Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We confirmed that the convergence to the fixed point in both logistic and cubic maps for a region close to the fixed point goes exponentially fast to the fixed point and with a relaxation time described by a power law of exponent -1. At the bifurcation point, the exponent is not universal and depends on the type of the bifurcation as well as on the nonlinearity of the map.Univ Estadual Paulista, Dept Fis, UNESP, BR-13506900 Rio Claro, SP, BrazilUniv Estadual Paulista, UNESP, BR-13874149 Sao Joao Da Bao Vista, SP, BrazilAbdus Salaam Int Ctr Theoret Phys, I-34151 Trieste, ItalyUniv Estadual Paulista, Dept Fis, UNESP, BR-13506900 Rio Claro, SP, BrazilUniv Estadual Paulista, UNESP, BR-13874149 Sao Joao Da Bao Vista, SP, Brazil