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 $\frac{1}{\sqrt{ϵ}}$ for $ϵ>0$ to $\exp \{\frac{1}{D}\mathrm{\mid }ϵ{\mathrm{\mid }}^{3\mathrm{/}2}\}$ 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.