Cosmic acceleration in the Randall-Sundrum II brane world

In this paper, we implement the dynamical system tools to study the dynamics of a self-interacting scalar field for suitable interactions of dark energy and dark matter in the Randall-Sundrum II brane scenario. Here we investigate three distinct forms of interaction strength namely, Iϕ=αρϕ˙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\phi}=\alpha\dot{\rho_{\phi}}$\end{document}, Im1=δϕ˙ρm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{m1}=\delta \dot{\phi} \rho_{m}$\end{document} and Im2=σϕ˙2ρmH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{m2}=\frac{\sigma\dot{\phi}^{2} \rho_{m}}{H}$\end{document}. We consider a homogeneous and isotropic Friedmann-Robertson-Walker brane model. The transformation equations are simplified to an autonomous system of ordinary differential equations by a suitable change of variables and hence a dynamical system analysis is performed for the respective interacting models in RS II brane cosmology. During the dynamical system analysis, we use the linear stability theory to study the stability of hyperbolic points, but these linearization techniques fail for non-hyperbolic points. So we apply the center manifold theory to determine the stability of non-hyperbolic points. For these specific interaction strengths, the 3D and 4D dynamical system analysis infers the existence of late-time attractors. These attractors are found for constant potential and inverse power law potential. The presence of these attractors results in late-time cosmic acceleration.