Cosmological dynamical systems

In this book are studied, from the perspective of the dynamical systems, several Universe models. In chapter 1 we give a bird's eye view on cosmology and cosmological problems. Chapter 2 is devoted to a brief review on some results and useful tools from the qualitative theory of dynamical systems. They provide the theoretical basis for the qualitative study of concrete cosmological models. Chapters 1 and 2 are a review of well-known results. Chapters 3, 4, 5 and 6 are devoted to our main results. In these chapters are extended and settled in a substantially different, more strict mathematical language, several results obtained by one of us in arXiv:0812.1013 [gr-qc]; arXiv:1009.0689 [gr-qc]; arXiv:0904.1577[gr-qc]; and arXiv:0909.3571 [hep-th]. In chapter 6, we provide a different approach to the subject discussed in astro-ph/0503478. Additionally, we perform a Poincar\'e compactification process allowing to construct a global phase space containing all the cosmological information in both finite and infinite regions for all the models.Comment: 321 pages, 41 figures. Draft of our book Cosmological Dynamical Systems: And their Applications, LAP LAMBERT Academic Publishing. 416 pp. ISBN:978-3-8473-0233-9. https://www.lap-publishing.com/catalog/details/store/gb/book/978-3-8473-0233-9/cosmological-dynamical-systems. arXiv admin note: text overlap with arXiv:astro-ph/0401547, arXiv:1101.0300, arXiv:hep-ph/0604152, arXiv:0911.1435, arXiv:1004.2474, arXiv:gr-qc/9910074, arXiv:gr-qc/9908067, arXiv:1001.1251, arXiv:1003.5637, arXiv:0707.2089 by other author

Phase space analysis of quintessence fields trapped in a Randall-Sundrum Braneworld: anisotropic Bianchi I brane with a Positive Dark Radiation term

In this paper we investigate, from the dynamical systems perspective, the evolution of an scalar field with arbitrary potential trapped in a Randall-Sundrum's Braneworld of type 2. We consider an homogeneous but anisotropic Bianchi I (BI) brane filled also with a perfect fluid. We also consider the effect of the projection of the five-dimensional Weyl tensor onto the three-brane in the form of a positive Dark Radiation term. Using the center manifold theory we obtain sufficient conditions for the asymptotic stability of de Sitter solution with standard 4D behavior. We also prove that there are not late time de Sitter attractors with 5D-modifications since they are always saddle-like. This fact correlates with a transient primordial inflation. We present here sufficient conditions on the potential for the stability of the scalar field-matter scaling solution, the scalar field-dominated solution, and the scalar field-dark radiation scaling solution. We illustrate our analytical findings using a simple $f$-deviser as a toy model. All these results are generalizations of our previous results obtained for FRW branes.Comment: 14 pages, 11 figures, one affiliation added, matches the published version at CQG. arXiv admin note: substantial text overlap with arXiv:1110.173