Excitable dynamics in the presence of time delay

The spiking properties of a subcritical Hopf oscillator with a time delayed nonlinear feedback is investigated. Finite time delay is found to significantly affect both the statistics and the fine structure of the spiking behavior. These dynamical changes are explained in terms of the fundamental modifications occurring in the bifurcation scenario of the system. Our mathematical model can find useful applications in understanding the dynamical behavior of various real life excitable systems where propagation delay effects are ubiquitous.     

Coherence and stochastic resonance in threshold crossing detectors with delayed feedback

We consider the dynamical behavior of threshold systems driven by external periodic and stochastic signals and internal delayed feedback. Specifically, the effect of positive delayed feedback on the sensitivity of a threshold crossing detector (TCD) to periodic forcing embedded in noise is investigated. The system has an intrinsic ability to oscillate in the presence of positive feedback. We first show conditions under which such reverberatory behavior is enhanced by noise, which is a form of coherence resonance (CR) for this system. Further, for input signals that are subthreshold in the absence of feedback, the open-loop stochastic resonance (SR) characteristic can be sharply enhanced by positive delayed feedback. This enhancement is shown to depend on the stimulus period, and is maximal when this period is matched to an integer multiple of the delay. Reverberatory oscillations, which are particularly prominent after the offset of periodic forcing, are shown to be eliminated by a summing network of such TCDs with local delayed feedback. Theoretical analysis of the crossing rate dynamics qualitatively accounts for the existence of CR and the resonant behavior of the SR effect as a function of delay and forcing frequency.     

Delayed stochastic systems

Noise and time delay are two elements that are associated with many natural systems, and often they are sources of complex behaviors. Understanding of this complexity is yet to be explored, particularly when both elements are present. As a step to gain insight into such complexity for a system with both noise and delay, we investigate such delayed stochastic systems both in dynamical and probabilistic perspectives. A Langevin equation with delay and a random-walk model whose transition probability depends on a fixed time-interval past (delayed random walk model) are the subjects of in depth focus. As well as considering relations between these two types of models, we derive an approximate Fokker-Planck equation for delayed stochastic systems and compare its solution with numerical results.     

时滞反馈控制扭转振动系统的振动

研究了采用时滞反馈来控制扭转振动系统的振动问题.在一个带有非线性动力吸振器的扭转振动系统中,采用时滞反馈来控制主系统的振动.研究了反馈增益系数和时滞对主系统振动的影响.研究结果表明,对某一固定的反馈增益系数,存在时滞的某段调节区间,可以通过调节时滞来抑制主系统的振动.在时滞的调节区间内存在一个最佳点,主系统的振动被抑制到最小值.可以同时调节反馈增益系数和时滞两参数,当反馈增益系数和时滞都调节到最佳值时,主系统振动的振幅由0.24减小到0.03,取得了很好的减振效果. 关键词： 时滞反馈 扭转振动 减振     

Chaotic breathers in delayed electro-optical systems

We show that in integro-differential delayed dynamical systems, a hybrid state of simultaneous fast-scale chaos and slow-scale periodicity can emerge subsequently to a sequence of Hopf bifurcations. The resulting time trace thereby consists in chaotic oscillations "breathing" periodically at a significantly lower frequency. Experimental evidence of this type of dynamics in delayed dynamical systems is achieved with a Mach-Zehnder modulator optically fed by a semiconductor laser and is subjected to a delayed nonlinear electro-optical feedback. We also propose a theoretical understanding of the phenomenon.     

A data-analysis method for identifying differential effects of time-delayed feedback forces and periodic driving forces in stochastic systems

In this work a method is developed for analyzing time series of periodically driven stochastic systems involving time-delayed feedback. The proposed data-analysis method yields dynamical models in terms of stochastic delay differential equations. On the basis of these dynamical models differential effects of driving forces and time-delayed feedback forces can be identified.     

Characterising chaos-hyperchaos transition using correlation dimension

Transition to hyperchaos is uaually studied by computing the spectrum of Lyapunov Exponents (LE). But such a procedure can be employed mainly when the equations governing the dynamical system are known. However, if the information available on the system is only through time series, the method becomes difficult to implement. We show that the transition to hyperchaos is followed by a sudden change in the topological structure of the underlying attractor. Our numerical results indicate that the transition to hyperchaos can be characterized accurately through the computation of correlation dimension (D ₂) from time series. We use two standard time delayed hyperchaotic systems as examples since, for such systems, D ₂ varies smoothly as a function of the time delay τ which can be used as the control parameter.     

Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay

Many natural and artificial systems are subject to some sort of delay, which can be in the form of a single discrete delay or distributed over a range of times. Here, we discuss the impact of this distribution on (thermo-)dynamical properties of time-delayed stochastic systems. To this end, we study a simple classical model with white and colored noise, and focus on the class of Gamma-distributed delays which includes a variety of distinct delay distributions typical for feedback experiments and biological systems. A physical application is a colloid subject to time-delayed feedback control, which is, in principle, experimentally realizable by co-moving optical traps. We uncover several unexpected phenomena in regard to the system’s linear stability and its thermodynamic properties. First, increasing the mean delay time can destabilize or stabilize the process, depending on the distribution of the delay. Second, for all considered distributions, the heat dissipated by the controlled system (e.g., the colloidal particle) can become negative, which implies that the delay force extracts energy and entropy of the bath. As we show here, this refrigerating effect is particularly pronounced for exponential delay. For a specific non-reciprocal realization of a control device, we find that the entropic costs, measured by the total entropy production of the system plus controller, are the lowest for exponential delay. The exponential delay further yields the largest stable parameter regions. In this sense, exponential delay represents the most effective and robust type of delayed feedback.     

Stability of superflow for ultracold fermions in optical lattices

We investigate the stability of superflow of paired fermions in an optical lattice. We show that there are two distinct dynamical instabilities which limit the superflow in this system. One dynamical instability occurs when the superfluid stiffness becomes negative; this evolves, with increasing pairing interaction, from the fermion pair breaking instability to the well-known dynamical instability of lattice bosons. The second, more interesting, dynamical instability is marked by the emergence of a transient atom density wave. Both dynamical instabilities can be experimentally accessed by tuning the pairing interaction and the fermion density.     

Dynamic instabilities in scalar neural field equations with space-dependent delays

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These nonlocal models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg-Landau equations describing a Turing-Hopf bifurcation with modulation group velocity of O(1). Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but also those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin-Feir instabilities.     

循环平稳声场的声源定位研究

旋转机械产生的辐射声场具有循环平稳特性,传统的平面近场声全息技术无法准确反映其调制特性,往往在边频带上出现虚假的能量的分布. 采用循环谱密度取代功率谱密度作为重建物理量,则可准确提取循环平稳声场的调制和载波信息. 考虑到循环谱密度的计算量以及特征提取的准确性,提出了循环谱密度组合切片分析法,并分析了加性白噪声对重建的影响. 仿真分析及实验结果表明,此方法有较强的噪声抑制能力,全息重建的结果可准确反映声源的位置. 关键词： 循环平稳声场 近场声全息 噪声源定位     

Complex dynamics in delay-differential equations with large delay

We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback.     

A low-dimensional dynamical system to describe low-frequency fluctuations in a semiconductor laser with optical feedback

We derive a low-dimensional dynamical system to describe a semiconductor laser with optical feedback. This model captures many features from the original time delayed dynamical system and explains the origin of the low-frequency instability observed in the time-delayed equation.     

基于延时光电反馈控制两段式半导体激光器的双稳及自脉动特性

提出一种将延时光电反馈引入两段式双稳半导体激光器（TBLD）的吸收区，从而达到控制其双稳及自脉动特性的方案.利用速率方程模型，分析了反馈增益及延迟时间对TBLD双稳及不稳定特性的控制作用.数值模拟的结果表明：TBLD的稳定区域随延迟时间呈周期性变化；双稳区域随延迟时间的增大而变窄；在一定的取值范围内，增大延迟时间有利于增强双稳的稳定性；负反馈更容易出现不稳定性. 关键词： 两段式双稳半导体激光器 动态双稳 自脉动 延时光电反馈     

Controlling a time-delay system using multiple delay feedback control

In this paper multiple delay feedback control (MDFC) with different and independent delay times is shown to be an efficient method for stabilizing fixed points in finite-dimensional dynamical systems. Whether MDFC can be applied to infinite-dimensional systems has been an open question. In this paper we find that for infinite-dimensional systems modelled by delay differential equations, MDFC works well for stabilizing (unstable) steady states in long-, moderate- and short-time delay regions, in particular for the hyperchaotic case.     

Recovery of delay time from time series based on the nearest neighbor method

We propose a method for the recovery of delay time from time series of time-delay systems. The method is based on the nearest neighbor analysis. The method allows one to reconstruct delays in various classes of time-delay systems including systems of high order, systems with several coexisting delays, and nonscalar time-delay systems. It can be applied to time series heavily corrupted by additive and dynamical noise.     

Phase synchronization of bursting neural networks with electrical and delayed dynamic chemical couplings

Diffusive electrical connections in neuronal networks are instantaneous, while excitatoryor inhibitory couplings through chemical synapses contain a transmission time-delay.Moreover, chemical synapses are nonlinear dynamical systems whose behavior can bedescribed by nonlinear differential equations. In this work, neuronal networks withdiffusive electrical couplings and time-delayed dynamic chemical couplings are considered.We investigate the effects of distributed time delays on phase synchronization of burstingneurons. We observe that in both excitatory and Inhibitory chemical connections, the phasesynchronization might be enhanced when time-delay is taken into account. This distributedtime delay can induce a variety of phase-coherent dynamical behaviors. We also study thecollective dynamics of network of bursting neurons. The network model presents theso-called Small-World property, encompassing neurons whose dynamics have two time scales(fast and slow time scales). The neuron parameters in such Small-World network, aresupposed to be slightly different such that, there may be synchronization of the bursting(slow) activity if the coupling strengths are large enough. Bounds for the criticalcoupling strengths to obtain burst synchronization in terms of the network structure aregiven. Our studies show that the network synchronizability is improved, as itsheterogeneity is reduced. The roles of synaptic parameters, more precisely those of thecoupling strengths and the network size are also investigated.     

Quantum study of information delay in electromagnetically induced transparency

Using electromagnetically induced transparency (EIT), it is possible to delay and store light in atomic ensembles. Theoretical modeling and recent experiments have suggested that the EIT storage mechanism can be used as a memory for quantum information. We present experiments that quantify the noise performance of an EIT system for conjugate amplitude and phase quadratures. It is shown that our EIT system adds excess noise to the delayed light that has not hitherto been predicted by published theoretical modeling. In analogy with other continuous-variable quantum information systems, the performance of our EIT system is characterized in terms of conditional variance and signal transfer.     

Instabilities and quasi-localized states in nonlinear Fano-like systems

We study the dynamical scattering in one-dimensional systems with a nonlinear side-coupled defect. Such structures exhibit the nonlinear Fano resonances, where nothing can propagate through. We developed a numerical model to study dynamical scattering. According to our analysis the scattering waves become dynamically unstable in the vicinity of the nonlinear Fano resonances, due to modulational instability caused by the presence of nonlinearity. It results in a time-growing amplitude of the nonlinear defect. We also demonstrate the existence of the nonlinear quasi-localized state, supported by such structures.     

Delay-Coordinate Maps and the Spectra of Koopman Operators

The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the point spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions in systems with pure point or mixed spectra. We illustrate our results with applications to product dynamical systems with mixed spectra.