三时间尺度下非光滑电路中的簇发振荡及机理

实际工程应用中存在着诸如冲击、干摩擦、切换等非光滑因素,以此建立的动力学模型是包含非光滑项的系统. 目前针对非光滑动力系统的研究大多基于单一尺度或者两尺度, 而含有更多尺度的非光滑动力系统可能会存在更复杂的动力学现象. 本论文旨在探讨非光滑动力系统中的多尺度效应及其分岔机制.基于典型的非光滑蔡氏电路, 引入一个与系统固有频率存在量级差的周期变化的激励项, 同时通过选取适当的参数值,建立了一个三时间尺度耦合下的、含有两个分界面的四维分段线性电路系统模型, 研究了该系统存在的簇发振荡行为及其分岔机制. 首先,将对应快尺度与中间尺度的变量合并作为快变量, 将对应慢尺度的变量看作慢变量, 重新划分了快慢子系统,从而将三时间尺度耦合问题转化为两时间尺度耦合问题去分析. 然后根据双参数下的Hopf分岔情况, 对应于慢子流形的不同稳定性,给出了不同参数下系统存在的两种典型的簇发振荡行为. 最后, 基于快慢分析法, 结合转换相图以及慢子流形在非光滑分界面上的非光滑动力学行为的详细讨论, 分析了不同簇发振荡相互转化的分岔机制, 发现了一个新的簇发振荡的演化路径, 即由破坏性的擦边分岔诱导的簇发振荡.      

串联式叉型滞后簇发振荡及其动力学机制

簇发振荡是自然界和科学技术中广泛存在的快慢动力学现象,其具有与通常的振荡显著不同的特性.根据不同的动力学机制可将其分为多种模式,例如,点-点型簇发振荡和点-环型簇发振荡等.叉型滞后簇发振荡是由延迟叉型分岔诱发的一类具有简单动力学特性的点-点型簇发振荡.研究以多频参数激励Duffing系统为例,旨在揭示一类与延迟叉型分岔相关的具有复杂动力学特性的簇发振荡,即串联式叉型滞后簇发振荡.考虑了一个参激频率是另一个的整倍数情形,利用频率转换快慢分析法得到了多频参数激励Duffing系统的快子系统和慢变量,分析了快子系统的分岔行为.研究结果表明,快子系统可以产生两个甚至多个叉型分岔点;当慢变量穿越这些叉型分岔点时,形成了两个或多个叉型滞后簇发振荡;这些簇发振荡首尾相接,最终构成了所谓的串联式叉型滞后簇发振荡.此外,分析了参数对串联式叉型滞后簇发振荡的影响.      

一类三维非线性系统的复杂簇发振荡行为及其机理

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.      

COMPLICATED BURSTING BEHAVIORS AS WELL AS THE MECHANISM OF A THREE DIMENSIONAL NONLINEAR SYSTEM 1)

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.     

Bursting generation mechanism in a three-dimensional autonomous system,chaos control,and synchronization in its fractional-order form

The bifurcation mechanism of bursting oscillations in a three-dimensional autonomous slow-fast Kingni et al. system (Nonlinear Dyn. 73, 1111–1123, 2013) and its fractional-order form are investigated in this paper. The stability analysis of the system is carried out assuming that the slow subsystem evolves on quasi-static state. It is reveaved that the bursting oscillations found in the system result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. We refer this class of bursting to “source/bursting.” The coexistence of symmetrical bursting limit cycles and chaotic bursting attractors is observed. In addition, the fractional-order chaotic slow-fast system is studied. The lowest order of the commensurate form of this system to exhibit chaotic behavior is found to be 2.199. By tuning the commensurate fractional-order, the chaotic slow-fast system displays Chen- and Lorenz-like chaotic attractors, respectively. The stability analysis of the controlled fractional-order-form of the system to its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Moreover, the synchronization of chaotic bursting oscillations in two identical fractional-order systems is numerically studied using the unidirectional linear error feedback coupling scheme. It is shown that the system can achieve synchronization for appropriate coupling strength. Furthermore, the effect of fractional derivatives orders on chaos control and synchronization is analyzed.     

一类混沌系统中的簇发振荡及其延迟叉形分岔行为

由于多时间尺度问题在实际工程系统中广泛存在,关于其复杂动力学行为及其产生机制的研究已成为当前国内外的热点课题之一.簇发振荡是多时间尺度系统复杂动力学行为的典型代表,而分岔延迟又是簇发振荡中的常见现象.本文为探讨非线性系统中分岔延迟所引发的簇发振荡的分岔机制,在一个三维混沌系统中引入参数激励,当激励频率远小于系统的固有频率时,系统产生了两时间尺度簇发振荡.将整个激励项看做慢变参数,激励系统转化为广义自治系统也即快子系统,分析快子系统平衡点的稳定性以及分岔条件,并运用快慢分析法和转换相图揭示了簇发振荡的动力学机理.文中考察了4组参数条件下系统的动力学行为,研究发现当慢变激励项周期性地通过分岔点时,系统产生了明显的超临界叉形分岔延迟行为,随着参数激励振幅的增大,分岔延迟的时间也逐渐延长,当这种延迟的动态行为终止于不同的参数区域时,导致系统轨线围绕不同稳定吸引子(平衡点,极限环)运动,从而得到了不同的簇发振荡行为.      

电磁场下神经元模型中混合簇的分岔机制

Pre-B?tzinger复合体是新生哺乳动物呼吸节律起源的关键部位, 是呼吸节律产生的中枢. 忆阻器的功能类似于神经元突触的可塑性, 可用其模拟磁通量.本文在Butera动力学模型的基础上引入刺激电流和磁通控制忆阻器, 分别研究这两个因素对单个pre-B?tzinger复合体神经元中混合簇放电模式的影响.通过无量纲化的方法对变量进行时间尺度分析, 结果表明, 模型包含3个不同的时间尺度.通过快慢分解和分岔分析研究了神经元混合簇放电产生和转迁的动力学机制.电流和磁通量都可以影响混合簇中胞体簇的个数, 减小电流和磁通量的值, 混合簇中胞体簇的个数也会相应减少, 并使簇的类型由"fold/homoclinic"型簇放电转迁为经由"fold/homoclinic"滞后环的"Hopf/Hopf"型簇放电.双参数分岔分析表明, 随着钙离子浓度的逐渐增加, 全系统轨线在鞍结分岔曲线和同宿轨分岔曲线之间来回跃迁, 是混合簇的产生分岔机制.全系统轨线在鞍结分岔曲线和同宿轨分岔曲线之间跃迁的次数, 与混合簇中胞体簇的个数相对应.      

参外联合激励下的簇发振荡及其机理

当周期激励频率远小于系统固有频率时，会存在快慢耦合效应，与单项激励不同，参外联合激励不仅会导致快子系统平衡曲线和分岔行为的复杂化，也会产生一些特殊的非线性现象，为此，本文以两耦合Hodgkin-Huxley细胞模型为例，引入周期参外联合激励，探讨在频域不同尺度耦合时该系统的簇发振荡的特点及其分岔机制．通过建立相应的快慢子系统，得到慢变参数变化下的快子系统的各种分岔模式以及相应的分岔行为，结合转换相图，揭示耦合系统随激励幅值变化时的动力学行为及其机理．研究表明，在激励幅值较小时，系统表现为概周期振荡，两频率分别近似于快子系统平衡曲线由Hopf分岔引起的两稳定极限环的振荡频率．概周期解随激励幅值的增加进入簇发振荡，导致这些簇发振荡的主要原因是在慢变参数变化的部分区间内，存在唯一稳定的平衡曲线，使得系统的轨迹逐渐趋向该平衡曲线，产生沉寂态，并随着慢变参数的变化，由分岔进入激发态．同时，快子系统中参与簇发振荡的稳定吸引子随激励幅值的变化也会不同，导致不同形式的簇发振荡．另外，与单项激励下的情形不同，联合激励时快子系统的部分稳定吸引子掩埋在其它稳定吸引子内，从而失去对簇发振荡的影响．     

Bursting phenomenon in a piecewise mechanical system with parameter perturbation in stiffness

In this paper the existence condition and generation mechanism of the possible bursting phenomenon in a piecewise mechanical system with different time scales are studied. As an example of mechanical systems, a piecewise linear oscillator with parameter perturbation in stiffness and subject to external excitation is examined. The order gaps between the time scales are considered in the model, which are related to the periodic excitation and the changing rates of the variables. The focus-type periodic bursting oscillation with two time scales is presented, and the corresponding generation mechanism is revealed by using slow–fast analysis method. Furthermore, the analytical solution of piecewise linear subsystem as well as the stability condition of the fast subsystem are explored to explain the transition of bursting behaviors coming from the variation of intrinsic parameter and external excitation. The results about bursting phenomenon and its generation mechanism would provide important theoretical basis on the mechanical manufacturing and engineering practice.     

3-torus,quasi-periodic bursting,symmetric subHopf/fold-cycle bursting,subHopf/fold-cycle bursting and their relation

In this paper, the dynamical behaviors of a perturbed hyperchaotic system is studied. The fast subsystem is examined using local stability and bifurcations, including simple bifurcation, Hopf bifurcation, and fold bifurcation of limit cycle. The results of these analysis are applied to the perturbed hyperchaotic system, where two types of periodic bursting, i.e., symmetric subHopf/fold-cycle bursting and subHopf/fold-cycle bursting, can be observed. In particular, the symmetric subHopf/fold-cycle bursting is new and has not been reported in previous work. With variation of the parameter, subHopf/fold-cycle bursting with symmetric structure may bifurcate into two coexisted subHopf/fold-cycle bursting symmetric to each other. Moreover, 3-torus and quasi-periodic bursting (2-torus) are presented. The relation among 3-torus, quasi-periodic bursting, and symmetric subHopf/fold-cycle bursting is discussed, which suggests that 3-torus may develop to quasi-periodic bursting, while quasi-periodic bursting may further evolve to symmetric subHopf/fold-cycle bursting.     

Canard Explosion in Delay Differential Equations

We analyze canard explosions in delay differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow–fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of ‘classical’ canard explosions ends as a Bogdanov–Takens bifurcation occurs at the folds points of the S-shaped critical manifold.     

Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity

This paper aims at offering an insight into the dynamical behaviors of incommensurate fractional-order singularly perturbed van der Pol oscillators subjected to constant forcing, especially when the forcing is close to Andronov–Hopf bifurcation points. These bifurcation points are predicted thanks to the theorem on stability of incommensurate fractional-order systems, as functions of the forcing and fractional derivative orders. When the forcing is chosen near Andronov–Hopf bifurcation, the dynamics of fractional-order systems show a static-looking transient regime whose length increases exponentially with the closeness to the bifurcation point. This peculiar phenomenon is not common in numerical simulation of dynamical systems. We show that this quasi-static transient behavior is due to the combine action of the slow passage effect at folded saddle-node singularity and fractional derivation memory effect on the slow flow around this singularity; this forces the system to remain for a long time in the vicinity of its equilibrium point, though unstable. The system frees oneself from this quasi-static transient state by spiraling before entering relaxation oscillation. Such a situation results in mixed mode oscillations in the oscillatory regime. One obtains mixed mode oscillations from a very simple system: A two-variable system subjected to constant forcing.     

Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models

A series of bifurcations from period-1 bursting to period-1 spiking in a complex (or simple) process were observed with increasing extra-cellular potassium concentration during biological experiments on different neural pacemakers. This complex process is composed of three parts: period-adding sequences of burstings, chaotic bursting to chaotic spiking, and an inverse period-doubling bifurcation of spiking patterns. Six cases of bifurcations with complex processes distinguished by period-adding sequences with stochastic or chaotic burstings that can reach different bursting patterns, and three cases of bifurcations with simple processes, without the transition from chaotic bursting to chaotic spiking, were identified. It reveals the structures closely matching those simulated in a two-dimensional parameter space of the Hindmarsh–Rose model, by increasing one parameter \(I\) and fixing another parameter \(r\) at different values. The experimental bifurcations also resembled those simulated in a physiologically based model, the Chay model. The experimental observations not only reveal the nonlinear dynamics of the firing patterns of neural pacemakers but also provide experimental evidence of the existence of bifurcations from bursting to spiking simulated in the theoretical models.     

快慢耦合振子的张驰簇发及其非光滑分岔机制

通过引入适当的参数值, 得到了两时间尺度下的快慢耦合振子, 分析了耦合系统及子系统的平衡点及其性质, 进而利用微分包含理论, 探讨了非光滑分界面上的奇异性, 指出在适当的参数条件下, 系统轨迹在穿越分界面时会产生由Hopf分岔和Fold分岔组合的非常规分岔. 给出了不同参数条件下的周期簇发行为, 分析了簇发过程的振荡特性, 指出激发态的频率取决于快子系统在非光滑分界面上的Hopf分岔频率, 而慢子系统的固有频率影响了簇发行为的振荡周期, 并进一步揭示了由非光滑分岔引起的不同周期簇发的分岔机制.     

由多平衡态快子系统所诱发的簇发振荡及机理

通过引入子电路模块, 并选取适当的参数及非线性电阻特性, 建立了多时间尺度下具有多平衡态的四维广义哈特利(Hartley) 电路模型. 基于快子系统的多平衡态及其稳定性, 给出了参数空间的分岔集, 得到了不同区域中的动力学特性及其相应的分岔模式和临界条件. 针对两种典型具有不同分岔特征的情形, 分别给出了多平衡态参与下的两种不同的周期簇发振荡行为, 结合快子系统的分岔分析, 揭示了沉寂态和激发态之间相互转化的产生机制, 指出多平衡态不仅会导致多种沉寂态和激发态同时参与同一周期簇发振荡, 也会导致簇发振荡模式的多样性.      

On occurrence of mixed-torus bursting oscillations induced by non-smoothness

Nonlinear Dynamics - The paper devotes to the slow–fast behaviors of a higher-dimensional non-smooth system with the coupling of two scales. Some novel bursting attractors and interesting...     

双频1:2激励下修正蔡氏振子两尺度耦合行为

不同尺度耦合系统存在的复杂振荡及其分岔机理一直是当前国内外研究的热点课题之一. 目前相关工作大都是针对单频周期激励频域两尺度系统,而对于含有两个或两个以上周期激励系统尺度效应的研究则相对较少. 为深入揭示多频激励系统的不同尺度效应,本文以修正的四维蔡氏电路为例,通过引入两个频率不同的周期电流源,建立了双频1:2周期激励两尺度动力学模型. 当两激励频率之间存在严格共振关系,且周期激励频率远小于系统的固有频率时,可以将两周期激励项转换为单一周期激励项的函数形式. 将该单一周期激励项视为慢变参数,给出了不同激励幅值下快子系统随慢变参数变化的平衡曲线及其分岔行为的演化过程,重点考察了3种较为典型的不同外激励幅值下系统的簇发振荡行为. 结合转换相图,揭示了各种簇发振荡的产生机理. 系统的轨线会随慢变参数的变化,沿相应的稳定平衡曲线运动,而fold分岔会导致轨迹在不同稳定平衡曲线上的跳跃,产生相应的激发态. 激发态可以用从分岔点向相应稳定平衡曲线的暂态过程来近似,其振荡幅值的变化和振荡频率也可用相应平衡点特征值的实部和虚部来描述,并进一步指出随着外激励幅值的改变,导致系统参与簇发振荡的平衡曲线分岔点越多,其相应簇发振荡吸引子的结构也越复杂.      

Complex dynamics of compound bursting with burst episode composed of different bursts

Episodic or compound bursting arises from a transition between a burst episode composed of a long burst and several short bursts and a relatively long subthreshold oscillation in this work. The minimal and generic phantom bursting model proposed by Bertram et al. is employed to produce compound bursting of a single pancreatic ??-cell and compound bursting synchronization with antiphase spikes of two electrical coupling pancreatic ??-cells. Two different fast/slow analysis for the moderate and the slower slow variables in three-dimensional spaces are combined to highlight better how these two slow variables with different time scales commonly or separately result in complex dynamic of the compound bursting of both the single ??-cell and the two electrical coupling ??-cells. For the compound bursting synchronization with antiphase spikes, we reveal how varying coupling strength leads to a change of the number of short bursts within the burst episode for different types of compound bursting.     

Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.