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:(x0 ∊ [−2,2],C ∊ [−2,5]) of the (x0, 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:(x0 ∊ [−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 ordern, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x0, 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 Di 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. 相似文献
Low pressure partial melting of basanitic and ankaramitic dykes gave rise to unusual, zebra-like migmatites, in the contact aureole of a layered pyroxenite–gabbro intrusion, in the root zone of an ocean island (Basal Complex, Fuerteventura, Canary Islands). These migmatites are characterised by a dense network of closely spaced, millimetre-wide leucocratic segregations. Their mineralogy consists of plagioclase (An32–36), diopside, biotite, oxides (magnetite, ilmenite), +/− amphibole, dominated by plagioclase in the leucosome and diopside in the melanosome. The melanosome is almost completely recrystallised, with the preservation of large, relict igneous diopside phenocrysts in dyke centres. Comparison of whole-rock and mineral major- and trace-element data allowed us to assess the redistribution of elements between different mineral phases and generations during contact metamorphism and partial melting.
Dykes within and outside the thermal aureole behaved like closed chemical systems. Nevertheless, Zr, Hf, Y and REEs were internally redistributed, as deduced by comparing the trace element contents of the various diopside generations. Neocrystallised diopside – in the melanosome, leucosome and as epitaxial phenocryst rims – from the migmatite zone, are all enriched in Zr, Hf, Y and REEs compared to relict phenocrysts. This has been assigned to the liberation of trace elements on the breakdown of enriched primary minerals, kaersutite and sphene, on entering the thermal aureole. Major and trace element compositions of minerals in migmatite melanosomes and leucosomes are almost identical, pointing to a syn- or post-solidus reequilibration on the cooling of the migmatite terrain i.e. mineral–melt equilibria were reset to mineral–mineral equilibria. 相似文献
Migmatites produced by low-pressure anatexis of basic dykes are found in a contact metamorphic aureole around a pyroxenite–gabbro
intrusion (PX2), on Fuerteventura. Dykes outside and inside the aureole record interaction with meteoric water, with low or
negative δ18O whole-rock values (+0.2 to −3.4‰), decreasing towards the contact. Recrystallised plagioclase, diopside, biotite and oxides,
from within the aureole, show a similar evolution with lowest δ18O values (−2.8, −4.2, −4.4 and −7.6‰, respectively) in the migmatite zone, close to the intrusion. Relict clinopyroxene phenocrysts
preserved in all dykes, retain typically magmatic δ18O values up to the anatectic zone, where the values are lower and more heterogeneous. Low δ18O values, decreasing towards the intrusion, can be ascribed to the advection of meteoric water during magma emplacement, with
increasing fluid/rock ratios (higher dyke intensities towards the intrusion acting as fluid-pathways) and higher temperatures
promoting increasing exchange during recrystallisation.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. 相似文献
A morphometric analysis was done to determine the drainage characteristics of Lules River basin using land-sat imageries and topographical maps. This catchment was divided into seven sub-basins for the analysis: Liquimayo, Hoyada, Ciénaga, De Las Tablas, Siambón, Potrerillo and San Javier. Yungas ecoregion covers almost all the watershed. The drainage patterns of the sub-basins are dendritic and parallel. The basin includes seventh order stream and lower streams order mostly dominate the basin. The development of stream segments is affected by slope and local relief. The mean bifurcation ratio indicates that the drainage pattern is not much influenced by geological structures. The shape parameters also reveal the elongation of the basin and sub-basins. 相似文献
The Precambrian massif of Ourika is crosscut by two systems of basic dykes, striking N40°E and N90–120°E. Using incompatible trace elements, the two systems form two distinct chemical groups, displaying a continental tholeiitic affinity. The composition variations between the two defined groups can be due to heterogeneities of mantle sources and to contamination, during the magma ascent, by the continental crust. The emplacement of these basic dykes, before the late-PIII formations, can be related to the Neoproterozoic distension generalised to the Anti-Atlas chain. To cite this article: A. Barakat et al., C. R. Geoscience 334 (2002) 827–833.相似文献