Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints

The Einstein constraint equations have been the subject of study for more than fifty years. The introduction of the conformal method in the 1970's as a parameterization of initial data for the Einstein equations led to increased interest in the development of a complete solution theory for the constraints, with the theory for constant mean curvature (CMC) spatial slices and closed manifolds completely developed by 1995. The first general non-CMC existence result was establish by Holst et al. in 2008, with extensions to rough data by Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC theory remains mostly open; moreover, recent work of Maxwell on specific symmetry models sheds light on fundamental non-uniqueness problems with the conformal method as a parameterization in non-CMC settings. In parallel with these mathematical developments, computational physicists have uncovered surprising behavior in numerical solutions to the extended conformal thin sandwich formulation of the Einstein constraints. In particular, numerical evidence suggests the existence of multiple solutions with a quadratic fold, and a recent analysis of a simplified model supports this conclusion. In this article, we examine this apparent bifurcation phenomena in a methodical way, using modern techniques in bifurcation theory and in numerical homotopy methods. We first review the evidence for the presence of bifurcation in the Hamiltonian constraint in the time-symmetric case. We give a brief introduction to the mathematical framework for analyzing bifurcation phenomena, and then develop the main ideas behind the construction of numerical homotopy, or path-following, methods in the analysis of bifurcation phenomena. We then apply the continuation software package AUTO to this problem, and verify the presence of the fold with homotopy-based numerical methods.Comment: 13 pages, 4 figures. Final revision for publication, added material on physical implication

ГЛУБИННОЕ СТРОЕНИЕ И МОДЕЛЬ ФОРМИРОВАНИЯ КОНТИНЕНТАЛЬНОЙ КОРЫ ВЕРХОЯНСКОГО СКЛАДЧАТО-НАДВИГОВОГО ПОЯСА В ПОЗДНЕМ МЕЗОЗОЕ

The article considers the geological framework of a large orogenic structure in northeastern Eurasia - the Verkhoyansk fold-and-thrust belt (VFTB), formed in the Late Mesozoic on the eastern margin of the Siberian craton. Zoning of geopotential fields and the authors' interpretation of frequency-energetic characteristics along the 3-DV reference geo-physical profile provided the basis for modeling the deep structure of the VFTB and adjacent structures of the Siberian craton. There were identified structural zones of different geodynamic nature: the outer zone of the fold belt, underlain by the dropped margin of the craton; the inner zone of the VFTB with the oceanic crust at the base; the rear-zone structures formed by the Verkhoyansk complex in the subduction zone of the Uyandina-Yasachnaya island arc. In the Earth's crust of the VFTB there are distinguished two layers of approximately equal thickness: the lower one comprises a duplex system of complexes of the oceanic crust, and the upper one is built up by formations of the Verkhoyansk terrigenous complex, which have also undergone folding and duplexing. In parallel with thrusting of the upper terrigenous layer over the craton in the zone of collision between the VFTB structures and the Siberian craton margin there also occurred subduction of the lower layer of the VFTB under its margin. This led to an increase in thickness of the craton's crust by 5-10 km from below. The development of the Uyandina-Yasachnaya island arc system comprises two stages associated with blocking of the subduction zone and its transition towards the Oimyakon Ocean, which increased its area and complicated the structure. The paleosubduction zones and blocking structures are well-traced on the deep sections of reference seismic profiles.Рассмотрено строение крупной орогенной структуры северо-востока Евразии - Верхоянского складчато-надвигового пояса (ВСНП), образованного в позднем мезозое на восточной окраине Сибирского кратона. На основе районирования геопотенциальных полей, совместно с авторской интерпретацией частотно-энергетических характеристик по опорному геофизическому профилю 3-ДВ, построена модель глубинного строения ВСНП и прилегающих структур Сибирского кратона. Выделены структурные зоны, имеющие различную геодинамическую природу: внешняя зона складчатого пояса, подстилаемая опущенной окраиной кратона, внутренняя зона ВСНП с корой океанического типа в основании и структуры тыловой зоны, сформированные верхоянским комплексом в зоне субдукции Уяндино-Ясачненской островной дуги. В земной коре ВСНП выделяются два слоя примерно равной мощности: нижний сложен дуплекс-системой из комплексов океанической коры, а верхний - образованиями верхоянского терригенного комплекса, также подвергшимися складчатости и дуплексированию. В зоне столкновения структур ВСНП с окраиной Сибирского кратона, наряду с надвиганием верхнего терригенного слоя на кратон, происходило и пододвигание нижнего слоя коры ВСНП под его окраину. Это привело к наращиванию земной коры кратона снизу на 5-10 км. В развитии Уяндино-Ясачненской островодужной системы выделены два этапа, связанные с блокировкой зоны субдукции и перескоком ее в сторону Оймяконского океана, что увеличило ее площадь и усложнило строение. Следы зон палеосубдукции и блокирующих структур хорошо читаются на глубинных разрезах опорных сейсмических профилей

Tectonic history of the Kolyvan–Tomsk folded zone (KTFZ), Russia : insight from zircon U / Pb geochronology and Nd isotopes

The Kolyvan-Tomsk folded zone (KTFZ) represents part of the Central Asian Orogenic Belt (CAOB). The KTFZ is mainly composed of detrital Late Palaeozoic sedimentary deposits, with minor intrusions. Detrital zircon geochronology on the Upper Devonian to Lower Permian sedimentary sequences of the KTFZ and the associated Gorlovo foreland basin yields four age peaks, reflecting the magmatic events in the source terranes. These events consist of (a) a minor Neoproterozoic peak (0.9-0.7 Ga), (b) a significant Early Palaeozoic peak (550-460 Ma), with a maximum at 500 Ma, and two well-defined Late Palaeozoic peaks during (c) the Middle-Late Devonian (385-360 Ma) and (d) the Carboniferous-Early Permian (360-280 Ma), with a maximum at 320 Ma. Older zircons (>1 Ga) are quite rare in the sampled sedimentary sequences. Slightly negative epsilon Nd values and associated relatively young Nd model ages were obtained (epsilon Nd(T) = -0.78, T (DM) ~1.1 Ga for Upper Devonian sandstones, epsilon Nd(T) = -1.1, T (DM) ~1.1 Ga for Lower Permian sandstones), suggesting only minor contribution of ancient continental crust to the main sedimentary units of the KTFZ. All intrusive and volcaniclastic rocks on the contrary are characterized by high positive epsilon Nd(T) values in the range of 3.78-6.86 and a Late Precambrian model age (T (DM) = 581-916 Ma), which corroborates its juvenile nature and an important depleted mantle component in their source. The oldest unit of the KTFZ, the Bugotak volcanic complex formed at the Givetian-Early Frasnian transition, at about 380 Ma. Upper Devonian detrital deposits of the KTFZ were formed in the Early Palaeozoic accretion belt of the Siberian continent and specifically in a passive continental margin environment. Deposits of the Gorlovo foreland basin, adjoining the KTFZ, were accumulated as a result of erosion of the Carboniferous-Early Permian volcanic rocks, which are now buried under the Meso-Cenozoic sedimentary cover of the West Siberian Basin. The magmatic events, recorded in the KTFZ zircon data, correspond to the most significant magmatic stages that affected the western part of the CAOB as a whole