The present work deals with the dynamical system investigation of interacting dark energy models (quintessence and phantom) in the framework of Loop Quantum Cosmology by taking into account a broad class of self-interacting scalar field potentials. The main reason for studying potentials beyond the exponential type is to obtain additional critical points which can yield more interesting cosmological solutions. The stability of critical points and the asymptotic behavior of the phase space are analyzed using dynamical system tools and numerical techniques. We study two class of interacting dark energy models and consider two specific potentials as examples: the hyperbolic potential and the inverse power-law potential. We found a rich and interesting phenomenology including the avoidance of big rip singularities due to loop quantum effects, smooth and non-linear transitions from matter domination to dark energy domination and finite periods of phantom domination with dynamical crossing of the phantom barrier.

The braneworld model of gravity is well-known for several notable cosmological features such as self-acceleration originating from a geometric and not matter source, effective dark energy behavior with phantom characteristics but not leading to a Big-Rip singularity, rough resemblance to the ΛCDM evolution, etc. The dynamical system tools usually allow us to obtain generic conclusions on the global dynamics of a system over a wide range of initial conditions. With this motivation, in order to recover the important features of the braneworld model from a more global perspective, here, we investigate the global cosmological dynamics of the braneworld model using dynamical system techniques. We first analyze the case where there is just a normal matter on the brane and then extend the analysis to the case with an extra scalar field also trapped on the brane. In the presence of a scalar field, potentials belonging to different classes are considered. The stability behavior of critical points is examined using linear stability analysis and when necessary center manifold theory as well as numerical perturbation techniques are also used. To understand the global dynamics of a dynamical system, we utilized the Poincaré compactification method to capture the properties of all possible critical points. Applying dynamical system analysis, we found that brane gravity is consistent with observed actions of the Universe. In particular, our analysis shows that important cosmological behaviors like the long-lasting matter-dominated era, late time acceleration as well as the avoidance of Big-Rip singularity can be realized in brane gravity for a wide range of initial conditions. *

The paper deals with dynamical system analysis of a coupled scalar field in the Randall-Sundrum(RS)2 brane world. The late time attractor describes the final state of the cosmic evolution.In RS2 based phantom model there is no late-time attractor and consequently there is uncertainty in cosmic evolution. In this paper, we have shown that it is possible to get late-time attractor when gravity is coupled to scalar field. Finally, in order to predict the final evolution of the universe, we have also studied classical stability of the model. It is found that there are late time attractors which are both locally as well as classically stable and so our model can realise the late time cosmic acceleration.

We extend the investigation of cosmological dynamics of the general non-canonical scalar field models by dynamical system techniques for a broad class of potentials and coupling functions. In other words, we do not restrict the analysis to exponential or power-law potentials and coupling functions. This type of investigation helps in understanding the general properties of a class of cosmological models. In order to better understand the phase space of the models, we investigate the various special cases and discuss the stability and viability issues. Performing a detailed stability analysis, we show that it is possible to describe the cosmic history of the universe at the background level namely the early radiation dominated era, intermediate matter dominated era and the late time dark energy domination. Moreover, we find that we can identify a broad class of potentials and coupling functions for which it is possible to get an appealing unified description of dark matter and dark energy. The results obtained here, therefore, enlarge the previous analyses wherein only a specific potential and coupling functions describes the unification of dark sectors. Further, we also observe that a specific scenario can also possibly explain the phenomenon of slow-roll inflationary exit. arXiv:1812.07836v2 [gr-qc]