We study the dynamics of Friedmann-Lemaître-Robertson-Walker models where a dark energy component with a quadratic equation of state (EoS) nonlinearly interacts with cold dark matter. Thus, two energy scales naturally come into play: ${\rho }_{*}$ is the scale at which the nonlinearity of the EoS becomes relevant; ${\rho }_i$ is the energy scale around which the interaction starts to play an important role in the dynamics. Our focus is to understand whether there are parameter ranges for this system that can produce nonsingular bouncing and emergent cosmologies for any initial condition. We complete a dynamical systems analysis, and find the parameter range such that trajectories always expand from a high energy nonsingular de Sitter state. For flat and negative curvature models this de Sitter state is represented by a fixed point, the asymptotic past from which the Universe emerges. We find a subset of positive curvature models that during contraction get arbitrarily close to the de Sitter state, thus having a quasi-de Sitter bounce, then emerge from the bounce and expand, evolving toward spatial flatness. We find that the dimensionless parameter $q\equiv {\rho }_{*}/{\rho }_i$, which measures the relative strength of the nonlinear terms in the system, plays a crucial role in the topology of the phase space. When $q<3$, some trajectories expand toward a singularity, while others evolve toward a low energy cosmological constant at late times, with a subset going through a decelerated matter dominate era before the final acceleration. When $q>3$, all trajectories are nonsingular, and evolve toward a late-time cosmological constant. We find a subclass in this case in which all trajectories have at least one decelerated matter dominated phase, and accelerate at late times. Therefore, the nonlinear interacting cosmology presented here allows for a subclass of models that are singularity-free and qualitatively realistic for any initial condition.

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• Dark energy
• Dark matter
• General relativity
• Dynamical systems