Insight into instabilities of fiber laser regimes leading to complex self-pulsing operations is an opportunity to unlock the high power and dynamic operation tunability of lasers. Though many models have been suggested, there is no complete covering of self-pulsing complexity observed experimentally. Here, I further generalized our previous vector model of erbium-doped fiber laser and, for the first time, to the best of my knowledge, map tunability of complex vector self-pulsing on Poincare sphere (limit cycles and double scroll polarization attractors) for laser parameters, e.g., power, ellipticity of the pump wave, and in-cavity birefringence. Analysis validated by extensive numerical simulations demonstrates good correspondence to the experimental results on complex self-pulsing regimes obtained by many authors during the last 20 years.