力学学报, 2021, 53(3): 855-864 DOI: 10.6052/0459-1879-20-331

动力学与控制

三时间尺度下非光滑电路中的簇发振荡及机理1)

毛卫红, 张正娣,2), 张苏珍

江苏大学数学科学学院, 江苏镇江 212013

BURSTING OSCILLATIONS AND ITS MECHANISM IN A NONSMOOTH SYSTEM WITH THREE TIME SCALES1)

Mao Weihong, Zhang Zhengdi,2), Zhang Suzhen

Faculty of Mathematical Sciences$,$ Jiangsu University$,$ Zhenjiang $212013$$,$ Jiangsu$,$ China

通讯作者: 2) 张正娣, 教授, 主要研究方向: 非线性动力学. E-mail:dyzhang@ujs.edu.cn

收稿日期: 2020-09-16   网络出版日期: 2021-03-18

基金资助: 1) 国家自然科学基金资助项目.  11872189

Received: 2020-09-16   Online: 2021-03-18

作者简介 About authors

摘要

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.

关键词: 非光滑系统 ; 三时间尺度 ; 非光滑分岔 ; 簇发振荡

Abstract

Dynamic models established from practical engineering application are non-smooth systems owing to non-smooth factors, such as impact, dry friction and switching, etc. Up to now, most studies are in terms of the non-smooth dynamic systems with a single scale or two scales. While more complex dynamic phenomena may be observed in the non-smooth dynamic systems with more scales. The main purpose of this work is to explore multiscale effect in a non-smooth electric system and the related bifurcation mechanism. Upon the traditional Chua's circuit, by introducing a periodically excited oscillator with an order gap from the natural frequency of the system and taking suitable parameter values, a coupled 4-dimensional piecewise linear dynamic system with three time-scales and two boundaries is established to study the bursting oscillations as well as the corresponding bifurcation mechanism under three time-scales. Merging the variables corresponding to the fast scale and the variables related to the intermediate scale into the fast variables, while regarding the variable corresponding to the slow scale as the slow variable, the coupled problem with three time-scales is transformed into that with two time-scales. According to the relevant Hopf bifurcation curve under two independent parameters and the stability analyses of the slow submanifold of the fast subsystem, two different bursting oscillations of the coupled dynamic system are given in the case of two different parameter values. On the basis of the fast-slow analysis method, the transformed phase portrait and the non-smooth dynamics of the slow submanifold occurring on the non-smooth boundaries, the bifurcation mechanism of the mutual transformation of different bursting oscillations is analyzed in details, in which some helpful numerical simulations are given to illustrate the validity of our study simultaneously. At the same time, a new evolution of bursting oscillations is found, i.e., the bursting oscillation induced by destructive grazing bifurcation.

Keywords: non-smooth system ; three time-scales ; non-smooth bifurcation ; bursting oscillations

PDF (1350KB) 元数据 多维度评价 相关文章 导出 EndNote| Ris| Bibtex  收藏本文

本文引用格式

毛卫红, 张正娣, 张苏珍. 三时间尺度下非光滑电路中的簇发振荡及机理1). 力学学报[J], 2021, 53(3): 855-864 DOI:10.6052/0459-1879-20-331

Mao Weihong, Zhang Zhengdi, Zhang Suzhen. BURSTING OSCILLATIONS AND ITS MECHANISM IN A NONSMOOTH SYSTEM WITH THREE TIME SCALES1). Chinese Journal of Theoretical and Applied Mechanics[J], 2021, 53(3): 855-864 DOI:10.6052/0459-1879-20-331

引言

多时间尺度问题[1-2]作为非线性科学的重要组成部分广泛存在于物理学、化学、生物学等领域中.例如在生物细胞的遗传中, 由较快的新陈代谢过程与相对较慢的遗传变化过程的结合导致的快慢耦合[3];在催化反应中, 不同量级反应速率之间的催化与自催化过程[4]; 在神经动力学中, 静息态和激发态交替出现的簇放电模式[5]等. 相比于单一时间尺度的系统, 多时间尺度耦合系统会表现出更为特殊的动力学行为,例如周期振荡中通常表现出的大幅振荡和微幅振荡的复合振荡[6-7]. 大幅振荡和微幅振荡可以分别看作快慢系统的激发态(spikingstate)和沉寂态(quiescent state). 这种连接快慢两个过程的行为也称为簇发(bursting)[8-10], 通常是由分岔产生的[11-12], 可以得到“点点”型簇发振荡、“点环”型簇发振荡等. 这类具有特殊结构的系统往往表现为复杂的动力学行为, 例如混沌簇发振荡[13],吸引着国内外学者的关注, 尤其是在Izhikevich等[14-16]引入Rinzel的快慢分析理论后,对于含多时间尺度的非线性动力系统, 许多学者从数值仿真、理论分析、数值实验等方面进行了深入的探索[17-20].

非光滑动力系统具有广泛的工程背景, 涉及到科学和工程技术的各个领域. 不同于光滑系统的分岔现象, 非光滑系统不仅可以产生光滑系统中的各种常规分岔, 而且还可能发生一些光滑系统所不具备的特有分岔, 统称为非光滑分岔[21].例如, 非光滑滑动分岔、擦边分岔、 角点碰撞分岔等[22-25], 导致系统产生复杂的振荡行为, 对系统的动力学行为产生重要的影响,从而提供了更多通向混沌的路径, 并且其研究方法与光滑系统也不尽相同[26-27], 是非光滑动力学研究的热点和难点.

目前, 对含多时间尺度非线性动力学行为的研究主要针对只含有一个慢变过程的快慢两尺度系统[28-29], 对于含有两个或两个以上慢变过程的、含3个及以上时间尺度的非光滑系统, 关于其沉寂态与激发态相互转迁而导致簇发振荡的非光滑分岔模式的研究相对较少, 而根据实际问题建立的动力系统通常含有不止两尺度.

本文工作如下: 基于典型的非光滑蔡氏电路, 通过选取适当的参数值, 建立了一个含三时间尺度的四维分段线性电路系统模型. 通过重新划分系统的快慢子系统, 将三时间尺度耦合问题转化为两时间尺度耦合问题. 分析了快子系统的动力学行为, 给出了不同参数下系统存在的两种典型的簇发振荡行为. 结合转换相图, 利用快慢分析法研究了系统的簇发振荡现象以及三尺度之间的相互作用机制, 探究了非常规分岔在多时间尺度效应下对系统簇发振荡行为的影响机理.

1 数学模型

蔡氏电路作为一种简单的非线性电子电路, 其制作的容易程度使它成为当前众多混沌电路中最具代表性的一种,成为许多研究的对象, 并广泛被人们在文献中引用[30-31]. 基于广义蔡氏电路模型, 本文建立了一个电路系统(图1),其中包含两个电感$L_{1}$和$L_{2}$, 两个电容$C_{1}$和$C_{2}$以及一个分段线性的非线性电阻$R_{G}$,同时并联一个周期变化的电流源$i_{G}$, 其相应的动力学模型可以表示为

图1

图1   电路原理图

Fig. 1   Schematic circuit diagram


$\left. {\begin{array}{l} {d}i_{L_{1} } /{d}t=(V_{C_{1} } -i_{L_{1} } R_{1} )/L_{1} \\ {d}i_{L_{2} } /{d}t=(V_{C_{2} } -V_{C_{1} } -i_{L_{2} } R_{2} )/L_{2} \\ {d}V_{C_{1} } /{d}t=(i_{L_{1} } +i_{L_{2} } )/C_{1} +i_{G} \\ {d}V_{C_{2} } /{d}t=(i_{L_{2} } -G(V_{C_{2} } ))/C_{2} \\ \end{array}} \right\}$

其中, 非线性电阻的伏安特性为$G(V_{C_{2} } )=P_{2} V_{C_{2} } +0.5(P_{1} -P_{2} )(\vert V_{C_{2} } +E_{0}\vert -\vert V_{C_{2} } -E_{0} \vert )$,周期变化的电流源特性为$i_{G} =I_{G} \sin (\omega t)$. 引入变换$t=R_{2} C_{1} \tau $, $x=R_{2} i_{L_{1} } $,$y=R_{2} i_{L_{2} } $, $u=V_{C_{1} } $, $v=V_{C_{2} } $, 则式(1)可表示为如下无量纲形式

$\left. {\begin{array}{l} \dot{{x}}=\alpha (-\beta x+u) \\ \dot{{y}}=\gamma (-y-u+v) \\ \dot{{u}}=x+y+A\sin (\varOmega \tau ) \\ \dot{{v}}=\delta (y-G(v)) \\ \end{array}} \right\}$

其中, $\alpha =C_{1}R_2^2/L_{1}$, $\beta=R_{1}/R_{2}$, $\gamma =C_{1}R_2^2/L_{2}$, $\delta =C_{1}/C_{2}$, $a=R_{2}C_{1}P_{1}/C_{2}$, $b=R_{2}C_{1}P_{2}/C_{2}$, $A=C_{1} R_{2}I_{G}$, $\varOmega =C_{1} R_{2}\omega $,$G(v)=bv+0.5(a-b)(|v+1|-|v-1|)$.

固定参数量级为$\alpha \equiv O(10^{-4})$, $\varOmega \equiv O(10^{-2})$. 由于系统的固有频率$\varOmega_{N}$ (可由在$\alpha=0$且$A=0$时系统(2)的平衡点特征值的虚部近似估计)的量级一般为1, 因此, 此时系统中存在明显的3个尺度,即$T_{1}\equiv O(1)$, $T_{2}\equiv O(10^{-2})$, $T_{3}\equiv O(10^{-4})$.如果将外激励项$A\sin(\varOmega\tau $)视为参数$w$, 则系统(2)分别对应着3个子系统, 表示为

$\begin{array}{l}\dot{y}=\gamma(-y-u+v) \\\dot{u}=x+y+w, \quad \text { fast scale } \\\dot{v}=\delta (y-G(v))\end{array}$
$w=A \sin (\Omega \tau), \text { middle scale }$
$\dot{x}=\alpha(-\beta x+u), \text { slow scale }$

将中间尺度变量所对应的子系统(4)与快尺度变量对应的子系统(3)合并为一个非自治系统,表示为

$\left. {\begin{array}{l} \dot{{y}}=\gamma (-y-u+v) \\ \dot{{u}}=x+y+A\sin (\varOmega \tau ) \\ \dot{{v}}=\delta (y-G(v)) \\ \end{array}} \right\}$

于是将原来的三时间尺度耦合系统(2)重新划分为快子系统(6)与慢子系统(5),从而可以将三时间尺度耦合问题(3)$\sim$(5)转化为两时间尺度耦合问题(5)与(6)来分析讨论.

2 快子系统稳定性分析

由非线性电阻$R_{G}$的分段线性的特性, 快子系统(6)中存在两个非光滑分界面$\sum_{\pm 1}: =H_{\pm }(v)=0$, 其中, $H_{\pm}(v)=v-(\pm 1)$, 从而其状态空间被$\Sigma_{\pm 1}$划分为3个不同的光滑区域, 表示为

$\begin{eqnarray*} D_{1}:=\{ (y, u, v)| v> 1\}\\ D_{2}:=\{ (y, u, v)| -1< v< 1\}\\ D_{3}:=\{ (y, u, v)| v < -1\} \end{eqnarray*}$

分别对应3个非自治的光滑子系统

$\left.\begin{array}{l}\dot{y}=\gamma(-y-u+v) \\\dot{u}=x+y+A \sin (\Omega \tau), \quad v>1 \\\dot{v}=\delta [y-a-b(v-1)]\end{array}\right\}$
$\left.\begin{array}{l}\dot{y}=\gamma(-y-u+v) \\\dot{u}=x+y+A \sin (\Omega \tau), \quad-1<v<1 \\\dot{v}=\delta [y-a v]\end{array}\right\}$
$\left.\begin{array}{l}\dot{y}=\gamma(-y-u+v) \\\dot{u}=x+y+A \sin (\Omega \tau), \quad v<-1 \\\dot{v}=\delta [y+a-b(v+1)]\end{array}\right\}$

其向量场依次记为$F_{1}$, $F_{2}$, $F_{3}$. 快慢耦合系统(2) 的慢子流形, 即系统(6)的平衡态, 这里称之为名义平衡轨道(nominal equilibrium orbits), 记为NEO, 其解析形式可设为

$\left. {\begin{array}{l} y=Y_{0} +A_{1} \sin (\varOmega \tau+\theta_{1} ) \\ u=U_{0} +A_{2} \sin (\varOmega \tau+\theta_{2} ) \\ v=V_{0} +A_{3} \sin (\varOmega \tau+\theta_{3} ) \\ \end{array}} \right\}$

其中, $Y_{0}$, $U_{0}$, $V_{0}$, $A_{i}$, $\theta_{i, }(i=1$, 2, 3)是常数, 可以通过把(10)分别代入系统(7)$\sim$(9)中得到. 例如,对区域$D_{3}$中的名义平衡轨道$NEO_{-}$有

$\begin{eqnarray*} Y_{0} =-x, \ \ U_{0} =x+(a-b-x)/b, \ \ V_{0} =(a-b-x)/b \\\theta_{1} =-\arctan \frac{H\gamma \varOmega }{-b^{2}\delta ^{2}\varOmega ^{2}+b^{2}\delta ^{2}\gamma -\delta \gamma \varOmega^{2}-\varOmega^{4}+\gamma \varOmega^{2}}\\ \theta_{2} =-\arctan \left[ {{{\varOmega L}/{\left( {H\gamma^{2}} \right)}}}\right] \\\theta_{3} =-\arctan \frac{(b\delta \gamma -\delta \gamma -\varOmega ^{2}+\gamma )\varOmega }{-b\delta \varOmega^{2}+b\delta \gamma -\gamma \varOmega ^{2}}\\ A_{1} =\frac{A\gamma }{k}\cdot \\\quad \sqrt{H^{2}\gamma^{2}\varOmega^{2}+(-b^{2}\delta ^{2}\varOmega ^{2}+b^{2}\delta ^{2}\gamma -\delta \gamma \varOmega^{2}-\varOmega^{4}+\gamma \varOmega^{2})^{2}} \\ A_{2} ={{A\sqrt {\varOmega^{2}L^{2}+H^{2}\gamma^{4}} }/k}\\ A_{3} =\frac{A\gamma \delta }{k}\cdot \\\quad \sqrt {\varOmega^{2}(b\delta \gamma -\delta \gamma -\varOmega ^{2}+\gamma )^{2}+(-b\delta \varOmega^{2}+b\delta \gamma -\gamma \varOmega ^{2})^{2}} \end{eqnarray*}$

其中

$\begin{eqnarray*} H=b^{2}\delta ^{2}-b\delta ^{2}+\varOmega^{2}\\ k=\varOmega^{2}(L-b^{2}\delta ^{2}\gamma -\delta \gamma^{2}-\gamma \varOmega^{2}+\gamma^{2})+b^{2}\delta ^{2}\gamma^{2} \\ L=b^{2}\delta ^{2}\gamma^{2}+b^{2}\delta ^{2}\varOmega^{2}-b^{2}\delta ^{2}\gamma -2b\delta ^{2}\gamma^{2}+\delta ^{2}\gamma^{2}+\\ 2\delta \gamma \varOmega^{2}+\gamma^{2}\varOmega ^{2}+\varOmega^{4}-\delta \gamma^{2}-\gamma \varOmega^{2} \end{eqnarray*}$

从名义平衡轨道的解析形式(10)可以发现, 外激励的作用是对无激励情形下对应的线性自治系统的平衡点位置的扰动, 因此,名义平衡轨道可以看作是由一系列周期为2$\pi /\varOmega $的极限环组成的平衡态. 由于名义平衡轨道在非光滑分界面所划分的子区域内满足处处等价, 所以对名义平衡轨道的稳定性的讨论可以转化为对名义平衡轨道上一个点的稳定性的讨论. 根据系统的对称性,$D_{1}$与$D_{3}$中的名义平衡轨道$NEO_{+}$, $NEO_{-}$具有相同的特性, 统一记作$NEO_{\pm}$, 与之相应的特征方程为

$\lambda^{3}+ n_{1}\lambda^{2}+n_{2}\lambda + n_{3}=0$

其中, $n_{1}=\delta b+\gamma $, $n_{2}=\gamma -\delta \gamma +\gamma \delta b$, $n_{3}=\gamma \delta b$.由Routh-Hurwitz判别准则知, 当参数满足条件$ n_{1}> 0$, $n_{3}> 0$, $n_{1}n_2-n_{3}> 0$时, $NEO_{\pm}$是渐近稳定的. 随着参数的变化, $NEO_{\pm }$可能存在以下两条失稳路径:

(1) $FB_{\pm }:= n_{3}=0\ (n_{1}> 0$, $n_{2}> 0$), 此时, 特征方程(11)有零单根出现, 系统可能产生fold分岔;

(2) $HB_{\pm }:= n_{1}n_2-n_{3}=0\ (n_{1}> 0$, $n_{2}> 0$, $n_{3}> 0$), 此时, 特征方程(11)有共轭纯虚根出现, 系统可能产生Hopf分岔.

类似可以讨论区域$D_{2}$中$NEO_{0}$的稳定性.

固定部分参数如下: $\alpha =0.000,1$, $\beta =1.2$, $a=-3.0$, $b=0.6$, $A=0.5$, $\varOmega =0.01$. 图2给出了快子系统(6)在($\gamma ,\delta )$参数平面上的 $\gamma > 0$, $\delta > 0$且慢变量$x =0$时的双参分岔情况. 图中曲线$HB$反映了名义平衡轨道$NEO_{\pm }$的一对复特征值通过穿越纯虚根失稳的临界情形. 取$\gamma =0.6$,表1给出了参数$\delta $分别取1.02, 1.1456和1.1584时非光滑分界面两侧的快子系统的名义平衡轨道$NEO_{\pm}$所对应的特征值.

图2

图2   快子系统(6)在($\gamma $, $\delta )$平面上的双参分岔图

Fig. 2   Two-parameter bifurcation diagram of fast subsystem (6) on ($\gamma$, $\delta )$ plane


表1   快子系统的名义平衡轨道对应的特征值

Table 1  Eigenvalues of the NEOs of the fast subsystems

新窗口打开| 下载CSV


结合表1知, 图2中分岔曲线$HB$将参数平面分为两部分: 当参数在曲线$HB$下方的区域I中取值时, 特征方程(11)的所有的根具有负实部,意味着名义平衡轨道$NEO_{\pm }$为稳定的, 即快子系统(6) 在区域$D_{1,3}$中存在两个稳定的极限环; 当参数从区域I穿越曲线$HB$进入区域II后, 名义平衡轨道$NEO_{\pm }$失稳, 意味着快子系统(6)的原来的两个稳定极限环将转变为对应的两个概周期运动, 如图3所示.

图3

图3   (a) 区域$D_{3}$中的概周期解; (b) 庞加莱截面图

Fig. 3   (a) Quasi-periodic solution in $D_{3}$; (b) Poincare section


随着慢变量$x$的连续变化, 无论是区域I中的名义平衡轨道$NEO_{\pm}$所对应的稳定极限环, 还是区域II中出现的概周期运动,都可能会接触到非光滑分界面$\varSigma_{\pm }$并出现非光滑分岔, 进而影响三时间尺度耦合系统(2)的动力学演化机制. 接下来, 将通过具体的数值, 进一步探讨分析非光滑分岔对簇发振荡的吸引子结构以及转迁机制的影响.

3 簇发振荡的动力学演化过程及机制分析

3.1 簇发振荡

图4图5分别给出了当参数$\delta =1.02$, $\delta =1.158,4$时系统(2)的簇发振荡的相图及其对应的时间历程图.

图4

图4   $\delta =1.02$时的周期簇发振荡

Fig. 4   Periodic bursting oscillation for $\delta =1.02$


图5

图5   $\delta =1.158,4$时的混沌簇发振荡

Fig. 5   Chaotic bursting oscillation for $\delta =1.158,4$


图5

图5   $\delta =1.158,4$时的混沌簇发振荡(续)

Fig. 5   Chaotic bursting oscillation for $\delta =1.158,4$ (continued)


当参数$\delta =1.02$时, 从($y$, $v)$平面中的相图4(a)中可知,整个周期运动关于原点对称, 运动方向如图中红色箭头所示. 根据时间历程图4(b)可计算得其频率近似为0.000,270,9. 接下来详细阐述此时的三时间尺度下周期簇发振荡的吸引子结构.

不失一般性, 设系统轨线从非光滑分界面$\varSigma_{+1}$上的点$A_{1}$出发. 随着时间的变化, 轨线进入光滑区域$D_{2}$中. 之后, 迅速向非光滑分界面$\varSigma_{-1}$运动并且在点$A_{2} $穿过$\varSigma_{-1}$而进入光滑区域$D_{3}$中. 此后, 轨线在区域$D_{3}$中首先呈现明显的密集的螺旋形振荡行为. 根据时间历程图可以发现, 轨线的振幅是逐渐减小的. 经计算,轨线的振荡频率近似为0.562,5. 随着时间继续延长, 轨线逐渐由密集的振荡形式转为舒缓的振荡形式, 且振幅基本保持不变,对应的近似振荡频率为0.01, 直到轨线抵达非光滑分界面$\varSigma_{-1}$上的点$A_{3}$. 之后, 轨线在$\overline{A_{3}A_{4}A_{1}}$的运动过程中, 将重复轨线在$\overline {A_{1}A_{2}A_{3}}$中的相似运动过程, 并最终返回出发点$A_{1}$, 完成一个周期的运动. 从上述结果发现, 周期簇发振荡的近似频率0.000,270,9与最慢的时间尺度$T_{3}\equiv O(10^{-4})$基本吻合; 从非光滑分界面跳入区域$D_{1,3}$并逐渐收敛时的近似振荡频率0.562,5与最快时间尺度$T_{1}\equiv O(1)$基本一致; 轨线的振幅稳定后的近似振荡频率0.01与外激励频率所对应的时间尺度$T_{2}\equiv O(10^{-2})$, 即与中间时间尺度相吻合. 可见, 此时的簇发振荡明显地具有3个不同量级的振荡频率, 体现了系统的三时间尺度效应.

而当参数$\delta =1.158,4$时, 快子系统的名义平衡轨道$NEO_{\pm}$由于所对应的特征值穿越纯虚根而失稳. 此时,系统(2)表现为明显的混沌行为(见图5), 且其轨线在来回穿越非光滑分界面的过程中, 连接了光滑区域$D_{1}$与$D_{3}$中的概周期运动,可以称作概周期-概周期型混沌簇发振荡.

3.2 簇发振荡的机制分析

主要探究$\delta =1.02$时的演化机制. 考虑到系统的对称性, 仅探究$\overline{A_{1} A_{2} A_{3}}$过程.

利用转换相图(TPP)与慢子流形的叠加图(图6(a)), 在($x$, $v)$平面内利用快慢分析法, 结合非光滑分界面处的分岔分析,进一步来探讨图4中具有的三时间尺度特性的周期簇发振荡的振荡机理, 也即系统在呈现三时间尺度的簇发振荡时, 轨线在不同时间尺度之间的转迁机制.

图6

图6   (a) $\delta =1.02$时转换相图与慢子流形的叠加图; (b)局部放大图

Fig. 6   (a) Overlay of TPP and the manifold of the fast subsystem for $\delta =1.02$; (b) enlarged figure


事实上, 在中间尺度系统(4)的激励作用下, 快子系统的具有解析形式(10)的名义平衡轨道$NEO_{-}$表现为区域$D_{3}$中的稳定极限环$LC_{1}$.随着慢变量$x$在时间尺度$T_{3}\equiv O(10^{-4})$上的连续变化, $LC_{1}$将在空间中形成环面结构形式的慢子流形.快子系统(6)的极限环$LC_{i}(i=1,2$)在$v$轴方向的极值分别记为$LC^{\max}_i$、$LC^{\min}_i$ ($i=1,2$), 如图6中的蓝色线条所示.

如上, 轨线仍从$\varSigma_{+1}$上的点$A_{1}$出发, 由于光滑区域$D_{2}$中名义平衡轨道的不稳定性, 轨线迅速向非光滑分界面$\sum_{-1}$运动并在点$A_{2}$穿过$\varSigma_{-1}$而进入光滑区域$D_{3}$, 在收敛到具有不变环面结构的稳定慢子流形的过程中出现大幅振荡, 形成激发态, 记作$SS_{1}$, 其振荡频率与慢子流形对应的平衡点的虚部基本吻合, 体现了系统的最快时间尺度$T_{1}\equiv O(1)$. 随着时间的增大, 轨线的振幅逐渐减小. 当系统轨线收敛到稳定慢子流形上后, 将沿着慢子流形呈现振幅保持不变的环面运动(对应的两条蓝色曲线之间的距离保持不变), 形成沉寂态, 记作$QS_{1}$, 其振荡频率与外激励保持一致, 体现了中间时间尺度$T_{2}\equiv O(10^{-2})$. 由于轨线在从激发态$SS_{1}$转迁到慢子流形上的沉寂态$QS_{1}$的转迁过程中并没有分岔出现,所以只是轨线收敛的暂态过程的体现.

然而, 从$D_{3}$中的沉寂态$QS_{1}$向$D_{1}$中的激发态SS$_{2}$的转迁是系统轨线通过依次穿越非光滑分界面上的点$A_{3}$与$A_{4}$的形式完成的. 但是, 通过快子系统(6)的稳定极限环$LC_{1}$可以发现(见图6(b)),$LC_{1}$在慢变量跨过$x=-2.5$ (对应图6(b)中的垂直方向的虚线)后消失了. 即随着慢变量的减小, 虽然慢子流形的稳定极限环$LC_{1}$消失了, 但是轨线并没有立刻向激发态转迁, 而是继续保持了一段时间的沉寂态,直到轨线穿越非光滑分界上的点$A_{3}$后才向激发态转迁. 因此, 慢子流形所对应的非光滑分岔将对此处的转迁机理起到至关重要的作用.

下面对快子系统在非光滑分界面上的非光滑动力学演化进行分析.

由于非光滑分界面$\varSigma_{-1}$两侧的向量场的连续性, 在非光滑分界面$\varSigma_{-1}$上存在一条分割曲线, 满足

$\varSigma^0_{-1}:=\langle \nabla H_{-1}, F_{3}\rangle =\langle H_{-1}, F_{2}\rangle =0$

其将非光滑分界面$\varSigma_{-1}$划分为两个向量场方向相反的不同区域, 如图7(a)所示. 代入参数得,$\varSigma ^0_{-1}:=\{(y, u, v)| y=3, v=-1|\}$. 又$\partial \varSigma^0_{-1}/\partial y=\delta > 0$, 因此, 分界面$\varSigma_{-1}$上在直线$\varSigma^0_{-1}$的左边的区域指向光滑区域$D_{2}$, 而在$\varSigma^0_{-1}$右边的区域指向光滑区域$D_{3}$.

图7

图7   (a) 非光滑分界面$\varSigma_{-1}$上的动力学分析; (b)$ x=-2.5$时与$\varSigma_{-1}$相切的$NEO_{-}$; (c) $x=-3$时与$\varSigma_{-1}$横截相交的$NEO_{-}$; (d) $x=-2.6$时的时间历程图

Fig. 7   (a) Dynamic analysis on non-smooth interface $\varSigma_{-1}$; (b) $NEO_{-}$ tangent to $\varSigma_{-1}$ for $x=-2.5$; (c) $NEO_{-}$ intersecting with $\varSigma_{-1}$ for $x=-3$; (d) time history for $x=-2.6$


图7

图7   (a) 非光滑分界面$\varSigma_{-1}$上的动力学分析; (b)$ x=-2.5$时与$\varSigma_{-1}$相切的$NEO_{-}$; (c) $x=-3$时与$\varSigma_{-1}$横截相交的$NEO_{-}$; (d) $x=-2.6$时的时间历程图(续)

Fig. 7   (a) Dynamic analysis on non-smooth interface $\varSigma_{-1}$; (b) $NEO_{-}$ tangent to $\varSigma_{-1}$ for $x=-2.5$; (c) $NEO_{-}$ intersecting with $\varSigma_{-1}$ for $x=-3$; (d) time history for $x=-2.6$ (continued)


另外, 由名义平衡轨道$NEO_{-}$的解析表达式可知, $\partial V_{0}/\partial x=-5/3< 0$, 即随着慢变量$x$的减小,$NEO_{-}$将向非光滑分界面$\varSigma_{-1}$移动. 经计算, 慢变量$x=-2.5$时, $NEO_{-}$与非光滑分界面$\varSigma_{-1}$相切于直线$\varSigma^0_{-1}$上的点$TP$, 如图7(b)所示,意味着对应的极限环$LC_{1}$此时也与分界面$\varSigma_{-1}$相切于点$TP$,且保持存在. 随着慢变量$x$的减小, 名义平衡轨道$NEO_{-}$将跨越分界面$\varSigma_{-1}$, 图7(c)给出了对应$x=-3$时的名义平衡轨道$NEO_{-}$. 此时,$NEO_{-}$被非光滑分界面$\varSigma_{-1}$分割为两部分:一部分位于光滑区域$D_{3}$内, 为真实存在的部分, 对应图7(c)中的黑色实线部分,记为$NEO^R_-$; 另一部分位于光滑区域$D_{2}$内, 不满足非光滑限制条件,对应图7(c)中的红色实线部分, 记为$NEO^{NR}_-$. 而此时, 与名义平衡轨道$NEO_{-}$相应的存在于快子系统(6)的极限环$LC_{1}$却在光滑区域$D_{3}$内到达非光滑分界面$\varSigma_{-1}$后进入了图7(a)中$\varSigma^0_{-1}$的左侧区域, 受到该分界面的非光滑动力学特性的影响,轨线将直接按照图中箭头所示的方向横截穿过$\varSigma_{-1}$与$\varSigma_{+1}$向光滑区域$D_{1}$内的稳定极限环$LC_{2}$转迁, 如图7(d)所示,意味着极限环$LC_{1}$消失.

鉴于极限环$LC_{1}$随着慢变量$x$的变化所展现的由光滑区域$D_{3}$内的稳定极限环,经过与非光滑分界面$\varSigma_{-1}$相切的临界情形, 再到与$\varSigma_{-1}$横截相交后消失的动力学演化过程, 这里称其为环的破坏性擦边分岔, 记为$CGB$.

回到系统(2)的轨线在非光滑分界面$\varSigma_{-1}$上的点$A_{3}$处由沉寂态$QS_{1}$到激发态$SS_{2}$的转迁机制.如图6(b), 当慢变量越过$x=-2.5$时, 虽然极限环$LC_{1}$ ($NEO_{-})$由于破坏性的擦边分岔而消失, 但是在区间$x\in (-2.5,-3.5$)上, 名义平衡轨道$NEO_{-}$在光滑区域$D_{3}$中仍然存在着$NEO^R_-$部分, 也即慢子流形在区间$x\in (-2.5, -3.5$)上仍然存在, 并且$NEO^R_-$仍然落在直线$\varSigma^0_{-1}$的左边(图7(c)). 因此, 沉寂态$QS_{1}$在慢变量$x$越过$x=-2.5$后会继续存在,直到轨线到达非光滑分界面$\varSigma_{-1}$上的点$A_{3}$. 此时, 受到极限环$LC_{1}$ ($NEO_{-})$的破坏性的擦边分岔CGB的影响, 轨线直接穿过非光滑分界面$\varSigma_{-1}$而进入光滑区域$D_{2}$, 意味着沉寂态$QS_{1}$的结束.

这样, 在$\overline{A_{1} A_{2}A_{3} } $过程中, 系统呈现三时间尺度的簇发振荡时, 轨线在不同时间尺度之间的转迁机制得到了完整的解释. 根据文献[11]中提供的对簇发振荡的命名和分类的方法,也即以导致沉寂态与激发态之间转迁的分岔对簇发振荡进行命名和分类,同时考虑到此时对应的周期簇发振荡中体现出的三时间尺度效应,图4中的周期簇发振荡可称为对称式$CGB/CGB$型三时间尺度周期簇发振荡.

当$\delta =1.158,4$时, 由于对应的快子系统的概周期运动同样会在跨越慢变量的临界值后失稳, 在非光滑分界面$\varSigma_{\pm 1}$上经历破坏性的擦边分岔, 导致概周期运动破裂, 表现了典型的由概周期运动破裂到混沌行为的动力学演化过程, 形成了此处的概周期-概周期型混沌簇发振荡.

4 结论

基于典型的蔡氏电路, 文章建立了一个含有三时间尺度的四维分段线性系统.以此系统为例, 将中间尺度变量视为对快尺度变量的扰动,并由此将中间尺度变量与快尺度变量看作一个整体,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. 在此方法的基础上,利用快慢分析法, 讨论了系统在两组不同数值情形下的簇发振荡行为并给出了相应的分岔机制. 发现,当名义平衡轨道为稳定的时候, 系统轨线能够落于慢子流形上并能够随着慢子流形穿越非光滑分界面,呈现周期簇发振荡的同时体现出三时间尺度的特性; 而当名义平衡轨道通过穿越纯虚根的路径失稳后, 轨线不能够落于慢子流形,而是表现为混沌运动. 经过分析, 非光滑分界面处的非常规分岔,即破坏性的擦边分岔是导致簇发振荡的主要诱因, 为全新的通向簇发振荡的演化路径.需要指出的是, 对于时间尺度具有量级差的多时间尺度耦合动力系统而言, 不同时间尺度之间差异到何种程度才能够出现簇发现象,这一点需要在后续工作中重点关注. 另外, 慢时间尺度在一定范围内变化时, 是否会导致不同的簇发振荡的出现或者其它的复杂动力学现象的产生, 这一点也需要在后续工作中关注.

参考文献

Abobda LT, Woafo P.

Subharmonic and bursting oscillations of a ferromagnetic mass fixed on a spring and subjected to an AC electromagnet

Communications in Nonlinear Science and Numerical Simulation, 2012,17(7):3082-3091

[本文引用: 1]

Courbage M, Maslennikov OV, Nekorkin VI.

Synchronization in time-discrete model of two electrically coupled spike-bursting neurons

Chaos, Solitons & Fractals, 2012,45(5):645-659

[本文引用: 1]

Pereda E, Cruz DMADL, Maas S, et al.

Topography of EEG complexity in human neonates: Effect of the postmenstrual age and the sleep state

Neuroscience Letters, 2006,394(2):152-157

DOI      URL     PMID      [本文引用: 1]

The topography of the EEG of human neonates is studied in terms of its power spectral density and its estimated complexity as a function of both the postmenstrual age (PMA) and the sleep state. The monopolar EEGs of three groups of seven neonates (preterm, term and older term) were recorded during active (AS) and quiet sleep (QS) from electrodes Fp1, Fp2, T3, T4, C3, C4, O1 and O2. The existence of changes between groups and sleep states in the power of delta, theta, alpha and beta bands and in the dimensional complexity of these electrodes was tested. Additionally, the nonlinearity of the EEG in each electrode and situation was analyzed. The results of the spectral measures show an increment of the power in the low frequency bands from AS to QS and with the PMA, which can be mainly traced on central and temporal electrodes. This change is shown as well by the dimensional complexity, which also presents the greatest differences in the central derivations. Moreover, the signals show evidence of nonlinearity in almost all the groups and situations, although a dynamic change from nonlinear to linear character is apparent in the central electrodes with increased PMA. As a result, it is concluded that nonlinear analysis methods provide a clear portrait of the integrated brain activity that complements the information of spectral analysis in the characterization of the brain development and the sleep states in neonates.

Bi QS.

The mechanism of bursting phenomena in Belousov-Zhabotinsky (BZ) chemical reaction with multiple time scales

Science China Technological Sciences, 2010, doi: CNKI:SUN:JEXK.0.2010-02-041

URL     PMID      [本文引用: 1]

The anti-virus and anti-bacteria active components were extracted from some Chinese medicine, such as the honeysuckle, forsythia and the licorice. Using a w/o/w emulsion method, the active components were fabricated to uniform particulate microcapsule with sustained-release properties. The polypropylene punched felt was finished with the finishing agent of microcapsule, nano ZnO and TiO2 and polymer adhesive, and the composite air filter with anti-virus and anti-bacteria properties were formed, staphylococcus aureus, colibacillus and candida albicans were applied to antibacterial experiments. The results indicate that the anti-bacteria rate are all 100%, and the virus inactivation rate also reaches 100% to pandemic influenza A virus.

Roussel C, Erneux T, Schiffmann SN, et al.

Modulation of neuronal excitability by intracellular calcium buffering: From spiking to bursting

Cell Calcium, 2006,39(5):455-466

DOI      URL     PMID      [本文引用: 1]

We have investigated the detailed regulation of neuronal firing pattern by the cytosolic calcium buffering capacity using a combination of mathematical modeling and patch-clamp recording in acute slice. Theoretical results show that a high calcium buffer concentration alters the characteristic regular firing of cerebellar granule cells and that a transition to various modes of oscillations occurs, including bursting. Using bifurcation analysis, we show that this transition from spiking to bursting is a consequence of the major slowdown of calcium dynamics. Patch-clamp recordings on cerebellar granule cells loaded with a high concentration of the fast calcium buffer BAPTA (15 mM) reveal dramatic alterations in their excitability as compared to cells loaded with 0.15 mM BAPTA. In high calcium buffering conditions, granule cells exhibit all bursting behaviors predicted by the model whereas bursting is never observed in low buffering. These results suggest that cytosolic calcium buffering capacity can tightly modulate neuronal firing patterns leading to generation of complex patterns and therefore that calcium-binding proteins may play a critical role in the non-synaptic plasticity and information processing in the central nervous system.

Mease KD.

Multiple time-scales in nonlinear flight mechanics: Diagnosis and modeling

Applied Mathematics & Computation, 2005,164(2):627-648

[本文引用: 1]

Shilnikov A.

Complete dynamical analysis of a neuron model

Nonlinear Dynamics, 2012,68(3):305-328

[本文引用: 1]

Yu Y, Zhang ZD, Han XJ.

Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system

Communications in Nonlinear Science and Numerical Simulation, 2018,56(3):380-391

[本文引用: 1]

Jia B, Wu Y, He D, et al.

Dynamics of transitions from anti-phase to multiple in-phase synchronizations in inhibitory coupled bursting neurons

Nonlinear Dynamics, 2018,93(3):1599-1618

Zhan FB, Liu SQ, Zhang XH, et al.

Mixed-mode oscillations and bifurcation analysis in a pituitary model

Nonlinear Dynamics, 2018,94:807-826

[本文引用: 1]

Izhikevich EM.

Neural excitability, spiking and bursting

International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 2000,10(6):1171-1266

[本文引用: 2]

Schaffer WM, Bronnikova TV.

Peroxidase-ROS interactions

Nonlinear Dynamics, 2012,68(3):413-430

[本文引用: 1]

Terman D.

Chaotic spikes arising from a model of bursting in excitable membranes

SIAM Journal on Applied Mathematics, 1991,51(5), doi: 10.1137/0151071

[本文引用: 1]

Rush ME, Rinzel J.

The potassium a-current, low firing rates and rebound excitation in Hodgkin-Huxley models

Bulletin of Mathematical Biology, l995, 57(6):899-929

URL     PMID      [本文引用: 1]

Izhikevich EM, Hoppensteadt F.

Classification of bursting mappings

International Journal of Bifurcation & Chaos, 2004,14(11):3847-3854

Izhikevich EM, Desai NS, Walcott EC, et al.

Bursts as a unit of neural information: Selective communication via resonance

Trends in Neurosciences, 2003,26(3):161-167

DOI      URL     PMID      [本文引用: 1]

What is the functional significance of generating a burst of spikes, as opposed to a single spike? A dominant point of view is that bursts are needed to increase the reliability of communication between neurons. Here, we discuss the alternative, but complementary, hypothesis: bursts with specific resonant interspike frequencies are more likely to cause a postsynaptic cell to fire than are bursts with higher or lower frequencies. Such a frequency preference might occur at the level of individual synapses because of the interplay between short-term synaptic depression and facilitation, or at the postsynaptic cell level because of subthreshold membrane potential oscillations and resonance. As a result, the same burst could resonate for some synapses or cells and not resonate for others, depending on their natural resonance frequencies. This observation suggests that, in addition to increasing reliability of synaptic transmission, bursts of action potentials might provide effective mechanisms for selective communication between neurons.

蒲刚, 章定国, 黎亮.

考虑尺度效应的夹层楔形多孔梁动力学分析

力学学报, 2019,51(6):1882-1896

[本文引用: 1]

( Pu Gang, Zhang Dingguo, Li Liang.

Dynamic analysis of sandwich tapered porous micro-beams considering size effect

Chinese Journal of Theoretical and Applied Mechanics, 2019,51(6):1882-1896 (in Chinese))

[本文引用: 1]

张毅, 韩修静, 毕勤胜.

串联式叉型滞后簇发振荡及其动力学机制

力学学报, 2019,51(1):228-236

( Zhang Yi, Han Xiujing, Bi Qinsheng.

Series-mode pitchfork-hysteresis bursting oscillations and their dynamical mechanism

Chinese Journal of Theoretical and Applied Mechanics, 2019,51(1):228-236 (in Chinese))

Yu Y, Zhang ZD, Bi QS.

Multistability and fast-slow analysis for Van der pol-Duffing oscillator with varying exponential delay feedback factor

Applied Mathematical Modelling, 2018,57:448-458

李航, 申永军, 李向红, .

Duffing 系统的主-亚谐联合共振

力学学报, 2020,52(2):514-521

[本文引用: 1]

( Li Hang, Shen Yongjun, Li Xianghong.

Primary and subharmonic simultaneous resonance of Duffing oscillator

Chinese Journal of Theoretical and Applied Mechanics, 2020,52(2):514-521 (in Chinese))

[本文引用: 1]

秦志英, 陆启韶.

非光滑分岔的映射分析

振动与冲击, 2009,28(6):79-81

[本文引用: 1]

( Qin Zhiying, Lu Qishao.

Map analysis for non-smooth bifurcations

Journal of Vibration and Shock, 2009,28(6):79-81 (in Chinese))

[本文引用: 1]

Li DH, Xie JH.

Absolutely continuous invariant measure of a map from grazing-impact oscillators

Physics Letter A, 2015,379(10-11):901-906

[本文引用: 1]

Chin W, Ott E, Nusse HE, et al.

Grazing bifurcations in impact oscillations

Physical Review E Statistical Physics Plasmas Fluids & Related Interdisciplinary Topics, 1996,53(6):134-139

张正娣, 刘亚楠, 李静, .

分段Filippov系统的簇发振荡及擦边运动机理

物理学报, 2018,67(11):110501

( Zhang Zhengdi, Liu Yanan, Li Jing, et al.

Bursting oscillations and mechanism of sliding movement in piecewise Filippov system

Acta Physica Sinica, 2018,67(11):110501 (in Chinese))

Qu R, Li SL, Bi QS.

Forced vibration of shape memory alloy spring oscillator and the mechanism of sliding bifurcation with dry friction

Advances in Mechanical Engineering, 2019,11(5):1-15

[本文引用: 1]

Chatzis MN, Chatzi EN, Triantafyllou SP.

A discontinuous extended kalman filter for non-smooth dynamic problems

Mechanical Systems and Signal Processing, 2017,92:13-29

[本文引用: 1]

张正娣, 毕勤胜.

非光滑系统的分岔

力学进展, 2012: 93

[本文引用: 1]

( Zhang Zhengdi, Bi Qinsheng.

Bifurcation of non-smooth systems

Advances in Mechanics, 2012: 93(in Chinese))

[本文引用: 1]

Hu DH, Yan YZ, Xu XM, et al.

Dynamics analysis of the hybrid powertrain under multi-frequency excitations with two time scales

AIP Advances, 2018,8(6):065212

[本文引用: 1]

马新东, 姜文安, 张晓芳, .

一类三维非线性系统的复杂簇发振荡行为及其机理

力学学报, 2020,52(6):1789-1799

[本文引用: 1]

( Ma Xindong, Jiang Wenan, Zhang Xiaofang, et al.

Complicated bursting behaviors as well as the mechanism of a three dimensional nonlinear system

Chinese Journal of Theoretical and Applied Mechanics, 2020,52(6):1789-1799 (in Chinese))

[本文引用: 1]

夏雨, 毕勤胜, 罗超 .

双频1:2激励下修正蔡氏振子两尺度耦合行为

力学学报, 2018,50(2):362-372

[本文引用: 1]

( Xia Yu, Bi Qinsheng, Luo Chao, et al.

Behaviors of modified chua's oscillator two time scales under two excitations with frequency ratio at 1:2

Chinese Journal of Theoretical and Applied Mechanics, 2018,50(2):362-372 (in Chinese))

[本文引用: 1]

吴天一, 陈小可, 张正娣, .

非对称型簇发振荡吸引子结构及其机理分析

物理学报, 2017,66(11):110501

[本文引用: 1]

( Wu Tianyi, Chen Xiaoke, Zhang Zhengdi, et al.

Structures of the asymmetrical bursting oscillation attractors and their bifurcation mechanisms

Acta Phys. Sin., 2017,66(11):35-45 (in Chinese))

[本文引用: 1]

/