Cosmology with interactions in the dark sector: Qualitative dynamics, singularities and applications

We will report here on our studies of cosmological models with interactions between dark energy and dark matter. Observational motivations for introduction of energy flows between the two dark fluids, as for example the Hubble tension, will be provided, and reasons for adopting adiabatic equations of state for the dark energy fluid other than the cosmological constant will be mentioned. A satisfactory understanding of the dynamics and physical properties of such frameworks requires an interplay between purely mathematical techniques, as the ones provided by dynamical system methods, and statistical analysis of appropriate datasets, one example being the redshift evolution of the cosmic rate of expansion: they will be briefly reviewed in this paper.