In this contribution we provide an in depth theoretical analysis of the bifurcations leading to nonlinear polarization dynamics in a free-running vertical-cavity surface-emitting laser (VCSEL). We detail the sequence of bifurcations that occurs when increasing the injection current, and which brings the laser from linear to elliptical polarization emission and then self-pulsating or even more complex chaotic dynamics of the light intensity. Continuation techniques allow us to follow the stable and unstable limit cycle solutions emerging from Hopf bifurcations, and therefore to interpret the frequency of the self-pulsating polarization dynamics. The fundamental frequency of the pulsating dynamics is either close to the laser relaxation oscillation frequency or close to the linear-birefringence-induced polarization mode frequency splitting, depending on the laser parameters. A systematic analysis of the parameter space allows us to identify two scenarios that are in excellent qualitative agreement with those reported in recent experiments. Our results provide, moreover, evidence for an interesting polarization mode hopping mechanism, i.e., a so-called deterministic mode hopping where the laser exhibits a chaotic and therefore random-like hopping between two states that are elliptically polarized and nonorthogonal.