The effect of a small second frequency component on a pump modulated neodymium fiber laser is investigated experimentally and theoretically. This term, whose frequency is exactly half the primary driver, incites an unfolding of the attractor. It breaks the period two pitchfork bifurcation and splits the period one orbit. The modification of the bifurcation diagram is studied analytically by employing a map derived from the class B laser rate equations. We determine specific conditions and scaling laws for this phenomenon. Our analytical predictions are in good agreement with recorded experimental data.