Introduction to the application of the dynamical systems theory in the study of the dynamics of cosmological models of dark energy

Abstract

The theory of the dynamical systems is a very complex subject which has brought several surprises in the recent past in connection with the theory of chaos and fractals. The application of the tools of the dynamical systems in cosmological settings is less known in spite of the amount of published scientific papers on this subject. In this paper a -- mostly pedagogical -- introduction to the application in cosmology of the basic tools of the dynamical systems theory is presented. It is shown that, in spite of their amazing simplicity, these allow to extract essential information on the asymptotic dynamics of a wide variety of cosmological models. The power of these tools is illustrated within the context of the so called $\Lambda$CDM and scalar field models of dark energy. This paper is suitable for teachers, undergraduate and postgraduate students from physics and mathematics disciplines.

arXiv comment15 pages, 2 figures

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• Selected topics in scalar-tensor theories and beyond
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• Kosambi-Cartan-Chern (KCC) theory for higher order dynamical systems
• Cosmological dynamics of scalar fields with kinetic corrections: Beyond the exponential potential
• New general parametrization of quintessence fields and its observational constraints
• Global asymptotic dynamics of Cosmological Einsteinian Cubic Gravity
• New perturbative method for analytical solutions in single-field models of inflation
• Brans--Dicke cosmology does not have the $\Lambda$CDM phase as an universal attractor
• Complete cosmic scenario in the Randall-Sundrum braneworld from the dynamical systems perspective
• Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background
• It's a dark, dark world: Background evolution of interacting $\varphi$CDM models beyond simple exponential potentials
• Dissipative Unified Dark Fluid Model
• On the quantum origin of inflation in the geometric inflation model
• Separatrices in the Hamilton-Jacobi Formalism of Inflaton Models
• The Chameleon Effect in the Jordan Frame of the Brans--Dicke Theory
• Late time cosmic acceleration from natural infrared cutoff?
• Inflationary equilibrium configurations of scalar-tensor theories of gravity
• Interacting dark energy model in the brane scenario: A Dynamical System Analysis
• Reconstructing the parameter space of non-analytical cosmological fixed points
• Exploring the Evolution of Nonlinear Electrodynamics in the Universe: A Dynamical Systems Approach
• On the equivalence between Sáez-Ballester theory and Einstein-scalar field system
• Phase space analysis of a FRW cosmology in the Maxwell$-$Cattaneo approach
• A dynamical systems perspective on the thermodynamics of late-time cosmology
• Revisiting dynamics of interacting quintessence