Classical and Loop Quantum Cosmology of Interacting Dark Energy: A Dynamical System Analysis with Superfluid Dark Matter and Dust Matter

We study the cosmological dynamics of interacting dark energy and dark matter in Classical Einstein Gravity and Loop Quantum Cosmology. Two dark matter scenarios are considered: superfluid dark matter described by a generalized cubic equation of state and the standard pressureless fluid. The dark energy component is modeled using both a generalized nonlinear equation of state and a constant equation of state. We examine two phenomenological interaction terms, $Q=\alpha\dot{\rho}_m$ and $Q=\beta\dot{\rho}_d$, which govern the energy transfer between the dark sectors. In classical gravity, the pressureless matter model exhibits stable late-time attractors, whereas the superfluid dark matter model admits only saddle and non-hyperbolic critical points. Extending the analysis to Loop Quantum Cosmology, quantum geometric corrections replace the Big Bang singularity with a nonsingular quantum bounce, and significantly modify the phase-space dynamics. As a result, the stable attractors of the classical pressureless matter model disappear, and all interacting models possess only saddle and non-hyperbolic critical points. These findings highlight the significant influence of both dark matter properties and quantum gravitational effects on the asymptotic evolution of interacting dark-sectors.