A dynamical system analysis of Tsallis holographic dark energy

In this paper we deal with the Tsallis holographic dark energy from a dynamical system approach. Here we assume the Hubble horizon as an infra red cutoff. We consider the cosmic evolution of the model in noninteracting, linearly, sign changeable and nonlinearly interacting cases. The analysis includes an autonomous system of equations and then we analyze the related phase spaces. We find that for a consistent picture of cosmic evolution in all cases we should take δ>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta >1$\end{document}. However in the linear interaction case there exists a chance for δ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta <1$\end{document} but the necessity of a decelerating matter dominated epoch leads to a restriction on the coupling constant b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b$\end{document}. This restriction rejects the chance of the δ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta <1$\end{document} in the latter case. Then we find that an observationally consistent description of the cosmic evolution for THDE in the presence of the discussed interaction terms is possible for δ>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta >1$\end{document}. In the linearly interacting case of interaction we also find the restriction b2<13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{2}<\frac{1}{3}$\end{document} for a consistent picture of cosmic evolution. Beside we study the squared sound speed analysis and find chances of stability for suitable choices of free parameters. We also discuss the impacts of the interaction terms on the age of the universe.