Hyperbolic sector

Abstract: This contribution summarizes the recent work carried out to analyze the behavior of the hyperbolic sector of the Fully Constrained Formulation (FCF) derived in Bonazzola et al. 2004. The numerical experiments presented here allows one to be confident in the performances of the upgraded version of CoCoNuT's code by replacing the Conformally Flat Condition (CFC) approximation of the Einstein equations by the FCF.

Abstract: The various admissible configurations of trajectories near a singular point are fan, hyperbolic sector, elliptic sector, center and focus. Except for the elliptic sector all of these configurations occur in the linear case. The trajectories in the neighborhood of an isolated singularity, described by the equations ẋ = ax n + by n and ẏ = cx n + dy n , have been examined and it is shown that the index for this singularity is +1 or −1 for n odd, and is zero for n even. Further, it is shown that indices of −1 and 0 and +1 correspond to four, two and zero hyperbolic sectors, and that elliptical sectors do not occur for the class of singularities under consideration. The index of +1 corresponds to a node, focus or center. This shows that the qualitative behavior of trajectories in the neighborhood of singular point for n odd is similar to the linear case (n = 1).

Abstract: Constant rescaling of a planar hyperbolic sector produces a one parameter family of pairwise not locally topologically conjugate sectors if and only if the planar hyperbolic sector is not locally topologically conjugate to a linear hyperbolic sector. Introduction We consider the effects of constant scaling on planar hyperbolic sectors. Shafer et al. [SSW] constructed a C°° planar hyperbolic sector that is not locally topologically conjugate to a linear hyperbolic sector. It was observed in that paper that constant rescaling of their example yields a one parameter family of pairwise not locally topologically conjugate hyperbolic sectors. A large collection of exotic planar hyperbolic sectors was constructed in [BSW], demonstrating the richness of the local topological conjugacy class structure of such sectors. For these examples the effect of constant rescaling is not so apparent. The main result of this paper is that a planar hyperbolic sector is not locally topologically conjugate to a linear hyperbolic sector if and only if constant rescaling of the sector produces a one parameter family of pairwise not locally topologically conjugate sectors. 1 . Preliminaries We will assume that the reader is familiar with the notion of a closed regular hyperbolic sector (cf. Hartman [H]). Such an object will be denoted by iX, D), or simply by X, where X is a planar vector field and D is a closed topological disk on which X is a regular hyperbolic sector. We will denote by H the collection of all planar closed regular hyperbolic sectors. As differentiability plays no role in this discussion we require only that X G H be sufficiently nice as to generate a well-defined (local) flow nxip, t). We say that iXx,Dx) G H and iX2,D2) G H are locally topologically conjugate if there is a homeomorphism y/:Ux -* U2 of a neighborhood Ux of Received by the editors July 6, 1988 and, in revised form, September 30, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 34C10, 34D30; Secondary 58F10, 58F15. Research partially supported by a grant from MONTS. ©1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use