First-passage failure of a business cycle model under time-delayed feedback control and wide-band random excitation

First-passage failure of a classical business cycle model subject to time-delayed feedback control and wide-band random excitation is investigated in this paper. First, we get the averaged equation by using a stochastic averaging method, and then a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations analytically and numerically with suitable initial and boundary conditions. The results show that time delay in the feedback control forces may remarkably reduce the conditional reliability and the mean first-passage time of the controlled economics system.     

Effect of time delay in FitzHugh-Nagumo neural model with correlations between multiplicative and additive noises

We study the effect of time delay in the FitzHugh-Nagumo neural model with correlations between multiplicative and additive noise terms. Based on the corresponding Fokker-Planck equation, the explicit expressions of the stationary probability distribution function (SPDF), the mean first passage time (MFPT) and the signal-to-noise ratio (SNR) are obtained, respectively. Research results show that: (i) the system undergoes a succession of two phase transitions (i.e., the reentrance phenomenon) as the noise correlation parameter is increased and a (single) phase transition as the time delay is increased. (ii) The MFPT as a function of the multiplicative noise intensity exhibits a maximum. This maximum for MFPT identifies the noise enhanced stability (NES) effect, the noise correlation parameter intensifies the NES effect while the time delay, and the additive noise intensity weakens it. (iii) The existence of a maximum in the SNR as a function of the multiplicative noise intensity is the identifying characteristic of the stochastic resonance (SR) phenomenon, the noise correlation parameter enhances the SR while the time delay, and the additive noise intensity weaken it.     

Stability and Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback

The stability and the Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback are studied.By considering the energy in the air-gap field of the AC motor,the dynamical equation of the electromechanical coupling transmission system is deduced and a time delay feedback is introduced to control the dynamic behaviors of the system.The characteristic roots and the stable regions of time delay are determined by the direct method,and the relationship between the feedback gain and the length summation of stable regions is analyzed.Choosing the time delay as a bifurcation parameter,we find that the Hopf bifurcation occurs when the time delay passes through a critical value.A formula for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is given by using the normal form method and the center manifold theorem.Numerical simulations are also performed,which confirm the analytical results.     

The influences of delay time on the stability of a market model with stochastic volatility

The effects of the delay time on the stability of a market model are investigated, by using a modified Heston model with a cubic nonlinearity and cross-correlated noise sources. These results indicate that: (i) There is an optimal delay time _τo${\tau }_o$ which maximally enhances the stability of the stock price under strong demand elasticity of stock price, and maximally reduces the stability of the stock price under weak demand elasticity of stock price; (ii) The cross correlation coefficient of noises and the delay time play an opposite role on the stability for the case of the delay time <_τo$<{\tau }_o$ and the same role for the case of the delay time >_τo$>{\tau }_o$. Moreover, the probability density function of the escape time of stock price returns, the probability density function of the returns and the correlation function of the returns are compared with other literatures.     

Estimation of time delay by coherence analysis

Using coherence analysis (which is an extensively used method to study the correlations in frequency domain, between two simultaneously measured signals) we estimate the time delay between two signals. This method is suitable for time delay estimation of narrow band coherence signals for which the conventional methods cannot be reliably applied. We show, by analysing coupled Rössler attractors with a known delay, that the method yields satisfactory results. Then, we apply this method to human pathologic tremor. The delay between simultaneously measured traces of electroencephalogram (EEG) and electromyogram (EMG) data of subjects with essential hand tremor is calculated. We find that there is a delay of 11–27 milli-seconds (ms) between the tremor correlated parts (cortex) of the brain (EEG) and the trembling hand (EMG) which is in agreement with the experimentally observed delay value of 15 ms for the cortico-muscular conduction time. By surrogate analysis we calculate error bars of the estimated delay.     

有界噪声激励下带有时滞反馈的随机Mathieu-Duffing系统的响应

研究了参数激励下带有时滞反馈的随机Mathieu-Duffing方程的主参数共振响应问题.运用多尺度方法分离了系统的快慢变量.分析了系统的分岔性质,发现调谐参数、时滞、时滞项的系数以及非线性项的强度等都可以影响系统的分岔行为,适当选择这些参数可以改变系统的分岔响应.同时,还讨论了非零解的稳定性,得到了非零解稳定的充要条件,而且发现在随机激励的带宽较小时,系统的多解现象仍然存在,分岔和跳跃现象仍会发生,数值模拟验证了理论推导的有效性. 关键词： 随机Mathieu-Duffing系统 多尺度 稳定性 分岔     

The critical phenomenon and the re-entrance phenomenon in the anti-tumor model induced by the time delay

The effects of time delay τ on an anti-tumor model driven by a multiplicative noise and a periodic signal are investigated. The results obtained from the small delay approximation and numerical simulations indicate: (i) For the absence of the periodic signal in the system, the two-peak structure of the stationary probability distribution transforms into the single-peak structure with the increasing τ, and τ exists a critical value τc. For τ<τc, the stationary mean value 〈x〉_st of the cell population decreases as the noise intensity D increases, however, for τ>τc, the 〈x〉_st increases as the D increases; (ii) For the presence of the periodic signal in the system, the structure of the signal-to-noise ratio with changes of the D exhibits the transitions of one peak → two peaks → one peak as τ increases.     

Economy with the time delay of information flow—The stock market case

Any decision process requires information about the past and present state of the system, but in an economy acquiring data and processing it is an expensive and time-consuming task. Therefore, the state of the system is often measured over some legal interval, analysed after the end of well defined time periods and the results announced much later before any strategic decision is envisaged. The various time delay roles have to be crucially examined. Here, a model of stock market coupled with an economy is investigated to emphasise the role of the time delay span on the information flow. It is shown that the larger the time delay the more important the collective behaviour of agents since one observes time oscillations in the absolute log-return autocorrelations.     

An approximation for prey-predator models with time delay

A prototype prey-predator (P-D) model in which the effective size of the predator population interacting with its prey follows an instantaneous time-delay τ regarding its total size is considered here. A simplified model was derived after substituting the approximation into the above time-delay model. In order to assess the reliability of the simplified model, we performed a comparative study of both models under a wide range of parameter values, focusing on the effect of τ on two issues: (i) the boundary (in parameter space) between the regions leading either to stable fixed points characteristic of the non-delay model, both the boundary and the periods and amplitudes obtained for the time-delay model can be fairly approximated by the corresponding results for the simplified model.     

基于直接延迟反馈的混沌反控制

在混沌有益时，有目的地产生混沌已经成为混沌学研究的热点问题，本文提出直接延迟反馈实现混沌的反控制，在非混沌系统中产生了混沌.该方法与间接延迟反馈控制方法相比，控制更加简单，更易于实现.该方法与Pyragas提出的延迟反馈混沌控制方法的控制器结构相同，因此，这种直接延迟反馈控制方法可以在需要混沌时产生混沌，不需要混沌时控制混沌，实现混沌控制和反控制的统一，为设计者提供最大的灵活性.针对参数处于非混沌区的Chen系统和Lorenz系统的仿真结果表明了该混沌反控制方法的有效性. 关键词： 混沌 反控制 直接延迟反馈     

分数阶时滞反馈对Duffing振子动力学特性的影响

研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.     

Stability and Neimark-Sacker bifurcation analysis of food-limited population model with time delay

In this paper, a kind of discrete delay food-limited model obtained by Euler method is investigated, where the discrete delay τ is regarded as a parameter. By analyzing the associated characteristic equation, the linear stability of this model is studied. It is shown that Neimark-Sacker bifurcation occurs when τ crosses some critical values. The explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form. Finally, numerical simulations are performed to verify the analytical results.     

双光反馈半导体激光混沌系统中外腔延时反馈特征的抑制

在一个双光反馈半导体激光混沌系统中,固定其中一个反馈腔镜(固定腔M₁)的反馈延迟时间以及反馈强度,实验研究了另一个反馈腔镜(可变腔M₂)的反馈延迟时间以及反馈强度对系统混沌输出的延时反馈特征的影响.研究结果表明:在可变腔M₂与固定腔M₁的反馈强度相等的条件下,改变M₂的腔长,当其腔长约等于(但不能严格等于)M₁的腔长或者腔长的一半时,系统将呈现较好地抑制延时反馈特征效果;当M 关键词： 半导体激光器 双光反馈 混沌输出 延时特征     

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.     

Stability and bifurcation of delayed bidirectional gene regulatory networks with negative feedback loops

This paper studies the stability and bifurcation of three cases of bidirectional gene regulatory networks with negative feedback loops and time delays. The uniqueness of the positive equilibrium point of the gene regulatory networks is verified. The delay independent stability conditions of the networks are established by analyzing their corresponding characteristic equations. The existence of the Hopf bifurcation of the networks with or without time delays is presented. Numerical simulations demonstrate the correctness of the obtained theoretical results.     

Stability analysis for coupled systems with time delay on networks

In this paper, the stability problem for some coupled systems with time delay on networks (CSDNs) is investigated. We provide a systematic method for constructing a global Lyapunov functional for CSDNs by using graph theory. The stability, uniform stability and global uniform stability of the systems are investigated. And by using the Lyapunov functional constructed, some sufficient conditions of stability are obtained.     

Stability and Neimark-Sacker bifurcation analysis of a food-limited population model with a time delay

In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the linear stability of this model is studied.It is shown that Neimark-Sacker bifurcation occurs when τ crosses certain critical values.The explicit formulae which determine the stability,direction,and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form.Finally,numerical simulations are performed to verify the analytical results.     

用于时延估计的一种相关峰内插算法

本文介绍一种基于余弦函数的相关峰内插算法，在时延估计系统中，为了减少运算量和增加观察时间，采用内插算法是必要的，因为在频率估算时应用了迭代算法，所以无论对于ｃｏｓｘ，ｓｉｘ／ｘ或实际的相关峰，都有相当高的内插精度。     

基于遗传算法的超声信号LMS自适应时延估计

为了克服ＬＭＳ自适应时延估计（ＬＭＳＴＤＥ）算法计算量大的问题,引进遗传算法进行ＬＭＳＴＤＥ的寻优规划,并采取了克服过早收敛的措施。对超声信号进行时延估计的实验表明,该方法大大减少了计算量,并有较高的时延估计精度。     

Logistic系统跃迁率的时间延迟效应

本文研究了时间延迟对Logistic系统从亚稳态跃迁到稳定态的跃迁率的影响. 假设在细胞演化过程中细胞衰减存在时间延迟,利用小延迟时间条件下一阶近似方法和最陡下降法导出了系统的跃迁率解析式. 数值计算结果表明,内噪声,外噪声和内外噪声之间的关联促使系统从亚稳态跃迁到稳定态, 而时间延迟阻碍了系统从亚稳态到稳定态的跃迁, 即时间延迟增强了系统亚稳态的稳定性. 关键词： Logistic模型 跃迁率 时间延迟 噪声