Characteristic relations of type-I intermittency in the presence of noise

Near the point of tangent bifurcation, the scaling properties of the laminar length of type-I intermittency are investigated in the presence of noise. Based on analytic and numerical studies, we show that the scaling relation of the laminar length is dramatically deformed from 1/√ε for ε>0 to exp(1/D)|ε|3/2 for ε<0 as ε passes the bifurcation point (ε=0). The results explain why two coupled R\"ossler oscillators exhibit deformation of the scaling relation of the synchronous length in the nearly synchronous regime.