一类准周期参激非线性相对转动动力系统的稳定性与时滞反馈控制

建立了一类含准周期参数激励和时滞反馈的相对转动非线性系统的动力学方程. 采用多尺度法求解1/2亚谐波主参数共振下的分岔响应方程,并分析了系统的稳定性. 在求解非受控系统的定常解的基础上,通过讨论系统的动力学特性,研究了准周期参数激励对系统响应的影响. 采用时滞反馈控制的方法对系统分岔和极限环(域)进行控制,数值模拟的结果表明通过改变时滞参数可以实现对系统分岔的控制,并能有效地控制极限环(域)的幅值和稳定性. 关键词： 相对转动 准周期参激 时滞反馈 极限环     

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     

Bursting dynamics in parametrically driven memristive Jerk system

Compared with general nonlinear systems, multi-time scale system has complex bursting dynamics and has received widespread attention. A memristor-based Jerk system with parametric excitation is proposed in this study. As the selected excitation frequency is far less than the natural frequency, implying the existence of an order gap between the excitation frequency and the natural one, the system can be considered as a classic fast-slow system with two timescales. In our system, when the slow-varying parameters periodically pass through the critical pitchfork bifurcation point periodically, a distinct time delay behavior can be observed. Complex bursting oscillations induced by the delayed pitchfork are revealed with different excitation amplitudes. By virtue of the fast-slow analysis method, the corresponding generation mechanisms are discussed by the transformed phase portraits, the time series, and the phase portraits. As the delay time interval induced by the pitchfork bifurcation is dependant not only on the excitation amplitude, but also on the excitation frequency, some excitation frequency related bursting patterns are also considered in our study. Finally, numerical simulations are provided to verify the validity of the study.     

一类相对转动时滞非线性动力系统的稳定性分析

建立了具有时变刚度、非线性阻尼和谐波激励的一类相对转动时滞非线性动力系统的动力学方程.采用多尺度法推导出时滞动力系统的分岔响应方程,运用奇异性理论研究系统结构稳定性,得到主共振稳态响应方程的转迁集以及不同参数下分岔曲线的拓扑结构.应用Hopf分岔理论讨论了时滞动力系统动态稳定性,给出了系统产生极限环的条件,最后用数值模拟的方法研究了时滞参数对系统极限环幅值的影响。     

Routes to bursting in a periodically driven oscillator

The dynamical behaviors of a periodic excited oscillator with multiple time scales in the form that order gap exists between the frequency of the excitation and the natural frequency, are investigated in this Letter. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamics. Different types of bursting phenomena, such as fold/Hopf bursting, fold/Hopf/homoclinic bursting and Hopf/homoclinic bursting, are presented, the mechanism of which is obtained based on the bifurcations of the generalized autonomous system as well as the introduction of the so-called transformed phase portraits. Furthermore, the evolution of the bursting is discussed in details, in which one may find that when the two limit cycles caused by the Hopf bifurcations of the two related equilibrium points interact with each other, homoclinic bifurcation may occur, leading to the merge of the two cycles to form a large amplitude cycle. The homoclinic bifurcation may cause the two asymmetric bursters to merge into a symmetric enlarged burster, in which the large amplitude of the spiking state agrees well with the amplitude of the cycle caused by the homoclinic bifurcation.     

Delay control of coherence resonance in type-I excitable dynamics

We consider a simple paradigmatic system of type-I excitability subject to noise and time-delayed feedback. This system is governed by a global bifurcation, namely a saddle-node bifurcation on a limit cycle. In the absence of noise, delay can induce complex dynamics including multiple stable and unstable periodic orbits. Random fluctuations result in coherence resonance in dependence on the noise strength. We show that this effect can be enhanced by delayed feedback control with suitably chosen feedback strength and time delay.     

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.     

Stimulus-evoked activity in clustered networks of stochastic rate-based neurons

Understanding the effect of network connectivity patterns on the relation between the spontaneous and the stimulus-evoked network activity has become one of the outstanding issues in neuroscience. We address this problem by considering a clustered network of stochastic rate-based neurons influenced by external and intrinsic noise. The bifurcation analysis of an effective model of network dynamics, comprised of coupled mean-field models representing each of the clusters, is used to gain insight into the structure of metastable states characterizing the spontaneous and the induced dynamics. We show that the induced dynamics strongly depends on whether the excitation is aimed at a certain cluster or the same fraction of randomly selected units, whereby the targeted stimulation reduces macroscopic variability by biasing the network toward a particular collective state. The immediate effect of clustering on the induced dynamics is established by comparing the excitation rates of a clustered and a homogeneous random network.     

一类非自治位置时滞反馈控制系统的亚谐共振响应

研究了弹性轨道条件下，控制回路中位置反馈信号存在时滞的磁浮系统在亚谐轨道激励作用下的响应问题. 将动力学模型在平衡点处线性化，以时滞为分岔参数，得到了系统出现Hopf分岔的条件. 用中心流形约化方法得到了包含轨道扰动系统的Poincaré规范型. 用多尺度法从理论上推导了时滞磁浮系统的亚谐共振周期解，得到了自由振动的分岔响应方程，分析了周期解中自由振动项的存在条件，研究了控制参数和激励参数与周期解的关系. 最后用数值仿真的方法分析了时滞参数、控制参数对系统响应的影响，分析结果指出，使系统保持稳定的亚谐响应的时滞边界小于无扰动时的时滞边界，时滞参数不但可以抑制亚谐响应，还能够控制混沌的产生，而控制参数可以控制系统响应中自由振动项的出现和受迫振动的幅值，适当选择这些参数可以有效抑制亚谐振动响应. 关键词： 亚谐共振响应 位置时滞反馈控制 非自治磁浮系统 分岔     

Novel design for bifurcation control in a delayed fractional dual congestion model

This letter puts forward an ingenious feedback control method with parametric delay to manipulate bifurcation control for a delayed fractional dual congestion model. By employing time delay as a bifurcation parameter, the local dynamics involving stability and Hopf bifurcation is examined. The control efforts can be realized with or without time delay in the strength of feedback control. It suggests that the stability performance is consumedly elevated by exploiting the parametric delay feedback controller, yet Hopf bifurcation engenders ahead of time in the event of the absence of the controller. Moreover, the impact of the order or linear feedback gain on the bifurcation point is numerically discussed via aborative calculation. Numerical simulations are eventually actualized to corroborate the proposed scheme.     

时滞与噪声影响 Notch信号通路动力学的模型化研究

本文基于Hill动力学与Michaelis-Menten方程,建立理论模型研究时滞与噪声影响Notch信号通路动力学.研究发现,her1、her7基因转录的时滞性在很大程度上调控着Notch信号通路的动力学行为.由于时滞性的调控,Notch系统动力学经历Hopf分岔,由稳态转变为周期演化特性.通过考察Notch信号通路的噪声效应,我们发现,由于噪声的扰动,Notch系统周期振荡动力学改变.在较小噪声幅值条件下,Notch信号通路中改变的周期节律性可以通过时滞得以平衡恢复,由此表明了her1、her7转录的时滞性促进了Notch信号通路的周期振荡.对于较强噪声环境,时滞效应很难改变Notch信号的巨大突变,其信号通路动力学行为被噪声影响.理论结果符合实验,并揭示了时滞与噪声对Notch信号通路动力学的一种调控机制,可为设计阻止Notch功能异常导致的多种疾病和癌症的通路治疗方案提供理论依据.     

Bursting oscillations as well as the bifurcation mechanism in a non-smooth chaotic geomagnetic field model

Based on the chaotic geomagnetic field model, a non-smooth factor is introduced to explore complex dynamical behaviors of a system with multiple time scales. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamic behaviors. Due to the existence of non-smooth factors, different types of bifurcations are presented in spiking states, such as grazing-sliding bifurcation and across-sliding bifurcation. In addition, the non-smooth fold bifurcation may lead to the appearance of a special quiescent state in the interface as well as a non-smooth homoclinic bifurcation phenomenon. Due to these bifurcation behaviors, a special transition between spiking and quiescent state can also occur.     

兴奋性作用诱发神经簇放电个数不增反降的分岔机制

非线性动力学在识别神经放电的复杂现象、机制和功能方面发挥了重要作用.不同于传统观念,本文提出了兴奋性作用可以降低而不是增加簇内放电个数的新观点.在簇放电模式休止期的适合相位施加强度合适的脉冲或自突触电流,能诱发簇内放电个数降低;电流的施加相位越早,所需的强度阈值越大,簇内放电个数越少.进一步,利用快慢变量分离获得的簇放电的动力学性质进行了理论解释.簇放电模式表现出低电位的休止期和高电位的放电的交替,存在于快子系统的鞍结分岔点和同宿轨分岔点之间;放电起始于鞍结分岔、结束于同宿轨分岔;越靠近同宿轨分岔从休止期跨越到放电所需的电流强度越大.因此,电流在休止期上的作用相位越早,就越靠近同宿轨分岔,因而从休止期跨越到放电需要的电流强度阈值越大,放电起始相位到同宿轨分岔之间的区间变小导致放电个数变少.研究结果丰富了非线性现象及机制,对兴奋性作用提出了新看法,给出了调控簇放电模式的新途径.     

Experimental unfolding of the nonlinear dynamics of a cable-mass suspended system around a divergence-Hopf bifurcation

Systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable-mass, subjected to anti-phase support motion at primary resonance, is accomplished. Upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics, an improved experimental apparatus is used to make it technically accessible. Results obtained by varying three control parameters, namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system, allow us to characterize in-depth various occurring classes of motion in terms of time and spatial complexity, to describe peculiar and/or persistent features of transition to nonregular dynamics, and to trace them back to a canonical scenario from bifurcation theory. Variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results, and overall behaviour charts are built in the excitation parameter space. Considering the temperature as a controllable parameter shows to be fundamental for: (i) indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent, (ii) qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence-Hopf bifurcation normal form, and (iii) determining system response not only in the strict neighbourhood of the organizing divergence-Hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low-dimensional homoclinic chaos.     

计算相位响应曲线的方波扰动直接算法

通过相位响应曲线可对具有极限环周期运动的动力系统的性质有更为深入的理解.神经元是一个典型的动力系统,因此相位响应曲线提供了一种研究神经元重复周期放电行为的新思路.本文提出一种求解相位响应曲线的方法,即方波扰动的直接算法,通过Hodgkin-Huxley,Fitz Hugh-Nagumo,Morris-Lecar和Hindmarsh-Rose神经元模型验证该算法可计算周期峰放电、周期簇放电的相位响应曲线.该算法克服了其他算法在运用过程中的局限性.利用该算法计算结果表明:周期峰放电的相位响应曲线类型是由其分岔类型所决定;在Morris-Lecar模型中发现一种开始于Hopf分岔终止于鞍点同宿轨道分岔的阈上周期振荡,其相位响应曲线属于第二类型.通过大量的相位响应曲线的计算发现相位响应的相对大小及正负性仅取决于扰动所施加的时间,而且周期簇放电的相位响应曲线比周期峰放电的相位响应曲线更为复杂.     

一类时滞非线性相对转动系统的Hopf分岔与周期解的稳定性

讨论了一类时滞非线性相对转动系统的Hopf分岔现象,给出了系统产生Hopf分岔的临界时滞条件,利用多尺度法研究了系统在主共振情况下时滞参数对分岔方向和周期解稳定性的影响,最后,用数值模拟对所得结论进行验证.     

Nonlinear analysis of a maglev system with time-delayed feedback control

This paper undertakes a nonlinear analysis of a model for a maglev system with time-delayed feedback. Using linear analysis, we determine constraints on the feedback control gains and the time delay which ensure stability of the maglev system. We then show that a Hopf bifurcation occurs at the linear stability boundary. To gain insight into the periodic motion which arises from the Hopf bifurcation, we use the method of multiple scales on the nonlinear model. This analysis shows that for practical operating ranges, the maglev system undergoes both subcritical and supercritical bifurcations, which give rise to unstable and stable limit cycles respectively. Numerical simulations confirm the theoretical results and indicate that unstable limit cycles may coexist with the stable equilibrium state. This means that large enough perturbations may cause instability in the system even if the feedback gains are such that the linear theory predicts that the equilibrium state is stable.     

Response of a strongly non-linear oscillator to narrowband random excitations

The principal resonance of a van der Pol-Duffing oscillator subject to narrowband random excitations has been studied. By introducing a new expansion parameter the method of multiple scales is adapted for the strongly non-linear system. The behavior of steady state responses, together with their stability, and the effects of system damping and the detuning, and magnitude of the random excitation on steady state responses are analyzed in detail. Theoretical analyses are verified by some numerical results. It is found that when the random noise intensity increases, the steady state solution may change form a limit cycle to a diffused limit cycle, and the system may have two different stable steady state solutions for the same excitation under certain conditions. The results obtained for the strongly non-linear oscillator complement previous results in the literature for weakly non-linear systems.     

Complex behavior of simple maps with fluctuating delay times

Delay systems used to model retarded actions are relevant in many fields such as optics, mechanical machining, biology or physiology. A frequently encountered situation is that the length of the delay time changes with time. In this study, we use a simple map system to investigate the influence of the fluctuating delay time on the system dynamics. For simplicity, we start from a case with the delay time taking only the value of zero or one discrete time steps, where the system dynamics reduces to one- and two-dimensional map, respectively. We study two situations, periodic or random variation of the delay. Rich dynamics including coexisting multiple attractors, strange nonchaotic attractors and on-off intermittency are observed. For a special case we can show analytically the existence of a dense set of singularities of the Lyapunov exponent. Finally we present results for higher dimensional generalizations to show the relevance of our findings to more general situations.     

Oscillatory Activities in Regulatory Biological Networks and Hopf Bifurcation

Exploiting the nonlinear dynamics in the negative feedback loop, we propose a statistical signal-response model to describe the different oscillatory behaviour in a biological network motif. By choosing the delay as a bifurcation parameter, we discuss the existence of Hopf bifurcation and the stability of the periodic solutions of model equations with the centre manifold theorem and the normal form theory. It is shown that a periodic solution is born in a Hopf bifurcation beyond a critical time delay, and thus the bifurcation phenomenon may be important to elucidate the mechanism of oscillatory activities in regulatory biological networks.