It is reported that the characteristic relations of type-II and type-III intermittencies, with respective local Poincaré maps of x n11 ͑1 1 e͒x n 1 ax 3 n and x n11 2͑1 1 e͒x n 2 ax 3 n , are both 2 ln͑e͒ under the assumption of uniform reinjection probability. However, the intermittencies have various characteristic relations such as e 2n (1͞2 # n # 1) depending on the reinjection probability. In this Letter the various characteristic relations are discussed, and the e 21͞2 characteristic relation is obtained experimentally in an electronic circuit, with uniform reinjection probability. [S0031-9007 (98)06383-2] PACS numbers: 05.45. + b, 07.50.EkIntermittency characterized by the appearance of intermittent short chaotic bursts between quite long quasiregular (laminar) periods is one of the critical phenomena that can be readily observed in nonlinear dynamic systems. The phenomenon was initially classified into three types according to the local Poincaré map (types I, II, and III) by Pomeau and Manneville [1]. The local Poincaré maps of type-I, type-II, and type-III intermittencies are y n11 y n 1 ay 2 n 1 e͑a, e . 0͒, y n11 ͑1 1 e͒y n 1 ay 3 n ͑a, e, y n . 0͒, and y n11 2͑1 1 e͒y n 2 ay 3 n ͑e, a . 0͒, respectively [2]. The characteristic relation of type-I intermittency is ͗l͘~e 21͞2 , where ͗l͘ is the average laminar length and e is the channel width between the diagonal and the local Poincaré map. Those of type-II and type-III intermittencies are ͗l͘~ln͑1͞e͒ where 1 1 e is the slope of the local Poincaré map around the tangent point under the assumption of uniform reinjection probability distribution (RPD). On the other hand, some monographs [3] suggested that the standard scaling should be ͗l͘~1͞e.Recently, however, it was found that the reinjection mechanism is another important factor of the scaling property of the intermittency. In the case of type-I intermittency, various characteristic relations appear dependent on the RPD for the given local Poincaré map, such as 2 ln e and e 2n ͑0 # n # 1͞2͒. When the lower bounds of the reinjection (LBR) are below and above the tangent point the critical exponent is always 21͞2 and 0, respectively, irrespective of the RPD. However, when the LBR is at the tangent point the characteristic relations have various critical exponents dependent on the RPD, such that when the RPDs are uniform, fixed, and of the form x 21͞2 , the characteristic relations are 2 ln e, e 21͞2 , and e 21͞4 respectively [4,5]. In the case of type-II and III intermittencies, the characteristic relations also have various critical exponents for a given local Poincaré map such as e 2n ͑1͞2 # n # 1͒ dependent on the RPD. When RPDs are uniform, of the form x 21͞2 around the tangent point, and fixed very close to the tangent point, the characteristic relations are e 21͞2 , e 23͞4 , and e 21 , respectively [6]. In this report we discuss the characteristic relations of type-III intermittency analytically, and obtain e 21͞2 characteristic relation experimentally in an electronic circuit that consists of ind...