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