2002/02/14 by H. P. de Oliveira, Alfredo M. Ozorio de Almeida, A. M. Ozorio de Almeida +2 · 3 citations
Mathematics · Physics and Astronomy · #Attractor #Bifurcation #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Critical point (mathematics) #Geometry #Homoclinic orbit #Mathematical analysis #Mathematical physics #Mathematics #Phase space #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Saddle #Saddle point #Singularity #Stable manifold #gr-qc
paper · pdf · doi:10.1103/physrevd.65.083511
published as Phys.Rev. D65 (2002) 083511 · 11 pages, 6 ps figures. Accepted for publication in Phys. Rev. D
arxiv created 2002/02/14 · openalex publication_date 2002/04/02 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The dynamics of a general Bianchi type-IX model with three scale factors is examined. The matter content of the model is assumed to be comoving dust plus a positive cosmological constant. The model presents a critical point of saddle-center-center type in the finite region of phase space. This critical point engenders in the phase space dynamics the topology of stable and unstable four dimensional tubes R\ifmmode×\else\texttimes\fiS3, where R is a saddle direction and S3 is the manifold of unstable periodic orbits in the center-center sector. A general characteristic of the dynamical flow is an oscillatory mode about orbits of an invariant plane of the dynamics which contains the critical point and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of tubes (one stable, one unstable) emerging from the neighborhood of the critical point towards the FRW singularity have homoclinic transversal crossings. The homoclinic intersection manifold has topology R\ifmmode×\else\texttimes\fiS2 and is constituted of homoclinic orbits which are biasymptotic to the S3 center-center manifold. This is an invariant signature of chaos in the model, and produces chaotic sets in phase space. The model also presents an asymptotic de Sitter attractor at infinity and initial conditions sets are shown to have fractal basin boundaries connected to the escape into the de Sitter configuration (escape into inflation), characterizing the critical point as a chaotic scatterer.