Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

Integration and magnitude homogenization of the Egyptian earthquake catalogue

The aim of the present work is to compile and update a catalogue of the instrumentally recorded earthquakes in Egypt, with uniform and homogeneous source parameters as required for the analysis of seismicity and seismic hazard assessment. This in turn requires a detailed analysis and comparison of the properties of different available sources, including the distribution of events with time, the magnitude completeness, and the scaling relations between different kinds of magnitude reported by different agencies. The observational data cover the time interval 1900–2004 and an area between 22°–33.5° N and 25°–36° E. The linear regressions between various magnitude types have been evaluated for different magnitude ranges. Using the best linear relationship determined for each available pair of magnitudes, as well as those identified between the magnitudes and the seismic moment, we convert the different magnitude types into moment magnitudes M _W, through a multi-step conversion process. Analysis of the catalogue completeness, based on the M _W thus estimated, allows us to identify two different time intervals with homogeneous properties. The first one (1900–1984) appears to be complete for M _W ≥ 4.5, while the second one (1985–2004) can be considered complete for magnitudes M _W ≥ 3.     

汶川特大地震后成都盆地内隐伏断层活动性分析

成都盆地内主要有3条隐伏的活动断裂带,包括大邑断裂带、蒲江-新津断裂带和龙泉山断裂带,它们在第四纪都表现出了一定的活动性."512"汶川特大地震后,笔者实地考察的结果表明,在汶川特大地震中成都盆地中的隐伏断裂没有产生新的活动性,目前成都盆地不存在发生特大地震的危险性,是安全的.     

Seismic hazard of Egypt

Earthquake hazard parameters such as maximum expected magnitude,M _max, annual activity rate,, andb value of the Gutenberg-Richter relation have been evaluated for two regions of Egypt. The applied maximum likelihood method permits the combination of both historical and instrumental data. The catalogue used covers earthquakes with magnitude 3 from the time interval 320–1987. The uncertainties in magnitude estimates and threshold of completeness were taken into account. The hazard parameter determination is performed for two study areas. The first area, Gulf of Suez, has higher seismicity level than the second, all other active zones in Egypt.b-values of 1.2 ± 0.1 and 1.0 ± 0.1 are obtained for the two areas, respectively. The number of annually expected earthquakes with magnitude 3 is much larger in the Gulf of Suez, 39 ± 2 than in the other areas, 6.1 ± 0.5. The maximum expected magnitude is calculated to be 6.5 ± 0.4 for a time span of 209 years for the Gulf of Suez and 6.1 ± 0.3 for a time span of 1667 years for the remaining active areas in Egypt. Respective periods of 10 and 20 years were reported for earthquakes of magnitude 5.0 for the two subareas.     

On some problems of seismic crustal phase

ＯｎｓｏｍｅｐｒｏｂｌｅｍｓｏｆｓｅｉｓｍｉｃｃｒｕｓｔａｌｐｈａｓｅＨｕａｎ－ＣｈｅｎｇＧＥ（葛焕称）（ＳｅｉｓｍｏｌｏｇｉｃａｌＢｕｒｅａｕｏｆＪｉａｎｇｓｕＰｒｏｖｉｎｃｅ,Ｎａｎｊｉｎｇ２１００１４,Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓ...     

New Generation of Probabilistic Seismic Hazard Assessment for the Area Cologne/Aachen Considering the Uncertainties of the Input Data

A new earthquake catalogue for central, northern and northwestern Europe with unified M_w magnitudes, in part derived from chi-square maximum likelihood regressions, forms the basis for seismic hazard calculations for the Lower Rhine Embayment. Uncertainties in the various input parameters are introduced, a detailed seismic zonation is performed and a recently developed technique for maximum expected magnitude estimation is adopted and quantified. Applying the logic tree algorithm, resulting hazard values with error estimates are obtained as fractile curves (median, 16% and 84% fractiles and mean) plotted for pga (peak ground acceleration; median values for Cologne 0.7 and 1.2 m/s² for probabilities of exceedence of 10% and 2%, respectively, in 50 years), 0.4 s (0.8 and 1.5 m/s²) and 1.0 s (0.3 and 0.5 m/s²) pseudoacclerations, and intensity (I₀ = 6.5 and 7.2). For the ground motion parameters, rock foundation is assumed. For the area near Cologne and Aachen, maps show the median and 84% fractile hazard for 2% probability of exceedence in 50 years based on pga (maximum median value about 1.5 m/s²), and 0.4 s (>2 m/s²) and 1.0 s (about 0.8 m/s²) pseudoaccelerations, all for rock. The pga 84% fractile map also has a maximum value above 2 m/s² and shows similarities with the median map for 0.4 s. In all maps, the maximum values fall within the area 6.2–6.3° E and 50.8–50.9° N, i.e., east of Aachen.     

Flood seasonality and generating conditions in the Tay catchment, Scotland from 1200 to present

The new maximum recorded river flows in Scotland since 1988 have triggered widespread interest in whether floods are becoming more frequent and in the conditions that generate floods of different magnitudes and frequencies. There are questions about the longer-term variability in flood-generating characteristics, and whether there are past analogues for present hydroclimatic variability. The present paper builds on previous work reconstructing a detailed historic flood chronology for the Tay, the largest catchment in Scotland, and its tributaries over the past 800 years, extending the gauged discharge record (1952 onwards). It categorizes flood-generating factors in the Tay catchment and analyses the hydro-meteorological conditions that have generated extreme and moderate floods over a historical period. This work is placed in a broader literature context of historical 'climaxes of storminess', periods of higher storm frequency, flood patterns observed in Scotland and Europe during the Little Ice Age and longer-term rainfall and temperature patterns. The paper concludes that the variability in flood-generating characteristics is highly dependent on the timescale of observation. Inevitably the relative dominance of winter and early spring flooding can vary from year to year and within specific time-periods, but so can the level of augmentation of the flood series with summer and autumn floods to produce notable 'flood years' and flood clusters. The Tay provides a good 'all-Scotland surrogate' for historical flood patterns, reflecting its gathering areas in eastern and western Scotland. The value of a historical approach to the assessment of flood seasonality and generating characteristics is clearly demonstrated.     

以2000年4月29日河南内乡地震为例谈谈关于计算震级误差的有关问题

2000年4月29日河南内乡发生地震，地这次地震，河南地震局台网和中国地震局台网震级量度之间有一定差异。采用不同量规函数和尾波持续时间震级公式计算发现，它们之间所产生的最大差异可达0.5-0.7级。综合分析后认为，2000年4月29日河南内乡地震震级定为ML5.0级较为适宜。     

黄海及其沿岸历史地震初步研究

黄海及其边缘，经历史地震资料整理，地震参数估定与编目，截至1949年8月，共有M≥3级地震2187次，其中M≥43/4级地震259次，资料显示大致有三个相对集中活跃区，16世纪曾有过历史地震活动高潮。