A Dynamical System-Based Framework for Dimension Reduction

We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems,which we call the dynamical dimension reduction(DDR).In the DDR model,each point is evolved via a nonlinear flow towards a lower-dimensional subspace;the projection onto the subspace gives the low-dimensional embedding.Training the model involves identifying the nonlinear flow and the subspace.Following the equation discovery method,we represent the vector field that defines the flow using a linear combination of dictionary elements,where each element is a pre-specified linear/nonlinear candidate function.A regularization term for the average total kinetic energy is also introduced and motivated by the optimal transport theory.We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method.We also show how the DDR method can be trained using a gradient-based optimization method,where the gradients are computed using the adjoint method from the optimal control theory.The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods,including the PCA,t-SNE,and Umap.

Computational fluid dynamics simulations of respiratory airflow in human nasal cavity and its characteristic dimension study

To study the airflow distribution in human nasal cavity during respiration and the characteristic parameters of nasal structure, three-dimensional, anatomically accurate representations of 30 adult nasal cavity models were recons- tructed based on processed tomography images collected from normal people. The airflow fields in nasal cavities were simulated by fluid dynamics with finite element software ANSYS. The results showed that the difference of human nasal cavity structure led to different airflow distribution in the nasal cavities and variation of the main airstream passing through the common nasal meatus. The nasal resistance in the regions of nasal valve and nasal vestibule accounted for more than half of the overall resistance. The characteristic model of nasal cavity was extracted on the basis of characteristic points and dimensions deduced from the original models. It showed that either the geometric structure or the airflow field of the two kinds of models was similar. The characteristic dimensions were the characteristic parameters of nasal cavity that could properly represent the original model in model studies on nasal cavity.

Lyapunov Exponents and General Dimensions of Strange Attractor of Electrodynamic Cone Loudspeaker

Lyapunov exponents,Lyapunov dimension and general dimensions have been determined from a time series of electrodynamic cone loudspeaker.It is found that a low-dimensional chaotic at tractor has one positive largest exponent,and the Lyapunov dimension is in concordance with the Hausdorff dimension which calculated by general correlation integrated method.

Dynamic Parallel Method for Eulerian Method of Elastoplastic Hydrodynamics in Three Dimension

Eulerian method is a main numerical simulation method in elastoplastic hydrodynamics, which is suitable for the problems with multi-component and large deformation. As the feature of the problems to be simulated, such as detonation and penetration, the dynamic parallel method （DPM） is designed to adjust the computational domain dynamically to get better load balance. Dynamic parallel method can be separated into two parts： one is division of initial computational domain and location of the data, the other is expansion of the computational domain and adjustment of the data location. DPM program can greatly shorten computational time and be preferable in simulating actual problems. The speedup of the DPM program is linear in parallel test. DPM can be popularized to parallel program of other multi-component high dimension Eulerian methods naturally.

A New Dynamics Analysis Model for Five-Axis Machining of Curved Surface Based on Dimension Reduction and Mapping

The equipment used in various fields contains an increasing number of parts with curved surfaces of increasing size.Five-axis computer numerical control(CNC)milling is the main parts machining method,while dynamics analysis has always been a research hotspot.The cutting conditions determined by the cutter axis,tool path,and workpiece geometry are complex and changeable,which has made dynamics research a major challenge.For this reason,this paper introduces the innovative idea of applying dimension reduction and mapping to the five-axis machining of curved surfaces,and proposes an efficient dynamics analysis model.To simplify the research object,the cutter position points along the tool path were discretized into inclined plane five-axis machining.The cutter dip angle and feed deflection angle were used to define the spatial position relationship in five-axis machining.These were then taken as the new base variables to construct an abstract two-dimensional space and establish the mapping relationship between the cutter position point and space point sets to further simplify the dimensions of the research object.Based on the in-cut cutting edge solved by the space limitation method,the dynamics of the inclined plane five-axis machining unit were studied,and the results were uniformly stored in the abstract space to produce a database.Finally,the prediction of the milling force and vibration state along the tool path became a data extraction process that significantly improved efficiency.Two experiments were also conducted which proved the accuracy and efficiency of the proposed dynamics analysis model.This study has great potential for the online synchronization of intelligent machining of large surfaces.

Assessment of flexural properties of different grade dimension lumber by ultrasonic technique

The dimension lumber （45mm×90mm×3700mm） of plantation Chinese fir （Cunninghamia lanceolata （Lamb.） Hook.） was graded to four different classes as SS, No. 1, No.2 and No.3, according to national lumber grades authority （NLGA） for structure light framing and structure joists and planks. The properties of apparent density was determined at 15% moisture content, bending strength and stiffness were tested according to American Society for Testing and Materials （ASTM） D198-99, and dynamic modulus of elasticity （Eusw） was measured by ultrasonic technique, for predicting the flexural properties of different grade lumbers. The results showed that Eosw was larger than the static MOE. The relationship between Eusw and static MOE was significant at 0.01 level, and the determination coefficients （R2） of the four grade lumbers followed the sequence as R^2No.2 （0.616）〉 R^2ss （0.567）〉 R^2No1 （0.366）〉 R^2No.3 （0.137）. The R^2 of Fusw and MOR were lower than that of the Etru and MOR for each grade. The Eusw of all the grade lumbers, except No.3-grade, had significant correlation with the static MOE and MOR, thus the bending strengthof those grade lumbers can be estimated by the E The Etru valuesof four grade lumbers followed a sequence as No.2-grade （10.701 GPa） 〉 SS-grade （10.359 GPa） 〉 No.l-grade （9.840 GPa） 〉 No.3-grade （9.554 GPa）. For the same grade dimension lumber, its Eusw value was larger than static MOE. Mean values of MOR for four grade lumbers follow a sequence as No.2-grade （48.67 MPa） 〉 SS-grade （48.16 MPa） 〉 No.3-grade （46.55 MPa） 〉 No. 1-grade （43.39MPa）.

Stability of Hausdorff dimension of Axiom A attractors under random perturbations

This paper proves that the Hausdorff dimension of an Axiom A attractor is stable under random perturbations.

THE RELATION OF DIMENSION, ENTROPY AND LYAPUNOV EXPONENT IN RANDOM CASE

We consider random systems generated by two-sided compositions of random surface diffeomorphisms, together with an ergodic Borel probability measure μ. Let D（μω） be its dimension of the sample measure, then we prove a formula relating D（μω） to the entropy and Lyapunov exponents of the random system, where D （μω） is dimHμω, dimBμm, or dimBμm.

含初始损伤饱水花岗岩的冲击破坏规律

为研究饱水和初始损伤对冲击荷载下花岗岩宏观和微观破坏特征的影响,开展了X射线衍射、霍普金森和扫描电镜试验,利用分形维数对花岗岩的破碎块度和断口形貌进行了分析,探讨了图像放大倍数对分形维数的影响,分析了冲击荷载下饱水后花岗岩的微观致裂机制。结果表明:饱水后花岗岩中角闪石、钠长石、微斜长石和石英的占比减少,高岭石占比显著提高;随着初始损伤的增大,花岗岩的动态峰值应力逐渐减小,而破碎程度和块度分形维数逐渐增大,且初始损伤对块度分形维数的影响大于饱水的影响;随着初始损伤的增加,断口出现更多的微裂纹和碎屑,断口图像的分形维数也逐渐增加;放大倍数在400~3200范围内时,断口图像分形维数随着图像放大倍数的增大而增加,超过3200后,分形维数减小。

Grey Relation between Nonlinear Characteristic and Dynamic Uncertainty of Rolling Bearing Friction Torque

The rolling bearing friction torque which is characterized by its uncertainty and nonlinearity affects heavily the dynamic performance of a system such as missiles, spacecrafts and radars, etc. It is difficult to use the classical statistical theory to evaluate the dynamic evaluation of the rolling bearing friction torque for the lack of prior information about both probability distribution and trends. For this reason, based on the information poor system theory and combined with the correlation dimension in chaos theory, the concepts about the mean of the dynamic fluctuant range （MDFR） and the grey relation are proposed to resolve the problem about evaluating the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque. Friction torque experiments are done for three types of the rolling bearings marked with HKTA, HKTB and HKTC separately; meantime, the correlation dimension and MDFR are calculated to describe the nonlinear characteristic and the dynamic uncertainty of the friction torque, respectively. And the experiments reveal that there is a certain grey relation between the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque, viz. MDFR will become the nonlinear increasing trend with the correlation dimension increasing. Under the condition of fewer characteristic data and the lack of prior information about both probability distribution and trends, the unitive evaluation for the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque is realized with the grey confidence level of 87.7%-96.3%.

A STUDY OF DYNAMIC CAUSTICS AROUND RUNNING INTERFACE CRACK TIP

Based on the first invariant of stress singular field in the vicinity of running tip of an interface crack, mapping equations of the caustic curve on the reference plane and the initial curve on the specimen plane are developed. The dynamic caustics are analyzed for the crack propagating along the interface between two bonded dissimilar materials. The variation of the caustic configurations is shown with the velocity change of the running crack and the ratio change of the stress intensity factors. Two characteristic dimensions are proposed that are not only practically measurable from optical caustic contours but also suitable to represent the behavior of transient caustics.

A METHOD FOR CALCULATING THE LYAPUNOV EXPONENT SPECTRUM OF A PERIODICALLY EXCITED NON-AUTONOMOUS DYNAMICAL SYSTEM

The relation between the Lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the definition of the Lyapunov dimension D-L((A)) of the autonomous system, the definition of the Lyapunov dimension D-L of the non-autonomous dynamical system is also given, and the difference between them is the integer 1, namely, D-L((A)) - D-L = 1. For a quasi-periodically excited dynamical system, similar conclusions are formed.

3-D fracture network dynamic simulation based on error analysis in rock mass of dam foundation

Accurate 3-D fracture network model for rock mass in dam foundation is of vital importance for stability,grouting and seepage analysis of dam foundation.With the aim of reducing deviation between fracture network model and measured data,a 3-D fracture network dynamic modeling method based on error analysis was proposed.Firstly,errors of four fracture volume density estimation methods(proposed by ODA,KULATILAKE,MAULDON,and SONG)and that of four fracture size estimation methods(proposed by EINSTEIN,SONG and TONON)were respectively compared,and the optimal methods were determined.Additionally,error index representing the deviation between fracture network model and measured data was established with integrated use of fractal dimension and relative absolute error(RAE).On this basis,the downhill simplex method was used to build the dynamic modeling method,which takes the minimum of error index as objective function and dynamically adjusts the fracture density and size parameters to correct the error index.Finally,the 3-D fracture network model could be obtained which meets the requirements.The proposed method was applied for 3-D fractures simulation in Miao Wei hydropower project in China for feasibility verification and the error index reduced from 2.618 to 0.337.

Stability and control of dynamic walking for a five-link planar biped robot with feet

During dynamic walking of biped robots, the underactuated rotating degree of freedom （DOF） emerges between the support foot and the ground, which makes the biped model hybrid and dimension-variant. This paper addresses the asymptotic orbit stability for dimension-variant hybrid systems （DVHS）. Based on the generalized Poincare map, the stability criterion for DVHS is also presented, and the result is then used to study dynamic walking for a five-link planar biped robot with feet. Time-invariant gait planning and nonlinear control strategy for dynamic walking with fiat feet is also introduced. Simulation results indicate that an asymptotically stable limit cycle of dynamic walking is achieved by the proposed method.

RESEARCH ON REBUILDING OZONE DYNAMIC SYSTEM OF ZONAL AVERAGE OVER THE TROPIC ZONE

The ozone data observed by TOMS in every 5°N are extended into the phase space to describe the characteristics of ozone with phase trace. First of all, the fractional dimension of the ozone layer is calculated. Then.the phase points are regarded as some discrete characteristics solution, and the parameters of mathematical model which describe the time variation of system state are retrieved, so that the nonlinear dynamic system which reflects the short-term variation of zonal average ozone layer over the tropics is rebuilt.

Research on dynamics patterns and parameter characters of crustal deformation field before and after strong earthquake

Based on the dynamic monitoring data of crustal deformation, the parameter evolution for the dynamics pattern and fractal dimension of crustal deformation field and the integral activity level of many faults etc. before and after the Tangshan (1976) and Lijiang (1996) strong earthquakes and others are studied by using the method of pattern dynamics. It is exposed that two time space characters, the ordered dimension drop of the deformation field and the accelerated motion of multi fault before an earthquake, are probably caused by the deformation localization and fault softening after the seismogenic process enters the nonlinear stage. They could be an important seismic precursor if they occurred repeatedly before strong earthquakes.

Nonlinear Dynamic Analysis of MPEG-4 Video Traffic

The main research motive is to analysis and to veiny the inherent nonlinear character of MPEG-4 video. The power spectral density estimation of the video trafiic describes its 1/f^β and periodic characteristics.The priraeipal compohems analysis of the reconstructed space dimension shows only several principal components can be the representation of all dimensions. The correlation dimension analysis proves its fractal characteristic. To accurately compute the largest Lyapunov exponent, the video traffic is divided into many parts.So the largest Lyapunov exponent spectrum is separately calculated using the small data sets method. The largest Lyapunov exponent spectrum shows there exists abundant nonlinear chaos in MPEG-4 video traffic. The conclusion can be made that MPEG-4 video traffic have complex nonlinear be havior and can be characterized by its power spectral density,principal components, correlation dimension and the largest Lyapunov exponent besides its common statistics.

Dynamic crossover in [VIO^(2+)][Tf_(2)N-]_(2) ionic liquid

Ionic liquids usually behave as fragile liquids,and the temperature dependence of their dynamic properties obeys supper-Arrhenius law.In this work,a dynamic crossover is observed in([VIO^(2+)][Tf_(2)N-]_(2)) ionic liquid at the temperature of 240-800 K.The diffusion coefficient does not obey a single Arrhenius law or a Vogel-Fulcher-Tammann(VFT) relation,but can be well fitted by three Arrhenius laws or a combination of a VFT relation and an Arrhenius law.The origin of the dynamic crossover is analyzed from correlation,structure,and thermodynamics.Ion gets a stronger backward correlation at a lower temperature,as shown by the fractal dimension of the random walk.The temperature dependence function of fractal dimension,heterogeneity order parameter,and thermodynamic data can be separated into three regions similar to that observed in the diffusion coefficient.The two crossover temperatures observed in the three types of data are almost the same as that in diffusion coefficient fitted by three Arrhenius laws.The results indicate that the dynamic crossover of[VIO2+][Tf2 N-]2 is attributed to the heterogeneous structure when it undergoes cooling.

A Topological Transformation of Quantum Dynamics

In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K?× S1?defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K?× S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.

循环冲击荷载作用下煤岩组合体力学响应和能量耗散特征研究

为探究循环冲击下煤岩组合体的力学响应与能量耗散特征,采用分离式霍普金森压杆试验系统,设计开展5种不同冲击速度的煤岩组合体循环冲击压缩试验,对比分析不同冲击速度下组合体动态应力应变关系、能量耗散特性和破坏特征。研究结果表明:循环冲击下煤岩组合体存在明显的临界效应,冲击速度低于临界速度时动态抗压强度在冲击前期迅速降低,峰值应变变化率随冲击速度增加而降低;循环冲击下试件总吸收能和分形维数逐渐增大,在临界速度时达到峰值,超过临界速度后两者皆降低;冲击速度超过临界速度后,总吸收能相同时循环冲击条件下煤岩体破坏程度高于单次冲击;当煤岩强度比为1∶2时,随着总吸收能的增加,岩体从积聚和传递能量作用转向于加剧组合体破坏;冲击作用下组合体呈张拉劈裂破坏,低速条件下宏观裂纹萌发主要发生在远离胶结面的煤体端面,随着冲击速度增大远离胶结面岩体一侧亦有裂纹出现,根据失稳发生区域组合体破坏类型分为煤体失稳型和组合体失稳型,循环冲击下均发生组合体失稳型破坏。研究结果为软弱岩层动压巷道与采场围岩稳定性控制提供基础研究。