Stability analysis of two-fluid neutron stars featuring twin star and ultradense configurations

We perform a detailed analysis of radial oscillations to discuss dynamical stability in two-fluid neutron stars composed of ordinary nuclear matter and a gravitationally coupled dark matter component. Using a fully relativistic two-fluid formalism, we solve the eigenvalue problem for a coupled system of equations with small-amplitude radial perturbations and derive the critical line corresponding to stability boundaries. We also compare these stability boundary lines obtained from the radial perturbations with those obtained from a generalized turning-point criterion based on extremization of mass and particle numbers, and find that the two methods agree to within better than $1\%$ across the parameter space explored. We consider both mirror dark matter and self-interacting fermionic dark matter models, and examine how microphysical properties$-$such as nuclear equations of state, dark matter mass, and vector coupling strength$-$reshape the topology of the stability boundary and gravitational mass contours. Our results reveal the emergence of ultra-dense and compact stars, with nuclear central densities exceeding single-fluid instability thresholds by factors of two or more, and the appearance of twin-star configurations with identical masses but distinct radii and internal fluid compositions. These findings have direct implications for the interpretation of neutron star observables and motivate future studies involving phase transitions, density discontinuities, or additional interactions in multi-component stellar systems. In particular, the emergence of exotic stable configurations beyond conventional stability limits underscores the need to reassess standard criteria in light of multi-fluid dynamics, with significant consequences for multimessenger probes of dense matter$-$including gravitational wave signals, mass-radius constraints, and post-merger remnants.