Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator

This paper is concerned with numerical continuation and analytical investigations of sliding bifurcations in Filippov systems. In particular, a methodology developed for the continuation of grazing bifurcations in impacting systems is used to continue sliding bifurcations in Filippov systems. A dry-friction oscillator is investigated from a sliding bifurcations point of view and a complex two-parameter bifurcation diagram of sliding bifurcations is presented. A number of codimension-two sliding bifurcation points that act as organising centres for codimension-one sliding bifurcations are revealed. Two representative codimension-two points are analysed and unfolded, and the analysis is used to explain the dynamics of the dry-friction oscillator in the neighbourhood of these points.     

Codimension-two bifurcation of axial loaded beam bridge subjected to an infinite series of moving loads

A novel model is proposed which comprises of a beam bridge subjected to an axial load and an infinite series of moving loads. The moving loads, whose distance between the neighbouring ones is the length of the beam bridge, coupled with the axial force can lead the vibration of the beam bridge to codimension-two bifurcation. Of particular concern is a parameter regime where non-persistence set regions undergo a transition to persistence regions. The boundary of each stripe represents a bifurcation which can drive the system off a kind of dynamics and jump to another one, causing damage due to the resulting amplitude jumps. The Galerkin method, averaging method, invertible linear transformation, and near identity nonlinear transformations are used to obtain the universal unfolding for the codimension-two bifurcation of the mid-span deflection. The efficiency of the theoretical analysis obtained in this paper is verified via numerical simulations.     

Multipulse excitability in a semiconductor laser with optical injection

An optically injected semiconductor laser can produce excitable multipulses. Homoclinic bifurcation curves confine experimentally accessible regions in parameter space where the laser emits a certain number of pulses after being triggered from its steady state by a single perturbation. This phenomenon is organized by a generic codimension-two homoclinic bifurcation and should also be observable in other systems.     

Spatial localization in rotating convection and magnetoconvection

Stationary spatially localized states are present in both rotating convection and magnetoconvection. In two-dimensional convection with stress-free boundary conditions, the formation of such states is due to the interaction between convection and a large scale mode: zonal velocity in rotating convection and magnetic potential in magnetoconvection. We develop a higher order theory, a nonlocal fifth order Ginzburg-Landau equation, to describe the effects of spatial modulation near a codimension-two point. Two different bifurcation scenarios are identified. Our results shed light on numerical studies of two-dimensional convective systems with stress-free boundary conditions. This paper is dedicated to Professor Helmut Brand on the occasion of his 60th birthday.     

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.     

Hopf bifurcation analysis of Chen circuit with direct time delay feedback

Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit (oscillation) in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit.     

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     

Blowout bifurcations of codimension two

We consider examples of loss of stability of chaotic attractors in invariant subspaces (blowouts) that occur on varying two parameters, i.e. codimension-two blowout bifurcations. Such bifurcations act as organising centres for nearby codimension-one behaviour, analogous to the case for codimension-two bifurcations of equilibria. We consider examples of blowout bifurcations showing change of criticality, blowouts that occur into two different invariant subspaces and interact, blowouts that occur with onset of hyperchaos, interaction of blowout and symmetry increasing bifurcations and collision of blowout bifurcations. As in the case of bifurcation of equilibria, there are many cases in which one can infer the presence and form of secondary bifurcations and associated branches of attractors. There is presently no generic theory of such higher codimension blowouts (there is not even such a theory for codimension-one blowouts). We want to present a number of examples that would need to be covered by such a theory.     

A memristor oscillator based on a twin-T network

A novel inductance-free nonlinear oscillator circuit with a single bifurcation parameter is presented in this paper. This circuit is composed of a twin-T oscillator, a passive RC network, and a flux-controlled memristor. With an increase in the control parameter, the circuit exhibits complicated chaotic behaviors from double periodicity. The dynamic properties of the circuit are demonstrated by means of equilibrium stability, Lyapunov exponent spectra, and bifurcation diagrams. In order to confirm the occurrence of chaotic behavior in the circuit, an analog realization of the piecewise-linear flux-controlled memristor is proposed, and Pspice simulation is conducted on the resulting circuit.     

基于分岔理论的突触可塑性对神经群动力学特性调控规律研究

神经群模型是典型的非线性系统,具有丰富而复杂的动力学行为模式.神经群兴奋性和抑制性突触具有可塑性,并对神经群动力学特性具有重要调控作用,研究突触可塑性对神经群动力学特性的调控规律具有重要意义.本文基于分岔理论,通过神经群模型兴奋性和抑制性突触增益的余维一分岔分析,分别给出了神经群运行于单稳、双稳、正常和异常极限环振荡状态的兴奋性和抑制性突触增益的单参数区间;进而通过兴奋性和抑制性突触增益的余维二分岔分析给出了神经群运行于上述多种状态的双参数区域.上述结果定量剖析了兴奋性与抑制性突触可塑性及二者的相互作用对神经群动力学特性的调控规律,揭示了兴奋性与抑制性的动态平衡在神经电活动调控中所扮演的关键角色,仿真结果验证了分岔分析的正确性.本文的研究对理解突触可塑性在脑功能的维持及各种神经疾病的诱发机制中所扮演的角色具有重要参考价值.     

Amplitude expansions for instabilities in populations of globally-coupled oscillators

We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite, in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens-Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable traveling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and by Okuda and Kuramoto predicted stable traveling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable traveling waves results from a failure to include all unstable modes.     

一个超混沌Lorenz吸引子及其电路实现

通过在三阶Lorenz系统中引入一个外加的状态变量构造了一个新的超混沌系统.对系统的一些基本特性，如耗散性、平衡点、稳定性、Hopf分叉进行了详细分析，且观察到了从周期到混沌、超混沌的演化.系统超混沌的存在性通过Lyapunov指数谱得到了验证.还设计了一个模拟电子电路，从电路实验中观察到了各种超混沌吸引子. 关键词： Lorenz系统 超混沌吸引子 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.     

Degenerate discontinuity-induced bifurcations in tapping-mode atomic-force microscopy

This paper documents the existence of degenerate bifurcation scenarios in the low-contact-velocity dynamics during tapping-mode atomic-force microscopy. Specifically, numerical analysis of a model of the microscope dynamics shows branch point and isola bifurcations associated with the emergence of two families of saddle–node bifurcation points along a branch of low-amplitude oscillations. The paper argues for the origin of the degenerate bifurcations in the existence of a periodic steady-state trajectory that (i) achieves tangential contact with a discontinuity surface in a piecewise smooth model of the cantilever response and (ii) retracts from the surface under variations in either direction along a line segment in parameter space. Specifically, the discontinuity-mapping technique is here rigorously applied to a general situation of such degenerate contact showing the codimension-two nature of these bifurcations for appropriately chosen parameter values. The discontinuity-mapping-based normal form derived here is a novel extension of that derived in Dankowicz and Nordmark (2000) [28] in the case that (ii) does not hold. In addition, the paper includes a quantitative reflection on the relative importance of discontinuities in the attractive and repulsive force components in producing the predicted bifurcations.     

Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a newhyperchaotic circuit system. The stability of the hyperchaotic circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. Animportant feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.     

Nonlinear feedback control of a novel hyperchaotic system and its circuit implementation

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system. 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 are studied. An effective nonlinear feedback control method is used to suppress hyperchaos to unstable equilibrium. Furthermore, a circuit is designed to realize this new hyperchaotic system by electronic workbench (EWB). Numerical simulations are presented to show these results.     

Dynamic analysis and synchronization control of an unusual chaotic system with exponential term and coexisting attractors

This paper reports a new four-dimensional chaotic system consisting of an exponential nonlinear term, two quadratic nonlinear terms and five linear terms. The system has only one equilibrium and performs stability, periodicity and chaos with the variation of the parameters. It losses its stability with the occurrence of Hopf bifurcation and goes into chaos via period-doubling bifurcation. One more interesting feature of the system is that it can generate multiple coexisting attractors for different initial conditions, such as two strange attractors with one limit cycle, one strange attractor with two limit cycles, etc. The dynamic properties of the system are presented by numerical simulation includes bifurcation diagrams, Lyapunov exponent spectrum and phase portraits. An electronic circuit is constructed to implement the chaotic attractor of the system. Based on the linear quadratic regulator (LQR) method, the synchronization control of the system is investigated.     

基于双缘调制的数字电压型控制Buck变换器离散迭代映射建模与动力学分析

建立了双缘调制数字电压型控制Buck变换器的离散迭代映射模型. 在该模型的基础上, 详细研究了双缘调制数字电压型控制Buck变换器的非线性动力学行为. 以输入电压、负载电阻等电路参数作为分岔参数, 绘制了输出电压和电感电流的分岔图, 并通过分岔图的分析发现了两种相似却又不同的Hopf分岔现象. 采用庞加莱截面、时域仿真波形和相轨图, 对比分析了两种不同的Hopf分岔和低频振荡现象, 并引入离散迭代映射模型的雅克比矩阵的特征值分析方法, 从理论上证明了两种Hopf分岔的存在性和差异性. 首次观察到基于双缘调制的数字电压型控制Buck变换器出现了奇数倍周期分岔现象, 并通过时域仿真波形和相轨图验证了该现象的真实性. 为更加接近实际电路, 考虑电容和电感的等效串联电阻, 使用Psim进行仿真, 其结果与理论仿真结果基本一致, 验证了理论仿真的正确性.     

电压型双频率控制开关变换器的动力学建模与多周期行为分析

在断续导电模式下, 建立了电压型双频率控制开关变换器的动力学模型, 并推导了相应的特征值方程. 根据动力学模型, 采用分岔图研究了电路参数变化时变换器存在的边界碰撞分岔行为和周期2, 周期3,周期4等多周期行为, 结果表明: 变换器经历了周期1态、多周期态、周期1态的分岔路由; 周期态的转变均是由边界碰撞分岔引起的. 根据特征值方程, 采用Lyapunov指数研究了变换器的稳定性, 结果表明: 随着电路参数的变化, Lyapunov指数始终小于零, 变换器一直工作于稳定的周期态, 验证了电压型双频率控制开关变换器的周期3行为并不意味着变换器会必然发生混沌. 通过电路仿真, 分析了负载变化时变换器的时域波形、相轨图和频谱图, 验证了动力学模型的可行性和理论分析的正确性. 实验结果验证了文中的仿真结果. 关键词： 开关变换器 双频率控制 边界碰撞分岔 多周期行为     

Chaotic memristive circuit: equivalent circuit realization and dynamical analysis

In this paper, a practical equivalent circuit of an active flux-controlled memristor characterized by smooth piecewise-quadratic nonlinearity is designed and an experimental chaotic memristive circuit is implemented. The chaotic memristive circuit has an equilibrium set and its stability is dependent on the initial state of the memristor. The initial state-dependent and the circuit parameter-dependent dynamics of the chaotic memristive circuit are investigated via phase portraits, bifurcation diagrams and Lyapunov exponents. Both experimental and simulation results validate the proposed equivalent circuit realization of the active flux-controlled memristor.