约束含参分岔问题的分类

约束边界与分岔参数有关的约束分岔问题,称为约束含参分岔问题．通过引入适当的变换,将约束含参分岔问题转化为新变量的非约束分岔问题,推导出了约束含参分岔问题转迁集的一般形式,结果表明只有约束分岔集受约束含参的影响,其它转迁集与不含参约束分岔的转迁集相同．以含参约束树枝分岔为例分析了此类问题的分岔分类,讨论了约束含参对分岔分类的影响．     

含约束非线性动力系统的分岔分类

讨论含约束非线性动力系统分岔的分类.研究表明,约束分岔的转迁集,除分岔集、滞后集和双极限点集外,还有三种转迁集是它特有的.在此基础上提出了一种约束分岔问题的奇异性分类方法.     

一类时变动力系统的高余维分岔及其控制

研究了一类时变动力系统的高余维分岔及其控制问题,首先利用新方法对时变分岔方程的两个方向的分岔转迁和跃迁现象进行分析,分别通过慢变解的线性化近似和量级平衡估计分岔转迁值,然后研究这类时变分岔方程的线性反蚀控制问题,通过分析相应的二维高次自治系统的Hopf分岔,在适当的条件下得到了稳定的动态滞后环,研究揭示出脉冲振动产生的机理是分岔参数随时间周期变化经过定常分岔值时所发生的分岔转迁的滞后和跃迁现象。     

时变参数系统的非完全分岔及其在Duffing方程中的应用

提出新的方法从本质上研究时变参数系统的非完全分岔问题。通过建立时变参数系统的解的线性近似定理去分析时变分岔方程运动的分岔转迁滞后和跃迁现象。利用Ｖ函数预测分岔转迁值,将新方法应用于Ｄｕｆｆｉｎｇ方程,获得一些新的分岔结果和关于解对初值和参数的敏感性结论。     

非对称碰撞约束下形状记忆合金梁的随机响应与分岔

形状记忆合金(SMA)是二十一世纪具有形状记忆效应的新型智能材料.针对具有非对称约束的SMA梁,本文构造了碰撞振动系统.在无碰撞和有碰撞两种情况下,利用随机平均法给出了近似解析结果.数值模拟作为验证解析结果的工具.结果表明,系统能量的概率响应曲线具有非光滑特性.当约束位置发生变化时,系统会出现随机P分岔和D分岔.     

次表面分岔裂纹的力学行为

在复杂荷载作用下，利用分布位错技术(DDT)对半无限大平面内的分岔裂纹问题进行研究，并进行了正确性验证.根据等效应力强度因子判据，初步解释了裂纹产生分岔的原因；研究了不同埋深、荷载比值、分支长度比值、分岔角度情况下的分岔裂纹尖端的应力强度因子；同时，研究了多分支分岔裂纹，计算结果与有限元结果吻合.结果显示：埋深越深，分岔裂纹扩展越困难，当埋深为d/a=1.5时，分支裂尖应力强度因子削弱程度可达15%左右；较长分支会极大地抑制短分支的扩展，当两分支裂纹长度比达到b/c=2以上时，屏蔽效应可达50%以上；另外，分岔角度和荷载比值会改变分岔裂纹主导的扩展模式.     

周期分岔解的鲁棒控制

根据C-L方法,可以得到非线性动力系统的分岔方程和拓扑分岔图．根据得到的分岔图,结合控制理论,提出了周期解的鲁棒控制方法．该方法将运动模式控制到目标模式．由于该方法对控制器的参数没有严格的控制,所以在设计和制造控制器方面是很方便的．数值研究验证了该方法的有效性．     

弹性约束浅拱的非线性动力响应与分岔分析

浅拱采用竖向、转动方向弹性约束时,自振频率和模态与理想的铰支/固结边界存在差异,不同约束刚度将改变外激励下的非线性响应及各种分岔产生的参数域．由浅拱基本假定建立无量纲动力学方程, 采用在频率和模态中考虑约束刚度大小的方法,通过Galerkin全离散和多尺度摄动分析导出极坐标、直角坐标形式的平均方程, 其中方程系数与约束刚度一一对应．用数值方法分析了周期激励下竖向弹性约束系统最低两阶模态之间1∶2内共振时的动力行为, 所得结果与有限元的对比以及平均方程系数的收敛性证明了所采用方法是可行的．随着激励幅值、频率的变化存在若干分岔点,分岔发生时的参数分布与约束刚度值有关,在由分岔点连接的不稳定区或共振区附近,存在一系列稳态解、周期解、准周期解和混沌解窗口,且随参数的变化可观测到倍周期分岔．     

具有饱和接触率和垂直传染的SIRS传染病模型分岔分析

研究了一类具有饱和发生率、脉冲生育、脉冲接种和垂直传染的SIRS传染病模型的复杂动力学行为,首先构造了一个庞卡莱映射,然后利用映射的不动点及其特征值,得到了系统无病周期解的存在和稳定的条件,接着详细讨论了系统的跨临界分岔、超临界分岔和倍周期分岔现象,最后给出了能很好验证理论分析的数值结果.     

一个微生物生态模型分岔的周期轨

本文研究一类与厌氧消化过程微生物生态模型有关的微分方程组在非双曲情形下的扰动。在其一维不变流形上讨论自治扰动下中心附近发生的Pionceare分岔,所涉及的闭轨存在性问题是许多熟知结果^[1-5]不能包含的情形。进而,利用Melnikov函数方法,给出周期扰动下发生次调和分岔的参数条件。     

Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions

We study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries.In previous work, we focused primarily on codimension one bifurcations. In this paper, we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifurcations, including resonant bifurcations and mode interactions. In particular, we present a general reduction scheme by which we relate bifurcations from periodic solutions to bifurcations from fixed points of twisted equivariant diffeomorphisms, which in turn are linked via normal form theory to bifurcations from equilibria of equivariant vector fields.We also obtain a general theory for bifurcation from relative periodic solutions and we show how to incorporate time-reversal symmetries into our framework.     

Bifurcation analysis of the flour beetle population growth equations

In this article, we study a discrete delayed flour beetle population equation. Firstly, we study the existence of period-doubling bifurcation and Neimark–Sacker bifurcations for the system by analysing its characteristic equations. Secondly, we investigate the direction of the two bifurcations and the stability of the bifurcation periodic solutions by using normal form theory. Finally, some numerical simulations are carried out to support the analytical results.     

强Duffing系统的局部分岔分析

本文通过坐标变化和近恒等变化，将强Ｄｕｆｆｉｎｇ方程化成范式，从而可以得到在不同共振条件下的分合方程以及其近似解，应用奇异性理论研究了强Ｄｕｆｆｉｎｇ在开折参数及物理参数平面上的转迁集及其局部分岔图．     

Synchronous Dynamics and Bifurcation Analysis in Two Delay Coupled Oscillators with Recurrent Inhibitory Loops

In this study, the dynamics and low-codimension bifurcation of the two delay coupled oscillators with recurrent inhibitory loops are investigated. We discuss the absolute synchronization character of the coupled oscillators. Then the characteristic equation of the linear system is examined, and the possible low-codimension bifurcations of the coupled oscillator system are studied by regarding eigenvalues of the connection matrix as bifurcation parameter, and the bifurcation diagram in the γ–ρ plane is obtained. Applying normal form theory and the center manifold theorem, the stability and direction of the codimension bifurcations are determined. Moreover, the symmetric bifurcation theory and representation theory of Lie groups are used to investigate the spatio-temporal patterns of the periodic oscillations. Finally, numerical results are applied to illustrate the results obtained.     

Hopf Bifurcation Analysis of a Class of Abstract Delay Differential Equation

The dynamics of a class of abstract delay differential equations are investigated. We prove that a sequence of Hopf bifurcations occur at the origin equilibrium as the delay increases. By using the theory of normal form and centre manifold, the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived. Then, the existence of the global Hopf bifurcation of the system is discussed by applying the global Hopf bifurcation theorem of general functional differential equation.     

迟滞型材料阻尼转轴的分岔

应用平均法研究迟滞型材料阻尼转轴的分岔.首先用Hamilton原理推导出复数形式的转轴运动微分方程,然后用平均法求出各阶模态主共振时的平均方程,并分析定常解的稳定性,最后用奇异性理论分析正常运动和失稳运动响应(异步涡动)的分岔.研究表明,一定参数条件下,转轴在通过各阶临界转速(主共振)时,可能会因受到冲击而失稳(Hopf分岔).正常运动响应在不平衡量较大时有滞后和跳跃现象,而失稳运动响应是一类余维数较高的非对称分岔.由于内阻尼的非线性,响应随转速增加时还可能产生二次Hopf分岔,对应原系统的双调幅运动.做好动平衡及提高外阻尼水平是避免这种大幅值自激振动的有效措施.     

Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations

This paper deals with local bifurcations occurring near singular points of planar slow-fast systems. In particular, it is concerned with the study of the slow-fast variant of the unfolding of a codimension 3 nilpotent singularity. The slow-fast variant of a codimension 1 Hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the Van der Pol system. Similarly, codimension 2 slow-fast Bogdanov–Takens bifurcations have been characterized. Here, the singularity is of codimension 3 and we distinguish slow-fast elliptic and slow-fast saddle bifurcations. We focus our study on the appearance on small-amplitude limit cycles, and rely on techniques from geometric singular perturbation theory and blow-up.     

Bifurcation analysis in a neutral differential equation

The dynamics of a neural network model in neutral form is investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. and a Bendixson's criterion for higher dimensional ordinary differential equations due to Li and Muldowney.     

Bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center

Hopf bifurcation which produces oscillations is a very important phenomena in the theory and application of dynamical systems. Almost all works available about Hopf bifurcations are related to a non-degenerate focus or center. For the case of a degenerate focus or center, the study of the bifurcations becomes challenge. In this paper, we consider the bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center. We take coefficients as parameters, then we can get six limit cycles.     

非线性颤振系统中既是超临界又是亚临界的Hopf分岔点研究

研究了二元机翼非线性颤振系统的Hopf分岔．应用中心流形定理将系统降维,并利用复数正规形方法得到了以气流速度为分岔参数的分岔方程．研究发现,分岔方程中一个系数不含分岔参数的一次幂,故使得分岔具有超临界和亚临界双重性质．用等效线性化法和增量谐波平衡法验证了所得结果．