The features of phase and group velocity were advanced by means of preparing samples with different shaping . the phenomenon of " cusp " in group surface of quasi - shear wave ( qsv ) was observed . the inversed velocity surface according to calculated elastic constants in real orthotropic media is consistent with practical measured velocity surface 用加工不同样品形态的方法给出了正交各向异性介质的相速度和群速度特征，观测到各向异性介质群速度面上的尖点现象，并用测试得到的正交介质的弹性常数进行了数学模拟，其波场特征与测试的速度面特征相符。
Phase space reconstruction technique is used to reconstruct the phase plane . the approach to detect the abnormal point of the tune - dependent of the crack by phase space is developed . combined with gray system and catastrophe theory , according to observation data of dam cracks , a gray cusp catastrophe model is established to diagnose the abnormality of the crack ( 5 )利用相空间重构技术重构裂缝时效变形的相平面，提出了裂缝时效转异点的相平面识别法；并且基于突变理论和灰色系统理论，建立了裂缝转异诊断的灰色尖点突变模型；并将以上方法和模型用于工程实际，由此对所提方法的可行性进行了验证。
Fourth , the theoretical model of optimum technological growth rate under discrete change is erected . technology is a unique production factor because technological growth experiences a mutation when technological increase accumulates to a certain extent . under this condition , the point - mutation model in tom ’ s mutation theory is applied to explain the phenomenon of technological 将技术作为不同于劳动和资本的特殊生产要素进行考察，即当技术量的积累达到一定程度时，技术增长会产生突变，在此条件下，运用汤姆突变理论中的尖点突变模型，解释了技术突变现象，讨论了劳动和资本的不同组合变化下，在临界点附近技术突然跃升或下降的突变性质；构建了技术突变完成后的技术增长率优化理论模型。
The strain softening of dam body and foundation is the necessary condition for dam instability . according to observation data of dam cracks , a gray cusp catastrophe model is established to judge the stability of the crack . according to the slope observation data of a certain hydropower plant , a nonlinear dynamic model is developed by means of reversion and the lyapunov exponent spectrum , and the lyapunov information dimension are calculated for judgement of the stability of the slope （ 5 ）基于变分原理，研究建立了能量形式的失稳准则，并说明坝体、岩基的材料具有应变软化的性质，是大坝失稳的必要条件；并利用大坝的裂缝实测资料，建立了相应的灰色尖点突变模型，由此判断裂缝的稳定性；根据某水电站库区滑坡体的变形实测资料，反演其非线性动力学模型，进而计算lyapunov指数谱、 lyapunov信息维来判定边坡的稳定性及稳定程度。
As a result , the curves of c3 and c4 continuity can be generated , and the shape of the curves can be adjusted by the parameters x . the quadratic non - uniform b - spline curves are further extended and the continuity of curves is improved in this paper ; with a local shape parameter in each piecewise curve , the shape of the curves can be controlled effectively ; moreover , cusps of curves can be generated conveniently on the curves while using multiple knots 对二次非均匀b样条作了进一步扩展，提高了曲线的连续性；曲线的每一段上都有一个局部控制参数，利用它们可以更有效的控制曲线的形状；同时，利用曲线的重节点可以很方便的在曲线上构造尖点。作为b样条扩展曲线的应用，作者将上面构造的各次调配函数应用到三次- b样条插值曲线上，得到下述结果。
Basing on the positional characteristics of double cusps produced in the track of link points of planar four - bar linkage and combining with the synthetic equation of conventional track generator , the constraint equation set for the position of track generator that could realize two cusps ; and by means of successive elimination to obtain the most simplified forms of constraint equation set 摘要基于平面四杆机构连杆点轨迹产生双尖点的位置特性，结合常规轨迹发生器综合方程，建立了能实现双尖点的轨迹发生器的位置约束方程组，并经过逐次消元，得到了约束方程组的最简形式。
On the basis of displacement - time series of the slope , a nonlinear dynamic model is set up according to backus generalized linear inversion theory in this paper . due to the equivalence beween autonomous gradient system and catastrophe model , a standard cusp catastrophe model can be obtained through variable substitution . the method is used in analysis of displacement data of huangci landslide and wolongsi landslide and in understanding how slopes evolve before sliding . the result shows that the nonlinear dynamic model can make satisfactory prediction result . is it most important that there is a sudden fall of d , which indicates the occurrence of catastrophe ( when d = 0 ) 研究表明，滑坡变形失稳过程具有混沌和分维特性，可以用分形理论来研究滑坡预测问题，基于对任一连续函数，至少在较小的邻域内可以用多项式任意逼近的数学理论，运用改进的backus广义线性反演理论，以斜坡位移时间序列为基础，反演了斜坡演化的非线性动力学模型。并利用自治梯度系统与突变模型的等价性，通过变量代换得到标准的尖点突变模型。
The curve of deformation against loads is lent to study the symbols , and monitoring models , time - effect exponent , catatrophic models are used to distinguish the abnormal symbols . based upon the study of deformation procedure and its abnormal symbols , structrual analysis method is used to determine the deformation monitoring quota 此外，应用监控模型、时效分量、尖点突变模型等进行变形转异识别；在此基础上，采用结构分析法拟定特殊结构混凝土坝的变形监控指标。
