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.     

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.      

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.      

余维三fold-fold-Hopf分岔下簇发振荡及其分类

不同尺度耦合系统存在着广泛的工程背景,通常表现为大幅振荡与微幅振荡交替出现的簇发振荡,其产生机理一直是当前国内外研究的前沿课题之一.传统的几何奇异摄动分析方法仅对时域上的两尺度耦合有效,无法揭示频域上不同尺度之间的相互作用,同时,当前相关研究仅针对余维一fold或Hopf分岔展开.本文针对频域两尺度耦合向量场存在余维三...     

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.      

Homoclinic bifurcation at resonant eigenvalues

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.Partially supported by DARPA and NSF.Partially supported by the Deutsche Forschungsgemeinschaft and by Konrad-Zuse-Zentrum für Informationstechnik Berlin.     

Bogdanov–Takens bifurcation in a tri-neuron BAM neural network model with multiple delays

In this paper, a tri-neuron BAM neural network model with multiple delays is considered. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. There is a wide range of different dynamical behaviors which can be produced by varying the coupling strength. By choosing the connected weights c ₂₁ and c ₃₁ (the connection weights through the neurons from J-layer to I-layer) as bifurcation parameters, the critical values where a Bogdanov–Takens bifurcation occurs are derived. Then, by computing the normal forms for the system, the bifurcation diagrams are obtained. Furthermore, some interesting phenomena, such as saddle-node bifurcation, pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation are found by choosing the different connection strengths. Some numerical simulations are given to support the analytic results.     

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.     

离心场中纵向悬臂梁的大范围分岔分析

采用打靶法研究了离心场中纵向悬臂梁的大范围失稳与分岔问题。分析结果证实：随着参数a（离心臂长与梁长之比）的变化,梁平衡解可能发生三种分岔现象。文中给出了平衡解的分岔形态,并发现了梁分岔解的单向跳跃现象,即突变现象。     

Orbits homoclinic to resonances in a harmonically excited and undamped circular plate

Orbits homoclinic to resonances in mode interactions of an imperfect circular plate with 1:1 internal resonance are investigated. The case of primary resonance is considered. The damping force is not included in the analysis. The energy-phase criterion is used to give a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonances. A saddle-node bifurcation of homoclinic orbits occurs. The existence of homoclinic orbits in the unperturbed system may lead to chaos in the sense of Smale horseshoes under perturbation.     

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR⁴{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR³{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS 1)

本文主要探究了一类含有两个慢变量的双稳态 Duffing 型系统,通过时间历程图、相图、分岔图等对系统进行数值模拟,然后从理论上分析不同参数下系统的动力学机理. 首先,研究发现当振幅参数取值大于 1 时,系统会表现出不动点混沌现象,并进一步解释了产生不动点混沌的机理. 其次, 介绍了参数空间中的簇发振荡现象,即系统穿过鞍结曲面的一侧到达另一侧所发生的行为,这里也称为鞍结簇发振荡. 事实上,当系统穿过鞍结曲面的时候,它的平衡点个数发生了变化. 然后,使用纵向抛物线路径说明了 Fold/Fold 簇发振荡产生的机理,发现无论常系数项和振幅的取值为多少,只要满足一定的关系,总会产生 Fold/Fold 簇发振荡,之后使用线性路径阐明了新增常系数项会使得系统发生簇发振荡的原因. 并且发现路径与鞍结曲面交点的位置会影响簇发振荡的对称性;路径的跨度会影响簇发振荡的大小. 最后,使用多拐折曲线路径讨论当两个激励项存在 $n$ 倍关系时系统产生的现象. 结果表明当 $n=3$ 时,常系数项的变化会使得系统表现出不同重数的 Fold/Fold 簇发振荡,最高可达到三重簇发振荡. 并且发现在理想状况下如果可以找到一条路径可以分割为 $n$ 段,并且每一段都会与鞍结曲面有交点,那么会产生 $n$ 重 Fold/Fold 簇发振荡.     

一类 Duffing 型系统的不动点混沌和Fold/Fold 簇发现象及机理分析

本文主要探究了一类含有两个慢变量的双稳态 Duffing 型系统,通过时间历程图、相图、分岔图等对系统进行数值模拟,然后从理论上分析不同参数下系统的动力学机理. 首先,研究发现当振幅参数取值大于 1 时,系统会表现出不动点混沌现象,并进一步解释了产生不动点混沌的机理. 其次, 介绍了参数空间中的簇发振荡现象,即系统穿过鞍结曲面的一侧到达另一侧所发生的行为,这里也称为鞍结簇发振荡. 事实上,当系统穿过鞍结曲面的时候,它的平衡点个数发生了变化. 然后,使用纵向抛物线路径说明了 Fold/Fold 簇发振荡产生的机理,发现无论常系数项和振幅的取值为多少,只要满足一定的关系,总会产生 Fold/Fold 簇发振荡,之后使用线性路径阐明了新增常系数项会使得系统发生簇发振荡的原因. 并且发现路径与鞍结曲面交点的位置会影响簇发振荡的对称性;路径的跨度会影响簇发振荡的大小. 最后,使用多拐折曲线路径讨论当两个激励项存在 $n$ 倍关系时系统产生的现象. 结果表明当 $n=3$ 时,常系数项的变化会使得系统表现出不同重数的 Fold/Fold 簇发振荡,最高可达到三重簇发振荡. 并且发现在理想状况下如果可以找到一条路径可以分割为 $n$ 段,并且每一段都会与鞍结曲面有交点,那么会产生 $n$ 重 Fold/Fold 簇发振荡.      

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.     

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.     

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.     

齿轮传动系统共存吸引子的不连续分岔

大量的多吸引子共存是引起齿轮传动系统具有丰富动力学行为的一个重要因素.多吸引子共存时,运动工况的变化以及不可避免的扰动都可能导致齿轮传动系统在不同运动行为之间跳跃变换,对整个机器产生不良的影响.目前,一些隐藏的吸引子没有被发现,共存吸引子的分岔演化规律没有被完全揭示.考虑单自由度直齿圆柱齿轮传动系统,构建由局部映射复合的Poincaré映射,给出Jacobi矩阵特征值计算的半解析法.应用数值仿真、延拓打靶法和Floquet特征乘子求解共存吸引子的稳定性与分岔,应用胞映射法计算共存吸引子的吸引域,讨论啮合频率、阻尼比和时变激励幅值对系统动力学的影响,揭示齿轮传动系统倍周期型擦边分岔、亚临界倍周期分岔诱导的鞍结分岔和边界激变等不连续分岔行为.倍周期分岔诱导的鞍结分岔引起相邻周期吸引子相互转迁的跳跃与迟滞,使倍周期分岔呈现亚临界特性.鞍结分岔是共存周期吸引子出现或消失的主要原因.边界激变引起混沌吸引子及其吸引域突然消失,对应周期吸引子的分岔终止.     

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.      

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.