A qualitative picture of development of a bifurcated plasma state in a single-null and double-null divertor configurations is discussed. The simple analytical considerations are verified with SOLPS code modeling in quasi-1D setup. In a more realistic, 2D model, these bifurcations cause self-sustained oscillations of the plasma parameters, leading to deep modulation of the power loading on the divertor targets. In the symmetric double-null case, the bifurcations prompt a spontaneous loss of the up-down symmetry, which results in a sharp, counter-intuitive increase of the power loading on one of the targets along with an increase of the particle content in the divertor plasma.