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.     

Point-to-curve ray tracing

Point-to-curve ray tracing is an attempt at dealing with multiplicity of solutions to a generic boundary-value problem of ray tracing. In a point-to-curve tracing (P2C) the input parameters of the boundary-value problem (BVP), such as the ends of the ray, are allowed to vary along a curve. The solutions of the BVP automatically wander from one solution branch to another generating a nearly complete multi-valued solution of the BVPs.A procedure for transforming an arbitrary iterative algorithm, solving a ray tracing BVP to a corresponding P2C algorithm, is presented. Bifurcations of the solution curve of the P2C problem at caustics are studied and an algorithm for obtaining the bifurcating branches is developed. In particular, transition from real rays to complex rays in a caustic shadow offers an additional link between otherwise disconnected solution curves of the P2C problem. The topological structure of a generic solution curve and its implications for the algorithm are studied.     

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.     

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.     

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.     

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。     

A three‐dimensional numerical model of sediment transport,erosion and deposition within a network of channel belts,floodplain and hill slope: extrinsic and intrinsic controls on floodplain dynamics and alluvial architecture

A three‐dimensional numerical model of sediment transport, erosion and deposition within a network of channel belts and associated floodplain is described. Sediment and water supply are defined at the upstream entry point, and base level is defined at the downstream edge of the model. Sediment and water are transported through a network of channels according to the diffusion equation, and each channel has a channel belt with a width that increases in time. The network of channels evolves as a result of channel bifurcation and abandonment (avulsion). The timing and location of channel bifurcation is controlled stochastically as a function of the cross‐valley slope of the floodplain adjacent to the channel belt relative to the down‐valley slope, and of annual flood discharge. A bifurcation develops into an avulsion when the discharge of one of the distributaries falls below a threshold value. The floodplain aggradation rate decreases with distance from the nearest active channel belt. Channel‐belt degradation results in floodplain incision. Extrinsic (extrabasinal, allogenic) and intrinsic (intrabasinal, autogenic) controls on floodplain dynamics and alluvial architecture were modelled, and sequence stratigraphy models were assessed. Input parameters were chosen based on data from the Rhine–Meuse delta. To examine how the model responds to extrinsic controls, the model was run under conditions of changing base level and increasing sediment supply. Rises and falls in base level and increases in sediment supply occurred over 10 000 years. Rising base level caused a wave of aggradation to move up‐valley, until aggradation occurred over the entire valley. Frequency of bifurcations and avulsions increased with rate of base‐level rise and aggradation rate. Channel‐belt width varied with water discharge and the lifespan of the channel belt. Wide, connected channel belts (and high channel‐deposit proportion) occurred around the upstream inflow point because of their high discharge and longevity. Less connected, smaller channel belts occurred further down‐valley. Such alluvial behaviour and architecture is also found in the Rhine–Meuse delta. During base‐level fall, valley erosion occurred, and the incised valley contained a single wide channel belt. During subsequent base‐level rise, a wave of aggradation moved up‐valley, filling the incised valley. Bifurcation and avulsion sites progressively moved upstream. Relatively thin, narrow channel belts bordered and cut into the valley fill. These results differ substantially from existing sequence stratigraphy models. The increase in sediment supply from upstream resulted in an alluvial fan. Most bifurcations and avulsions occurred at the fan apex (nodal avulsion), and channel belts were the widest and the thickest here (giving high channel‐deposit proportion) due to their high discharge and longevity. The width and thickness of channel belts decreased down‐valley due to decreased discharge, longevity and aggradation rate. This behaviour occurs in modern alluvial fans. Intrinsic controls also affect floodplain dynamics and alluvial architecture. Variation of aggradation rate, bifurcation frequency and number of coexisting channel belts occurred over periods of 500 to 2000 years, compared with 10 000 years for extrinsic controls. This variation is partly related to local aggradation and degradation of channel belts around bifurcation points. Channel belts were preferentially clustered near floodplain margins, because of low floodplain aggradation rate and topography there.     

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。     

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。