The Evolution of the Stable and Unstable Manifold of an Equilibrium Point

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter,ν. The eigenvalues of the linearized system are complex for ν < 0 and purely imaginary for ν > 0. Thus for ν < 0 the equilibrium has a two‐dimensional stable manifold and a two‐dimensional unstable manifold, but for ν > 0 these stable and unstable manifolds are gone. We study the system defined by the truncated generic normal form in this situation. One of two things happens depending on the sign of a certain quantity in the normal form expansion. In one case the two families detach as a single invariant manifold and recedes from the equilibrium as ν tends away from 0 through positive values. In the other case the stable and unstable manifold are globally connected for ν < 0 and the whole structure of these manifolds shrinks to the equilibrium as ν → 0 and disappears. These considerations have interesting implications about Strömgren's conjecture in celestial mechanics and the blue sky catastrophe of Devaney.     

Period Doubling Bifurcation of Stress Drop for Stick-slip and Its Physical Implication

It is revealed in frictional experiments on medium-scale samples that period doubling bifurcation of stress drop for stick-slip occurs due to macroscopic heterogeneity of the sliding surface under conditions for typical stick-slip.The observed data show that the period doubling bifurcation of stress drop results from the alternate occurrence of strain release along the whole fault and along part of fault.This implies that complicated nonlinear behavior corresponds to clear physical implication in some cases.     

Nonlinear Dynamics in Atmospheric Chemistry Rate Equations

Nonlinear terms in the gas-phase rate equations of atmospheric trace constituents give rise to unexpected oscillations in the concentrations. For a simplified chemical scheme of the troposphere that contains only the generic reaction paths, the underlying dynamical structure is investigated. It is shown that the source strengths of CO and NO are the controlling parameters of the system. A linear stability analysis reveals that the steady state solutions lose stability due to the occurrence of two Hopf bifurcations. Furthermore, it appears that the dynamical behaviour of the oscillatory solutions is dominated by O₃ and CO. Based on the two-variable system (CO–O₃) it is shown that the oscillatory solution involves an autocatalytic ozone production phase which is followed by a phase in which CO is oxidised quickly. A simple expression is presented from which the period of the oscillation can be obtained. The implications for the present troposphere are unclear, since other hydrocarbons are present and transport is taking place. Nevertheless, the system nicely shows the general nonlinear mechanisms that operate in the tropospheric chemistry equations.     

The Sequence-of-Bifurcations Approach towards Understanding Turbulent Fluid Flow

The sequence-of-bifurcations approach for the analysis of transitionsfrom simple to complex forms of fluid flow is described.Formulated for fluid systemswith external conditions that are homogeneous in two spatial dimensionsand in time,this approach determines bifurcations through theirsymmetry breaking properties.Solutions that are periodic in the homogeneous dimensionsand in time are generatedin this way. While the secondary solution induced by theinstability of the basichomogeneous state generically assumes the form of rolls orstripes, higher bifurcationsintroduce solutions exhibiting structures specific for thephysical system underconsideration. These structures often persist in the formof coherent structures inthe turbulent state of the system. Examples from the caseof thermal convection in alayer heated from below are used to illustrate thesequence-of-bifurcations approach.     

油轮“鱼尾”运动及Hopf分叉算法

本文使用一个六阶三自由度的非线性数学模型，研究单点系泊（ＳＰＭ）系统“鱼尾”运动，对系统的平衡状态，及其在风向角、风速、流速、缆绳长度等参数变化时规律，进行了一系列的分析，同时，指出了静平衡系泊力的变化规律。为了研究系统平衡态的稳定性，文中选择缆绳长度ｌ为分叉参数，分析了在不同参数值下，系统Ｊａｃｏｂｉａｎ矩阵的各组特征值的变化情况，并求出了系统的Ｈｏｐｆ分叉点。最后，本文利用Ｈｏｐｆ分叉理论，求出了系统的Ｈｏｐｆ分叉周期解。     

小间距隧道爆破动力特性试验研究

分岔隧道是一种新型隧道结构形式。以漆树槽分岔隧道为工程背景,进行了掘进爆破的围岩震动效应试验,通过不同掏槽结构下爆破震动作用隧道围岩衬砌质点振动速度的频谱分析与质点振动速度幅值的分布情况分析,对小间距公路隧道混凝土衬砌及其围岩在爆破地震波作用下的力学特征进行论述,重点研究了小间距隧道中隔墙及爆破掌子面的震动特性及变化规律。试验与分析结果表明,（1）中墙迎爆侧为振速峰值区,以水平径向振速最大,且沿着中墙走向呈现规律性变化;（2）掏槽孔爆破产生的地震效应最为强烈,其震动强度是扩槽眼和周边眼的2倍左右;（3）衬砌振速主振频率主要分布在低频段,以20～95 Hz为集中频段,高频地震波衰减极快;（4）同一测试断面中最大振速峰值出现在拱顶和边墙中部位置的概率最大,且大小比较接近,爆破掌子面附近的振速不适合萨道夫斯基预测公式,受爆破夹制及自由面的影响较大。     

铁晶砂胶结新型岩土相似材料的研制及其应用

根据地质力学模型试验的相似原理,经过大量力学试验,研制出一种新型铁晶砂胶结岩土相似材料。该材料由铁矿粉、重晶石粉、石英砂、石膏粉和松香酒精溶液按规定配比均匀拌和压实而成,其中铁矿粉、重晶石粉和石英砂作为主料,松香酒精溶液作为胶结剂,石膏粉作为调节剂。材料力学参数测试表明：该材料具有重度高、力学参数变化范围广、性能稳定、价格低廉、干燥快速、制作工艺简单、无毒无害等显著优点,可用来模拟从软岩到硬岩的大部分岩体材料。将铁晶砂胶结岩土相似材料应用于沪―蓉―西高速公路大型分岔隧道三维地质力学模型试验研究,有效地揭示了分岔隧道围岩的力学变形特性。     

顺层边坡岩体结构的非完善屈曲性态研究

视顺层边坡岩体自重的法向分量为结构的初始缺陷,应用初始后屈曲理论和尖点突变模型探讨了缺陷结构在临界状态下的屈曲性态,建立了边坡岩体结构的平衡稳定模态幅值方程,给出了边坡岩体结构分叉点附近的邻域及其稳定性判据。研究结果表明,表征结构初始缺陷的修正项与尖点突变模型中的剖分因子之间具有确定的关系;与理想化完善结构不同的是,顺层边坡岩体结构屈曲的临界值远大于其分叉点处的载荷值,是由其突变模型的分叉集确定的;只有考虑了边坡的实际缺陷,才可得出边坡岩体结构在真正临界意义上的临界值。     

考虑损伤效应的岩石类材料局部化特性分析

在各向同性损伤条件下,考虑岩石损伤过程中的刚度降低和体积扩容,通过理论推导建立了岩石发生分叉失稳时的最大硬化模量和临界局部化方向,探讨了局部化方向角对于岩石的损伤程度和初始泊松比的依赖关系,并对平面应力与平面应变两种条件下单轴拉伸试件的分叉特性进行了对比分析。研究表明,岩石的初始泊松比与损伤程度越大,局部化方向角越大;尽管平面应力与平面应变条件下单轴拉伸问题的局部化方向角具有相似的变化趋势,但平面应力条件下的局部化方向角低于平面应变条件下同样情况下的局部化方向角。     

南海北部次表层盐度的变化及其与北赤道流分叉点的关系

吕宋海峡是南海与外界水交换的重要通道,黑潮作为北太平洋最强的1支西边界流,在经过吕宋海峡时会对南海北部的环流和环境产生重要影响。本文用1991—2011年期间CTD断面实测资料和高度计资料,提取23.0~25.5 kg/m³等密度面之间的盐度极大值,研究了南海北部不同年月盐度极大值变化、黑潮入侵方式与强弱,以及盐度极大值变化与北赤道流分叉点南北移动的关系,结果表明:(1)黑潮入侵南海方式多样,既有分支形式,也有弯曲、流套形式。(2)不同年月间,黑潮入侵南海的强弱存在较大差别,120°E断面的次表层盐度极大值的变动可超过0.3。(3)北赤道流分叉点位置的南北变动对黑潮入侵南海的强弱具有重要影响:北赤道流分叉点位置偏北,黑潮入侵南海较强;北赤道流分叉点位置偏南,则黑潮入侵相对较弱。