Characteristic Relations of Type-III Intermittency in an Electronic Circuit

It is reported that the characteristic relations of type-II and type-III intermittencies, with respective local Poincar$\acute{e}$ maps of $x_{n+1}= (1+\epsilon)x_n + a x^3_n$ and $x_{n+1}= - (1+\epsilon)x_n - a x^3_n$, are both $- ln (\epsilon)$ under the assumption of uniform reinjection probability. However, the intermittencies have various characteristic relations such as $\epsilon^{-\nu}$ ($1/2 \leq \nu \leq 1$) depending on the reinjection probability. In this Letters the various characteristic relations are discussed, and the $\epsilon^{-1/2}$ characteristic relation is obtained experimentally in an electronic circuit, with uniform reinjection probability.