Cosmological Dynamical Systems
Preface xviiIntroduction 11 A bird’s eye view on cosmology and cosmological problems 131.1 Cosmological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.1 Homogeneity and isotropy: the Roberson-Walker metric . . . . . 161.1.2 Dynamics: the Friedmann equations . . . . . . . . . . . . . . . . 171.1.3 Flat universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.1.4 Including curvature . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.5 Geometry, destiny and dark energy . . . . . . . . . . . . . . . . . 221.1.6 Scalar ﬁelds and Dark Energy . . . . . . . . . . . . . . . . . . . 232 Qualitative theory of dynamical systems 452.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2 Qualitative theory of dynamical systems . . . . . . . . . . . . . . . . . . 482.2.1 Deﬁnitions and basic results . . . . . . . . . . . . . . . . . . . . 4 82.2.2 Desirable stability properties of nonlinear vector ﬁ elds . . . . . . 512.2.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2.3.1 Linear stability theorem . . . . . . . . . . . . . . . . . 552.2.4 Flow for autonomous vector ﬁelds . . . . . . . . . . . . . . . . . 5 72.2.5 Invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2.5.1 Stable, Unstable and Center subspaces of singularpoints of linear autonomous vector ﬁelds . . . . . . . . 602.2.5.2 Stable, Unstable and Center manifolds of singularpoints of nonlinear autonomous vector ﬁelds . . . . . . 642.2.5.3 Center Manifold Theory . . . . . . . . . . . . . . . . . 662.2.5.4 Normal Forms . . . . . . . . . . . . . . . . . . . . . . 682.2.5.5 Asymptotic behavior . . . . . . . . . . . . . . . . . . . 712.3 Procedure for analyzing cosmological dynamical systems . . . . . . . . . 76i