Transition scenario to turbulence in thin vibrating plates

A thin plate, excited by a harmonic external forcing of increasing amplitude, shows transitions from a periodic response to a chaotic state of wave turbulence. By analogy with the transition to turbulence observed in fluid mechanics as the Reynolds number is increased, a generic transition scenario for thin vibrating plates, first experimentally observed, is here numerically studied. The von Kármán equations for thin plates, which include geometric non-linear effects, are used to model large amplitude vibrations, and an energy-conserving finite difference scheme is employed for discretisation. The transition scenario involves two bifurcations separating three distinct regimes. The first regime is the periodic, weakly non-linear response. The second is a quasiperiodic state where energy is exchanged between internally resonant modes. It is observed only when specific internal resonance relationships are fulfilled between the eigenfrequencies of the structure and the forcing frequency; otherwise a direct transition to the last turbulent state is observed. This third, or turbulent, regime is characterized by a broadband Fourier spectrum and a cascade of energy from large to small wavelengths. For perfect plates including cubic non-linearity, only third-order internal resonances are likely to exist. For imperfect plates displaying quadratic nonlinearity, the energy exchanges and the quasiperiodic states are favored and thus are more easily obtained. Finally, the turbulent regime is characterized in the light of available theoretical results from wave turbulence theory.     

Topology of trajectories of the 2D Navier-Stokes equations

In spectral form the 2D incompressible Navier-Stokes equations in a square periodic region will be represented by 430 complex Fourier amplitudes which correspond to isotropic truncation of the upper wave number 16. For small viscosity, we have found five equilibrium states I-V in the entire range of forcing; I-fixed point, II-circle, III-closed orbit, IV-torus, and V-chaos. The fixed-point equilibrium state is the laminar flow. As the forcing passes through a critical value, the fixed point evolves directly to equilibrium state III under a typical multimode forcing. The chaotic transition takes place on a 2-torus-like manifold (equilibrium state IV) which is the product space of a circle and the closed orbit of equilibrium state III, similar to the quasiperiodic 2-torus of Ruelle and Takens. For sufficiently large forcing, the evolution of equilibrium state V is nothing but a simulation of quasistationary 2D turbulence. From the Lyapunov exponents of turbulent flows, we have evaluated the constants in the theoretical results of Foias and his colleagues, which relate the determining mode and fractal dimension with the enstrophy dissipation wave number of 2D turbulence.     

Statistics of power injection in a plate set into chaotic vibration

A vibrating plate is set into a chaotic state of wave turbulence by either a periodic or a random local forcing. Correlations between the forcing and the local velocity response of the plate at the forcing point are studied. Statistical models with fairly good agreement with the experiments are proposed for each forcing. Both distributions of injected power have a logarithmic cusp for zero power, while the tails are Gaussian for the periodic driving and exponential for the random one. The distributions of injected work over long time intervals are investigated in the framework of the fluctuation theorem, also known as the Gallavotti-Cohen theorem. It appears that the conclusions of the theorem are verified only for the periodic, deterministic forcing. Using independent estimates of the phase space contraction, this result is discussed in the light of available theoretical framework.     

Noise-induced intermittency in the quasiperiodic regime of Rayleigh-Bénard convection

Thermal measurements on a converting dilute³He-superfluid⁴He solution in the quasiperiodic regime show a transition from a mode-locked periodic state to chaotic time dependence via intermittency. The type of intermittency is discussed in the context of standard models of the phenomenon. In a region just below the onset of intermittency, injection of external multiplicative noise with noise amplitude above a certain threshold level induces the chaotic state. This noise-induced transition can be understood to be due to perturbations of a system with a barely stable attractor; the noise causes the system to escape the weakly attracting periodic points. We present a numerical simulation of a 1D map with external noise which explains some aspects of the noise-induced behavior, and a 2D map which has certain features of the intermittency.     

噪声在外加周期信号控制强湍中的作用研究

研究了噪声在控制中的作用.研究发现,较小强度的偏噪声信号能够使系统失去层流态,重新进入无序态,此时模式之间的广义相同步也失去了;而噪声强度较大时,系统则能够维持准周期的层流态,模式相位之间依然能够达到广义相同步. 关键词： 相同步 时空混沌 湍流 非线性漂移波     

Organizing multiple femtosecond filaments in air

We show that it is possible to organize regular filamentation patterns in air by imposing either strong field gradients or phase distortions in the input-beam profile of an intense femtosecond laser pulse. A comparison between experiments and 3+1 dimensional numerical simulations confirms this concept and shows for the first time that a control of the transport of high intensities over long distances may be achieved by forcing this well ordered propagation regime. In this case, deterministic effects prevail in multiple femtosecond filamentation, and no transition to the optical turbulence regime is obtained [Phys. Rev. Lett. 83, 2938 (1999)]].     

Clustering of arrays of chaotic chemical oscillators by feedback and forcing

Feedback and external forcing are applied to an array of chaotic electrochemical oscillators through variations in the applied potential. We see transitions from intermittent clusters to stable chaotic clusters to stable periodic clusters to synchronized states as the feedback gain and forcing amplitude, respectively, are varied. With forcing up to four clusters are observed in stable states. The transition to synchronization with feedback occurs by the increase in the size of one cluster at the expense of the others.     

Resonant activation in a stochastic Hodgkin-Huxley model: Interplay between noise and suprathreshold driving effects

The paper considers an excitable Hodgkin-Huxley system subjected to a strong periodic forcing in the presence of random noise. The influence of the forcing frequency on the response of the system is examined in the realm of suprathreshold amplitudes. Our results confirm that the presence of noise has a detrimental effect on the neuronal response. Fluctuations can induce significant delays in the detection of an external signal. We demonstrate, however, that this negative influence may be minimized by a resonant activation effect: Both the mean escape time and its standard deviation exhibit a minimum as functions of the forcing frequency. The destructive influence of noise on the interspike interval can also be reduced. With driving signals in a certain frequency range, the system can show stable periodic spiking even for relatively large noise intensities. Outside this frequency range, noise of similar intensity destroys the regularity of the spike trains by suppressing the generation of some of the spikes.     

一类新混沌系统的线性状态反馈控制

研究了一类新混沌系统的控制问题.利用Routh-Hurwitz准则对受控系统进行了稳定性分析，结合线性状态反馈方法理论上严格证明了达到控制目标反馈系数的选择原则.数值研究证明了该方法能够有效地控制混沌系统到失稳的平衡点或周期解，同时控制效果在弱噪声干扰下具有很强的鲁棒性. 关键词： 新混沌系统 线性状态反馈控制 Routh-Hurwitz准则 噪声     

Attractors of convex maps with positive Schwarzian derivative in the presence of noise

One-dimensional maps have proved to be useful models for understanding the transition to turbulence. We investigate a smooth perturbation of the well-known logistic system in order to examine numerically the change in the bifurcation behavior which is observed, when the Schwarzian derivative is allowed to become positive. We find coexistence of a fixed point attractor and a periodic or chaotic two-band-attractor. The chaotic two-band attractor can disappear by yielding a preturbulent state which will asymptotically settle down to a fixed-point. The chaotic behavior of some systems can be destroyed by arbitrarily small amounts of external noise. The concept of (ε, δ)-diffusions is used to describe the sensitivity of attractors against external noise. We also observe a direct transition from a fixed-point to a chaotic one-band attractor. This can be interpreted as type-III-intermittency of Pomeau and Manneville but with an almost linear scaling behavior of the Lyapunov exponent.     

Noise-sustained structure,intermittency, and the Ginzburg-Landau equation

The time-dependent generalized Ginzburg-Landau equation is an equation that is related to many physical systems. Solutions of this equation in the presence of low-level external noise are studied. Numerical solutions of this equation in thestationary frame of reference and with anonzero group velocity that is greater than a critical velocity exhibit a selective spatial amplification of noise resulting in spatially growing waves. These waves in turn result in the formation of a dynamic structure. It is found that themicroscopic noise plays an important role in themacroscopic dynamics of the system. For certain parameter values the system exhibits intermittent turbulent behavior in which the random nature of the external noise plays a crucial role. A mechanism which may be responsible for the intermittent turbulence occurring in some fluid systems is suggested.     

Wall pressure fluctuation spectra due to boundary-layer transition

Boundary-layer transition has been expected to be an important contributor to sensor flow-induced self-noise. The pressure fluctuations caused by this spatially bounded, and intermittent, phenomenon encompass a very wide range of wavenumbers and temporal frequencies. Here, we analyze the wavevector–frequency spectrum of the wall pressure fluctuations due to subsonic boundary-layer transition as it occurs on a flat plate under zero-pressure gradient conditions. Based on previous measurements of the statistics of the boundary-layer intermittency, it is found that transition induces higher low-streamwise wavenumber wall pressure levels than does a fully developed turbulent boundary layer that might superficially exist at the same location and at the same Reynolds number. The transition zone spanwise wavenumber pressure components are virtually unchanged from the fully developed turbulent boundary-layer case. The results suggest that transition may be more effective than the fully developed turbulent boundary layer in forcing structural excitation at low Mach numbers, and it may have a more intense radiated noise contribution. This may help explain increases in measured sensor self-noise when the sensors are placed near the transition zone. We believe, based on the presented analytical calculation and numerical simulation, that the rapid growth and subsequent decay of turbulent spots in the intermittent transition zone causes the higher low-(streamwise) wavenumber spectra.     

Forcing and control of localized states in optical single feedback systems

Under feedback extended nonlinear optical systems spontaneously show a variety of periodic patterns and structures. Control gives new insight into these phenomena and it can open the way for potential application of nonlinear optical structures. We briefly review methods to control localized states in single feedback experiments. Application of a Fourier control method allows to modify interaction behavior of the localized states. As a further approach we study a forcing method, using externally created light fields as additional input to the system. Recent experiments show that the forcing method enables to favor addressing positions for localized structures. We demonstrate static addressing and favoring of addressing positions. We extend the forcing method to a dynamic forcing scheme, which allows to move and reposition localized states. Additionally forcing is used to balance experimental imperfections. PACS 05.45.Gg; 42.60.Jf; 42.65.Tg     

Transition to turbulence for doubly periodic flows

We construct a typical model for the Poincaré map of doubly periodic flows, which presents numerically a transition to chaotic behavior. After the frequency locking phenomenon, we observe two types of transitions to turbulence. The first one involves successive subharmonic instabilities of a periodic solution. The second one occurs after the disappearance of a periodic solution and can be either intermittent or discontinuous with hysteresis.     

Stability of GOY Model Under Modulation of Periodic External Force

There is a phase transition between quasi-periodic state and intermittent chaos in GOY model with a critical value δ0. When we add a modulated periodic externa/force to the system, the phase transition can also be found with a critical value δe. Due to coupling between the force and the intrinsic fluctuation of the velocity on shells in GOY model, the stability of the system has been changed, which results in the variation of the critical value. For proper intensity and period of the force, δe is unequal to δ0. The critical value is a nonlinear function of amplitude of the force, and the fluctuation of the velocity can resonate with the external force for certain period Te.     

Stochastic resonance and energy optimization in spatially extended dynamical systems

We investigate a class of nonlinear wave equations subject to periodic forcing and noise, and address the issue of energy optimization. Numerically, we use a pseudo-spectral method to solve the nonlinear stochastic partial differential equation and compute the energy of the system as a function of the driving amplitude in the presence of noise. In the fairly general setting where the system possesses two coexisting states, one with low and another with high energy, noise can induce intermittent switchings between the two states. A striking finding is that, for fixed noise, the system energy can be optimized by the driving in a form of resonance. The phenomenon can be explained by the Langevin dynamics of particle motion in a double-well potential system with symmetry breaking. The finding can have applications to small-size devices such as microelectromechanical resonators and to waves in fluid and plasma.     

Suppression of the Spiral Wave and Turbulence in the Excitability-Modulated Media

Periodical forcing is used to control the spiral wave and turbulence in the modified Fithzhugh-Nagumo equation (MFHNe) when excitability is changed. The decisive parameter ε of (MFHNe), which describes the ratio of time scales of the fast activator u and the slow inhibitor variable v, is supposed to increase linearly to simulate the excitability modulation in the media. In the numerical simulation, a local periodical stimulus is imposed on the left border of the media and the periods of external forcing are adjusted according to the approximate formula ω ∝1/ε ^1/3 so that using the most appropriate frequency for the external forcing can approach a shorter transient period. It is found that the spiral wave and turbulence can be removed successfully by using an appropriate periodical forcing on the left border of the media. The mean activator and distribution of frequency of all the sites are also used to analyze the transition of spiral wave.     

Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force

The effects of stochastic perturbations and periodic excitations on the eutrophicated lake ecosystem are explored. Unlike the existing work in detecting early warning signals, this paper presents the most probable transition paths to characterize the regime shifts. The most probable transition paths are obtained by minimizing the Freidlin-Wentzell (FW) action functional and Onsager-Machlup (OM) action functional, respectively. The most probable path shows the movement trend of the lake eutrophication system under noise excitation, and describes the global transition behavior of the system. Under the excitation of Gaussian noise, the results show that the stability of the eutrophic state and the oligotrophic state has different results from two perspectives of potential well and the most probable transition paths. Under the excitation of Gaussian white noise and periodic force, we find that the transition occurs near the nearest distance between the stable periodic solution and the unstable periodic solution.     

Brownian motors and stochastic resonance

We study the transport properties for a walker on a ratchet potential. The walker consists of two particles coupled by a bistable potential that allow the interchange of the order of the particles while moving through a one-dimensional asymmetric periodic ratchet potential. We consider the stochastic dynamics of the walker on a ratchet with an external periodic forcing, in the overdamped case. The coupling of the two particles corresponds to a single effective particle, describing the internal degree of freedom, in a bistable potential. This double-well potential is subjected to both a periodic forcing and noise and therefore is able to provide a realization of the phenomenon of stochastic resonance. The main result is that there is an optimal amount of noise where the amplitude of the periodic response of the system is maximum, a signal of stochastic resonance, and that precisely for this optimal noise, the average velocity of the walker is maximal, implying a strong link between stochastic resonance and the ratchet effect.