Period doubling is a universal bifurcation of central importance in all disciplines of nonlinear science, which generally signals the existence of chaotic dynamics in the vicinity of the system parameters. Although observed in diverse ultrafast laser configurations, there is still no consensus on its physical origin. Real-time spectral characterization techniques have recently allowed revisiting pulsating dynamics, from period-2 to long-period pulsations. Following a contextual review, this article presents a variety of bifurcation sequences entailing the spectral pulsations of dissipative solitons. These experiments, using ultrafast fiber lasers operated in both chromatic dispersion regimes, are confronted with numerical simulations to demonstrate that self-phase modulation represents a general mechanism triggering period-2 bifurcations. In addition, by ramping up the pump power, original sequences of period-doubling bifurcations intertwined with more complex bifurcations are presented, where entrainment phenomena are unveiled. These findings provide a more general understanding of the period-doubling bifurcation in ultrafast laser systems while highlighting their potentially intricate combinations with complex bifurcations, which may be exclusively observable from the spectral domain.