Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle

A detailed asymptotic study of the effect of small Gaussian white noise on a relaxation oscillator undergoing a supercritical Hopf bifurcation is presented. The analysis reveals an intricate stochastic bifurcation leading to several kinds of noise-driven mixed-mode oscillations at different levels of amplitude of the noise. In the limit of strong time-scale separation, five different scaling regimes for the noise amplitude are identified. As the noise amplitude is decreased, the dynamics of the system goes from the limit cycle due to self-induced stochastic resonance to the coherence resonance limit cycle, then to bursting relaxation oscillations, followed by rare clusters of several relaxation cycles (spikes), and finally to small-amplitude oscillations (or stable fixed point) with sporadic single spikes. These scenarios are corroborated by numerical simulations.     

Bifurcation analysis of steady states and limit cycles in a thermal pulse combustor model

Oscillatory behaviour of state variables is desirable in pulse combustors, as properly designed pulsations lead to improved performances, such as higher thermal efficiency and lower emissions compared to steady combustors. In the present work, we perform a systematic investigation of the stability of steady states and limit cycles of a pulse combustor model through numerical continuation. Different bifurcation parameters such as tailpipe friction factor, wall temperature, convective heat transfer coefficient, inlet temperature and inlet fuel mass fraction are varied to identify the complete ranges of those parameters at which limit cycles can be expected. This analysis identifies alternative stable periodic regimes in parameter space (e.g. friction factor). In addition, a few performance indicators such as amplitude of oscillations, cycle-averaged heat transfer and cycle-averaged specific thrust are compared between different ranges of friction factor yielding limit cycle oscillations. The comparison clearly shows that, depending upon the application, friction factor can be chosen from different regimes. The time-integration of the model is also performed to support the bifurcation results obtained from numerical continuation, wherever appropriate. The complete stability margin of limit cycle oscillations for those bifurcation parameters can be useful for improved design of the combustor and for determining the optimal operating conditions of pulse combustors.     

A theoretical study of limit cycle oscillations of plenum air cushions

Air cushion vehicles (ACV) are prone to the occurrence of dynamic instabilities which frequently appear as stable finite amplitude oscillations. The aim of this work is to ascertain if the non-linearities characteristics of ACV dynamics generate limit cycle oscillations for cushion systems operating at conditions for which a linear theory predicts instability. The types of non-linearity that can occur are discussed, and an analysis is presented for a single cell flexible skirted plenum chamber constrained to move in pure heave only. Two cushion feed cases are considered: a plenum box supply and a duct. The results obtained by a Galerkin/describing function analysis are compared with those generated by a full numerical simulation. For the plenum box supply system, it is shown that the limit cycles can be suppressed by using a piston to introduce high frequency small amplitude volume oscillations into the plenum chamber.     

Synchronization,stickiness effects and intermittent oscillations in coupled nonlinear stochastic networks

Long distance reactive and diffusive coupling is introduced in a spatially extended nonlinear stochastic network of interacting particles. The network serves as a substrate for Lotka-Volterra dynamics with 3rd order nonlinearities. If the network includes only local nearest neighbour interactions, the system organizes into a number of local asynchronous oscillators. It is shown that (a) Introduction of all-to-all coupling in the network leads the system into global, center-type, conservative oscillations as dictated by the mean-field dynamics. (b) Introduction of reactive coupling to the network leads the system to intermittent oscillations where the trajectories stick for short times in bounded regions of space, with subsequent jumps between different bounded regions. (c) Introduction of diffusive coupling to the system does not alter the dynamics for small values of the diffusive coupling p_diff, while after a critical value p_diff ^c the system synchronizes into a limit cycle with specific frequency, deviating non-trivially from the mean-field center-type behaviour. The frequency of the limit cycle oscillations depends on the reaction rates and on the diffusion coupling. The amplitude σ of the limit cycle depends on the control parameter p_diff.     

Contagion effects in a chartist-fundamentalist model with time delays

In this paper two models of speculative markets are developed to study the effects of feedback mechanisms in financial markets. In the first model, a crash market model couples a linear chartist-fundamentalist model with time delays with a log-periodic market index I(t) through direct coupling. Numerical solutions to the model show that asset prices exhibit significant persistence as a result of the coupling to the log-periodic market index. An extension to include endogenous wealth dynamics shows that the chartists benefit from the persistent dynamics induced by the coupling. The second model is a two-asset model represented by a 2-dimensional delay-differential equation. Asset one price exhibits limit cycle dynamics while in the second market asset prices follow stable damped oscillations. The markets are coupled through a diffusive coupling term. Solutions to the coupled model show that the dynamics of asset two changes fundamentally with the price now exhibiting a limit cycle. The stable converging dynamics is replaced with limit cycle oscillations around the fundamental.     

光受激发射的稳定性

本文推导出描述三能级Laser工作过程的准经典方程组,并分析了输出振动的稳定性。在阈值以上,当T₁?T₂,q^-1时,只在1/(qT₂)>1时,输出振幅是稳定的(其中T₁,T₂,q^-1分别是分子纵向、横向及谐振腔的弛豫时间)。在稳定区域,趋向平衡的时间与T₁成正比。当分子线宽小于谐振腔宽度时,输出是不稳定的,而在1/(qT₂)减小时,平衡点由稳定变到不稳定时产生一个稳定的极限环,即输出振幅逐渐开始振动。关于稳定性的结论在气体Laser中是可以检验的。本文指出,在红宝石Laser中看到的输出不稳定,可能就是谐振腔的q很大的结果。     

耦合时间精度对模拟飞行器自由运动特性的影响

将亚迭代技术引入流体动力学和刚体动力学方程的耦合求解,获得细长三角翼极限环运动的规律.探讨耦合时间精度对飞行器非定常运动特性的影响,细长三角冀的大迎角自由滚运动最终形成极限环振荡的周期性自维持运动,不同攻角自由滚振幅阶跃式的变化特点较好地吻合了自由滚试验的规律.对于多系统耦合问题,亚迭代耦合求解(耦合时间精度为二阶)对物理时间步长的依赖性不明显;而存在一阶时间滞后的解耦推进方法的数值结果强烈地依赖于物理时间步长选取,稍大的时间步长将导致非物理的数值结果.     

Resurgence of oscillation in coupled oscillators under delayed cyclic interaction

This paper investigates the emergence of amplitude death and revival of oscillations from the suppression states in a system of coupled dynamical units interacting through delayed cyclic mode. In order to resurrect the oscillation from amplitude death state, we introduce asymmetry and feedback parameter in the cyclic coupling forms as a result of which the death region shrinks due to higher asymmetry and lower feedback parameter values for coupled oscillatory systems. Some analytical conditions are derived for amplitude death and revival of oscillations in two coupled limit cycle oscillators and corresponding numerical simulations confirm the obtained theoretical results. We also report that the death state and revival of oscillations from quenched state are possible in the network of identical coupled oscillators. The proposed mechanism has also been examined using chaotic Lorenz oscillator.     

Nonlinear non-spherical oscillations of gas bubble under a periodic variation of ambient liquid pressure

A mathematical model is constructed for the bubble dynamics, in which the interphase surface variation is presented in the form of a series in spherical harmonics, and the equations are written with the accuracy up to the squared amplitude of the distortion of the spherical shape of the bubble. In the oscillation regimes close to periodic sonoluminescence of a single bubble in a standing acoustic wave, the character of air bubble oscillations in water was studied depending on the bubble initial radius and the amplitude of the liquid pressure variation. It was found that non-spherical oscillations of bounded amplitude can take place outside the region of linearly stable spherical oscillations. Both the oscillations with a period equal to one or several periods of the liquid pressure variation and aperiodic oscillations are observed. It is shown that neglecting the distortions in the form of spherical harmonics with large numbers (i > 3) may lead to a change of oscillation regimes. The influence of distortions on the bubble surface shape for the harmonics with i > 8 is insignificant.     

二次耦合光力学系统的一类高维可控自持振荡行为

光力学系统通常的耦合是光压耦合, 是光场强度和纳米振子位移的一次耦合, 但在光场很强和振子振幅较大的光力学系统中, 非线性的耦合效应会变得非常明显和重要, 而且其所产生的非线性效应对制造具有特殊功能的光力学器件具有重要意义. 本文在二次耦合模型的基础上研究了光腔和振子之间通过二次耦合作用达到能 量平衡状态时系统所产生的自持振荡现象, 给出了二次耦合光力学系统的一般模型, 并通过数值方法研究了系统的定态行为和远离定态的极限环动力学行为, 标定了系统定态响应的稳定区域到极限环行为的分岔点. 发现在调节输入场参数(改变耦合系数)以及光腔和振子的弛豫系数时, 系统的相空间会出现一些稳定的高维自持振荡极限环. 通过数值分析发现该四维极限环在三维相空间的投影都趋于稳定的三维周期轨道, 并且该极限环轨道会随外部调控参数的改变发生扭动, 出现类似二维李萨如图样的稳定纽结结构. 该现象表明: 通过光场与振子的能量耦合, 利用一定强度的外部驱动可以有效控制振子的定态响应和振动, 可以让微振子锁定在具有一定振幅和频率的自发振动上, 为开发物理器件提供了可靠的光力学控制系统. 关键词： 光力系统 二次耦合 自持振荡 极限环     

NON-LINEAR AEROELASTIC ANALYSIS USING THE POINT TRANSFORMATION METHOD, PART 1: FREEPLAY MODEL

A point transformation technique is developed to investigate the non-linear behavior of a two-dimensional aeroelastic system with freeplay models. Two formulations of the point transformation method are presented, which can be applied to accurately predict the frequency and amplitude of limit cycle oscillations. Moreover, it is demonstrated that the developed formulations are capable of detecting complex aeroelastic responses such as periodic motions with harmonics, period doubling, chaotic motions and the coexistence of stable limit cycles. Applications of the point transformation method to several test examples are presented. It is concluded that the formulations developed in this paper are efficient and effective.     

Noise-induced generation of saw-tooth type transitions between climate attractors and stochastic excitability of paleoclimate

Motivated by important paleoclimate applications we study a three dimensional model ofthe Quaternary climatic variations in the presence of stochastic forcing. It is shown thatthe deterministic system exhibits a limit cycle and two stable system equilibria. Wedemonstrate that the closer paleoclimate system to its bifurcation points (lying either inits monostable or bistable zone) the smaller noise generates small or large amplitudestochastic oscillations, respectively. In the bistable zone with two stable equilibria,noise induces a complex multimodal stochastic regime with intermittency of small and largeamplitude stochastic fluctuations. In the monostable zone, the small amplitude stochasticoscillations localized in the vicinity of unstable equilibrium appear along with the largeamplitude oscillations near the stable limit cycle. For the analysis of thesenoise-induced effects, we develop the stochastic sensitivity technique and use theMahalanobis metric in the three-dimensional case. To approximate the distribution ofrandom trajectories in Poincare sections, we use a method of confidence ellipses. Aspatial configuration of these ellipses is defined by the stochastic sensitivity and noiseintensity. The glaciation/deglaciation transitions going between two polar Earth’s stateswith the warm and cold climate become easier and quicker with increasing the noiseintensity. Our stochastic analysis demonstrates a near 100 ky saw-tooth type climate selffluctuations known from paleoclimate records. In addition, the enhancement of noiseintensity blurs the sharp climate cycles and reduces the glaciation-deglaciation periodsof the Earth’s paleoclimate.     

Synchronization of diffusively coupled oscillators near the homoclinic bifurcation

It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the in-phase synchronization and also that it is the only stable state in the weak-coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which often occurs especially for the neuronal oscillators. In this paper we propose a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the sychronization behaviors between the distinctive regimes as well. The dephasing mechanism is analyzed both qualitatively and quantitatively in the weak-coupling limit. A general form of coupling is introduced and the synchronization behaviors over a wide range of the coupling parameters are explored to construct the phase diagram using the bifurcation analysis.     

NON-LINEAR AEROELASTIC ANALYSIS USING THE POINT TRANSFORMATION METHOD, PART 2: HYSTERESIS MODEL

This paper investigates the dynamic response of a two-dimensional aeroelastic system with structural non-linearity represented by hysteresis. The formulations of the point transformation method developed in Part 1 of this study for the aeroelastic system with a freeplay model is extended for a hysteresis model. These formulations can be applied not only to predict the amplitude and frequency of limit cycle oscillations, but also to detect complex aeroelastic responses such as periodic motion with harmonics, period doubling, chaotic motion and the coexistence of stable limit cycles. It is shown that the point transformation technique is the most suitable to analyze the aeroelastic response of systems containing piecewise continuous restoring forces.     

Bifurcation analysis of a two-degree-of-freedom aeroelastic system with freeplay structural nonlinearity by a perturbation-incremental method

A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of limit cycle oscillations (LCO) of a two-degree-of-freedom aeroelastic system containing a freeplay structural nonlinearity. Both stable and unstable LCOs can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Thus, the present method is capable of detecting complex aeroelastic responses such as periodic motion with harmonics, period-doubling (PD), saddle-node bifurcation, Neimark-Sacker bifurcation and the coexistence of limit cycles. Emanating branch from a PD bifurcation can be constructed. This method can also be applied to any piecewise linear systems.     

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.     

Variability in the noise-induced modes of climate dynamics

New features of noise-induced climate variability are revealed on the basis of the three-dimensional model derived by Saltzman and Maasch. It is shown that the climate system can be highly noise excitable and it possesses the large-amplitude fluctuations even in those regions where its akin deterministic model does not contain any self-sustained oscillations. Intermittency in small- and large amplitude climate fluctuations between different basins of attraction of a limit cycle and stable equilibria substantially influencing the climate state (from warm to cold and vice versa) are found at various noise intensities. Suddenly occurring jumps between the basins of attraction of two stable equilibria corresponding to the warm and cold climate states are statistically confirmed under a certain diapason of noise intensities. The climate system undergoes transitions between its equilibria in the presence of noise in its prognostic variables. In addition, such transitions become more likely with increasing the noise intensity.     

Transmission phase of a singly occupied quantum dot in the Kondo regime

We report on the phase measurements on a quantum dot containing a single electron in the Kondo regime. Transport takes place through a single orbital state. Although the conductance is far from the unitary limit, we measure directly, for the first time, a transmission phase as theoretically predicted of pi/2. As the dot's coupling to the leads is decreased, with the dot entering the Coulomb blockade regime, the phase reaches a value of pi. Temperature shows little effect on the phase behavior in the range 30-600 mK, even though both the two-terminal conductance and amplitude of the Aharonov-Bohm oscillations are strongly affected. These results also suggest that previous phase measurements involved transport through more than a single level.     

From simple to complex oscillatory behavior in metabolic and genetic control networks

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.