Dynamical analysis of the Tsallis holographic dark energy models with event horizon as cut-off and interaction with matter

The model of generalized Tsallis holographic dark energy (which is known to be particular representative of Nojiri-Odintsov HDE) with event horizon as cut-off is investigated using methods of dynamical analysis. We take into consideration possible interaction with dark energy and matter in various forms. Critical points are determined. Cosmological evolution of the Universe depends from interaction parameters. If we use event horizon scale as cutoff quasi-de Sitter expansion is possible only for interaction of type $\sim H(\alpha {\rho }_{de}+\beta {\rho }_m)$ (where $H$ is the Hubble parameter). For interactions $\sim {\rho }_m{\rho }_{de}\mathrm{/}H$ and $\sim H{\rho }_m^{\alpha }{\rho }_{de}^{1-\alpha }$ Universe eventually stops ($H\to 0$) or ends its existence in final singularity ($H\to \mathrm{\infty }$). In first case fraction of dark energy tends to $1$ or constant value lesser than 1 because dynamical equilibrium between matter and dark energy is established on late times.