态时的边界碰撞分岔的形成机理. 研究结果表明, 随参数变化, PT控制DCM Buck变换器始终运行在不同的周期态, 各周期态的切换由边界碰撞分岔引起, 李雅谱诺夫指数始终小于零. 利用PSIM电路仿真软件, 给出了不同负载电阻时的时域波形和相轨图. 实验结果验证了理论分析和仿真结果的正确性, 同时说明了本文动力学建模的可行性.