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 $1/\sqrt{\epsilon }$ for $\epsilon >0\mathrm{}$ to $\exp \{\mathrm{}(1/D)\mathrm{}|\epsilon {|}^{3/2}\}$ for $\epsilon <0\mathrm{}$ as $\epsilon$ passes the bifurcation point $(\epsilon =0).\mathrm{}$ The results explain why two coupled Rössler oscillators exhibit deformation of the scaling relation of the synchronous length in the nearly synchronous regime.