正弦锁相环路方程的调和解、浑沌与分支

讨论调频输入正弦锁相环路方程的调和解、浑饨与分支。利用Melnikov方法确定了产生浑沌与次谐波分支及其稳定性的条件，同时我们证明了当方程的参数适当小时，它必存在调和解。     

An automated strategy for calculation of phase diagram sections and retrieval of rock properties as a function of physical conditions

We formulate an algorithm for the calculation of stable phase relations of a system with constrained bulk composition as a function of its environmental variables. The basis of this algorithm is the approximate representation of the free energy composition surfaces of solution phases by inscribed polyhedra. This representation leads to discretization of high variance phase fields into a continuous mesh of smaller polygonal fields within which the composition and physical properties of the phases are uniquely determined. The resulting phase diagram sections are useful for understanding the phase relations of complex metamorphic systems and for applications in which it is necessary to establish the variations in rock properties such as density, seismic velocities and volatile‐content through a metamorphic cycle. The algorithm has been implemented within a computer program that is general with respect to both the choice of variables and the number of components and phases possible in a system, and is independent of the structure of the equations of state used to describe the phases of the system.     

Melting of β-quartz up to 2.0 GPa and thermodynamic optimization of the silica liquidus up to 6.0 GPa

Melting relations of β-quartz were experimentally determined at 1.0 GPa (1900±20 °C), 1.5 GPa (2033±20 °C), and 2.0 GPa (2145±20 °C) using a new high-pressure assembly in a piston–cylinder apparatus and substantial differences were found with data previously reported. The new melting data of β-quartz were combined and optimized with all available thermodynamic, volumetric, and phase equilibria data for β-cristobalite, β-quartz and coesite to produce a P–T liquidus diagram for silica valid up to 6.0 GPa. Using the new optimized thermodynamic parameters, the invariant point β-cristobalite+β-quartz+liquid and β-quartz+coesite+liquid were determined to lie at 1687±17 °C and 0.457 GPa, and 2425±25 °C and 5.00 GPa, respectively.     

Practical VTI approximations: a systematic anatomy

Transverse isotropy (TI) with a vertical symmetry axis (VTI) often provides an appropriate earth model for prestack imaging of steep-dip reflection seismic data. Exact P-wave and SV-wave phase velocities in VTI media are described by complicated equations requiring four independent parameters. Estimating appropriate multiparameter earth models can be difficult and time-consuming, so it is often useful to replace the exact VTI equations with simpler approximations requiring fewer parameters. The accuracy limits of different previously published VTI approximations are not always clear, nor is it always obvious how these different approximations relate to each other. Here I present a systematic framework for deriving a variety of useful VTI approximations. I develop first a sequence of well-defined approximations to the exact P-wave and SV-wave phase velocities. In doing so, I show how the useful but physically questionable heuristic of setting shear velocities identically to zero can be replaced with a more precise and quantifiable approximation. The key here to deriving accurate approximations is to replace the stiffness a₁₃ with an appropriate factorization in terms of velocity parameters. Two different specific parameter choices lead to the P-wave approximations of Alkhalifah (Geophysics 63 (1998) 623) and Schoenberg and de Hoop (Geophysics 65 (2000) 919), but there are actually an infinite number of reasonable parametrizations possible. Further approximations then lead to a variety of other useful phase velocity expressions, including those of Thomsen (Geophysics 51 (1986) 1954), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Harlan (Stanford Exploration Project Report 89 (1995) 145), and Stopin (Stopin, A., 2001. Comparison of v(θ) equations in TI medium. 9th International Workshop on Seismic Anisotropy). Each P-wave phase velocity approximation derived this way can be paired naturally with a corresponding SV-wave approximation. Each P-wave or SV-wave phase velocity approximation can then be converted into an equivalent dispersion relation in terms of horizontal and vertical slownesses. A simple heuristic substitution also allows each phase velocity approximation to be converted into an explicit group velocity approximation. From these, in turn, travel time or moveout approximations can also be derived. The group velocity and travel time approximations derived this way include ones previously used by Byun et al. (Geophysics 54 (1989) 1564), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Tsvankin and Thomsen (Geophysics 59 (1994) 1290), Harlan (89 (1995) 145), and Zhang and Uren (Zhang, F. and Uren, N., 2001. Approximate explicit ray velocity functions and travel times for P-waves in TI media. 71st Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, 106–109).     

Computational study of tetrahedral Al–Si and octahedral Al–Mg ordering in phengite

 As part of a wider study of the nature and origins of cation order–disorder in micas, a variety of computational techniques have been used to investigate the nature of tetrahedral and octahedral ordering in phengite, K₂ ^[6](Al₃Mg)^[4](Si₇Al)O₂₀(OH)₄. Values of the atomic exchange interaction parameters J _n used to model the energies of order–disorder were calculated. Both tetrahedral Al–Si and octahedral Al–Mg ordering were studied and hence three types of interaction parameter were necessary: for T–T, O–O and T–O interactions (where T denotes tetrahedral sites and O denotes octahedral sites). Values for the T–T and O–O interactions were taken from results on other systems, whilst we calculated new values for the T–O interactions. We have demonstrated that modelling the octahedral and tetrahedral sheets alone and independently produces different results from modelling a whole T–O–T layer, hence justifying the inclusion of the T–O interactions. Simulations of a whole T–O–T layer of phengite indicated the presence of short-range order, but no long-range order was observed. Received: 8 August 2002 / Accepted: 14 February 2003 Acknowledgements The authors are grateful to EPSRC (EJP) and the Royal Society (CIS) for financial support. Monte Carlo simulations were performed on the Mineral Physics Group's Beowulf cluster and the University of Cambridge's High Performance Computing Facility.     

Separate Analysis of Remote Sensing Information of Structures of Different Geological Periods and Quantitative Study of Corresponding Tectonic Stress Fields

The structural feature shown on a remote sensing image is a synthetic result ofcombination of the deformations produced during the entire geological history of an area.Therefore, the respective tectonic stress field of each of the different stages in the complexdeformation of an area can be reconstructed in three steps: (1) geological structures formed atdifferent times are distinguished in remote sensing image interpretation; (2) structuraldeformation fields at different stages are determined by analyzing relationships betweenmicrostructures (joints and fractures) and the related structures (folds and faults); and (3)tectonic stress fields at different stages are respectively recovered through a study of the featuresof structural deformation fields in different periods. Circular structures and related circlular and radial joints are correlated in space to con-cealed structural rises. The authors propose a new method for establishing a natural model ofthe concealed structural rises and calculating the tectonic stress field by using quantitative dataof the remote sensing information of circular structures and related linear structures.     

High-T phase transition of synthetic ANaB(LiMg)CMg5Si8O22(OH)2 amphibole: an X-ray synchrotron powder diffraction and FTIR spectroscopic study

The synthetic amphibole Na_0.95(Li_0.95Mg_1.05)Mg₅Si₈O₂₂(OH)₂ was studied in situ at high-T, using IR OH-stretching spectroscopy and synchrotron X-ray powder diffraction. At room-T the sample has P2₁ /m symmetry, as shown by the FTIR spectrum. It shows in the OH region two well-defined and intense absorptions at 3,748 and 3,712 cm⁻¹, respectively, and two minor bands at 3,667 and 3,687 cm⁻¹. The main bands are assigned to the two independent O–H groups in the primitive structure. The two minor bands evidencing the presence of small amount of vacant A-site (^A□_0.05). With increasing T, these bands shift continuously and merge into a unique absorption at high temperature. A change as a function of increasing T is revealed by the evolution of the refined unit-cell parameters, whose trend shows a transition to C2/m at about 320–330°C. The spontaneous scalar strain, fitted with a tricritical 2–6 Landau potential, gives a T _c of 325(10)°C (β parameter = 0.27). Comparison with the second-order P2₁ /m ⇔ C2/m phase transition at 255°C for synthetic amphibole ^ANa_0.8^B(Na_0.8Mg_1.2)^CMg₅Si₈O₂₂(OH)₂ indicates that the substitution of Na with Li at the B-sites strongly affects the thermodynamic character and the T _c of the phase transition. The comparison of LNMSH amphiboles with cummingtonitic ones shows that the high-T thermodynamic behaviour is affected by A-site occupancy.