Lévy稳定噪声激励下的Duffing-van der Pol振子的随机分岔

研究了Lévy稳定噪声激励下的双稳Duffing-van der Pol振子,利用Monte Carlo方法,得到了振幅的稳态概率密度函数.分析了Lévy稳定噪声的强度和稳定指数对概率密度函数的影响,通过稳态概率密度的性质变化,讨论了噪声振子的随机分岔现象,发现了不仅系统参数和噪声强度可以视为分岔参数,Lévy噪声的稳定指数 α 的改变也能诱导系统出现随机分岔现象. 关键词： Lévy稳定噪声 Duffing-van der Pol振子 稳态概率密度函数 随机分岔     

A note on the SU(2) bloch coherent states

It is shown that the Bloch coherent states for a system of spins can be derived from the SU(2) van der Waerden invariant.     

太阳强迫厄尔尼诺/南方涛动充电振子模型的Hopf分岔与混沌

本文研究了一类太阳强迫的厄尔尼诺/南方涛动(ENSO)充电振子数理模型,通过数学变换将此ENSO振子方程组变换为有周期强迫项的van der Pol-Duffing方程,利用谐波平衡法定性分析得到此ENSO系统发生Hopf分岔的条件并做简单数值模拟,结果发现随着强迫作用增大,11年周期太阳循环强迫的ENSO系统经历准周期、倍频锁相到混沌的过程.     

Nonlinear oscillations of the FitzHugh-Nagumo equations under combined external and two-frequency parametric excitations

The continuous FitzHugh-Nagumo (FHN for short) model is transformed into modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations. At the first, the dependence of the solutions on a combined external and two-frequency parametric stimulus forcing is investigated. By using the multiple scale method, ranges of applied current and/or parametric forcing in which nonlinear oscillations are observed are described. Second, when the multiple scale method cannot be used, we numerically prove that in the modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations, chaos and periodic solution depending on the combination between different frequencies of the model should appear. We also show that the amplitude of the oscillations can be reduced or increased. To do this, we perform the study of the FHN model by choosing a range of parameters exhibiting Hopf bifurcation and two qualitative different regimes in phase portrait.     

Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. The probability density functions of the first-passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.     

Nonvolatile graphene nanoflake shuttle memory

We present the possible schematics of nanoscale graphene-flake shuttle-memory, discuss its basic principles, and investigate the energetic and the dynamic properties via classical molecular dynamics simulations. Graphite flake surface, where the nanoscale graphene-flake’s stage of activity takes place, has a dumbbell shape, and a nanoscale graphene-flake is placed on its surface. The van der Waals interactions between the graphene-flake and the patterned graphite make the bistable potential energy wells in the larger surface area regions of the patterned graphite, and then the graphene-flake keeps its seat on one of the bistable positions, which is the place where the binary data is archived. Since the movable graphene-flake can be also shuttled between the bistable states through the narrow passageway, this proposed nano-graphene-flake shuttle-memory can be utilized from nanoscale switchable nonvolatile memory.     

一类随机van der Pol系统的Hopf 分岔研究

研究了一类随机van der Pol 系统的Hopf分岔行为.首先根据Hilbert空间的正交展开理论,含有随机参数的van der Pol系统被约化为等价确定性系统,然后利用确定性分岔理论分析了等价系统的Hopf分岔,得出了随机van der Pol 系统的Hopf 分岔临界点,探究了随机参数对系统Hopf分岔的影响.最后利用数值模拟验证了理论分析结果. 关键词： 随机van der Pol系统 Hopf分岔 正交多项式逼近     

Dynamical switching and memory via incoherent pump assisted optical bistability

We analyze the dynamical behavior of all-optical switching and memory based on tunable optical bistability in a three-level Λ-type atomic system driven by a probe field circulating in an optical ring cavity, a coherent coupling field and an incoherent pump field outside cavity. Owing to the incoherent pump process, the absorption and Kerr nonlinearity near the atomic resonance change dramatically as a result of varied population distribution and atomic coherence, and then the switch-up and switch-down thresholds, as well as the width (or area) of bistable curve, can be manipulated effectively. By tuning the intensity of either coherent coupling field or incoherent pump field, we can make the system output flip between the lower and upper branches of different bistable hysteresis loops while the cavity input keeps to a constant value. Accordingly, all-optical switching and memory can be implemented via dynamical control of the bistable behavior under the assistance of incoherent pump. The proposed scheme can find potential applications in all-optical communication and computation.     

Bifurcations of a parametrically excited oscillator with strong nonlinearity

A parametrically excited oscillator with strong nonlinearity, including van der Pol and Duffing types, is studied for static bifurcations. The applicable range of the modified Lindstedt－Poincaré method is extended to 1/2 subharmonic resonance systems. The bifurcation equation of a strongly nonlinear oscillator, which is transformed into a small parameter system, is determined by the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analysed.     

Pressure broadening and shift of argon emission lines

Resonance and van der Waals broadening coefficients, as well as width to shift ratios, have been obtained by scanning argon emission lines produced in a high-pressure (～ 1 atm) glow discharge at a temperature of 1130°K. The resonance broadening coefficients yield absolute oscillator strengths of 0·283 ± 0·024 and 0·076 ± 0·008 for the two resonance lines at 1048 Å and 1067 Å. These oscillator strengths are in good agreement with previous measurements. The importance of higher-order terms in the van der Waals potential expansion is demonstrated by the measured coefficients.     

Tunable all-optical bistability in a semiconductor quantum dot damped by a phase-dependent reservoir

We propose a method of frequency and phase control of optical bistability in a unidirectional ring cavity containing a semiconductor structure which is characterized as a ladder three-level system. The system interacts with a coherent probe field, and a control field which consists of a strong coherent field and a weak amplitude-fluctuating stochastic field. A perturbative solution of the master equation of the system allows to eliminate the stochastic field and provides a physical picture in terms of correlation properties of the stochastic field. We find that the bistable response can be modified strongly by means of the amplitude, the frequency and the phase of the stochastic field. In order to illustrate the feasibility of the results, we use parameter values corresponding to an semiconductor quantum dot (QD). This investigation may be used to optimize and control the optical switching process in the QD solid-state system, which is much more practical than that in atomic systems.     

Controlling bistability by linear augmentation

In many bistable oscillating systems only one of the attractors is desired to possessing certain system performance. We present a method to drive a bistable system to a desired target attractor by annihilating the other one. This shift from bistability to monostability is achieved by augmentation of the nonlinear oscillator with a linear control system. For a proper choice of the control function one of the attractors disappears at a critical coupling strength in an control-induced boundary crisis. This transition from bistability to monostability is demonstrated with two paradigmatic examples, the autonomous Chua oscillator and a neuronal system with a periodic input signal.     

A memory device sustained by noise

We propose a memory device consisting of a ring of two bistable coupled oscillators. We show that this system is capable of storing a single bit and its performance improves with noise, in agreement with previous experimental results obtained with discrete bistable electronic circuits. The stored bit can be retrieved asynchronously and, after a certain synchronization time, the probability of error does not depend on the interrogated oscillator. Memory persistence is shown to be maximized for the same noise range that both minimizes the probability of error and ensures synchronization.     

The residue harmonic balance for fractional order van der Pol like oscillators

When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time.     

一种广义三模腔光机械系统的相干完美吸收与透射

提出了一种广义的三模式腔光机械系统,系统的中间是一个反射率为100%的可移动的全反射机械振子,两侧均由一个部分透射的固定光学腔镜构成.其中两个光学腔由一束较强的控制场和一束较弱的信号场驱动与同一个机械振子实现耦合.较弱的信号场将会被该系统完全吸收而不产生任何能量输出,并且当相干完美吸收产生时,输入信号场的能量将由两个腔场和机械模共同分担;较弱的输入信号场由一个腔完美透视到另一个腔而不产生任何的能量损耗.分析与数值结果显示,在不同参数机制下,在该三模光机械系统中可以实现相干完美吸收、相干完美透射和相干完美合成的量子现象.此外,改变腔与腔之间的耦合度,可以实现输出探测场在相干完美吸收和相干完美透射之间转换;通过简单的相位调制,可以实现探测场左腔-右腔的输出和输入方向的互换.这些动态控制在量子信息网络可用来构造光子开关、光子路由、光子交换机等一些特殊功能的光子学器件.     

Enhancement of feasibility of macroscopic quantum superposition state with the quantum Rabi-Stark model

We propose an efficient scheme to generate a macroscopical quantum superposition state with a cavity optomechanical system, which is composed of a quantum Rabi-Stark model coupling to a mechanical oscillator. In a low-energy subspace of the Rabi-Stark model, the dressed states and then the effective Hamiltonian of the system are given. Due to the coupling of the mechanical oscillator and the atom-cavity system, if the initial state of the atom-cavity system is one of the dressed states, the mechanical oscillator will evolve into a corresponding coherent state. Thus, if the initial state of the atom-cavity system is a superposition of two dressed states, a coherent state superposition of the mechanical oscillator can be generated. The quantum coherence and their distinguishable properties of the two coherent states are exhibited by Wigner distribution. We show that the Stark term can enhance significantly the feasibility and quantum coherence of the generated macroscopic quantum superposition state of the oscillator.     

Bistability in coherent two-photon processes

Using the coupled Maxwell-Bloch equations for coherent two photon processes, we show the existence of bistable behavior of systems undergoing coherent two photon resonant processes such as two photon absorption and Raman process. The system is taken to be contained in a ring cavity. The dependence of the bistable behavior on detuning parameter and the Stark shifts is also discussed.     

A note on stochastic phase lockings in a relaxation oscillator forced by a sine input with additive noise

Summary Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. Using the probability density functions of the first-passage time, the operator which governs a transition of an input phase density after one cycle of the oscillator is defined. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator. Paper presented at the International Workshop ?Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena?, Elba, 5–10 June 1994.     

Recurrence analysis and synchronization of oscillators with coexisting attractors

The method of recurrence plots (RPs) has been traditionally used for experimental time series analysis with no comparison with the mathematical model. This is in part because of lack of nonlinear analysis of mathematical model based on the recurrence quantification analysis (RQA) parameters. The paper provides substantial information about the mathematical and numerical analysis and synchronization of a multi-limit cycle oscillator from the RQA perspective. The recurrence quantification analysis parameters are used to discuss the birhythmic behavior of the system, as well as various bifurcations (quasi-periodicity, periodicity and chaos) in the system response. Finally, the results of the method of RPs are compared to those of phase diagrams and the problem of synchronization of limit cycle and chaotic response is discussed by the mean of cross recurrence.