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 Poincaré maps of xn+1 = (1+ǫ)xn+ax 3 n and xn+1 = −(1 + ǫ)xn − ax3n, are both −ln(ǫ) under the assumption of uniform reinjection probability. However, the intermittencies have various characteristic relations such as ǫ−ν (1/2 ≤ ν ≤ 1) depending on the reinjection probability. In this Letters the various characteristic relations are discussed, and the ǫ−1/2 characteristic relation is obtained experimentally in an electronic circuit, with uniform reinjection probability. Typeset using REVTEX 1 Intermittency 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 (type-I, II, and III) by Pomeau and Manneville [1]. The local Poincaré maps of type-I, II, and III intermittencies are yn+1 = yn + ay 2 n + ǫ (a, ǫ > 0), yn+1 = (1 + ǫ)yn + ay 3 n(a, ǫ, yn > 0), and yn+1 = −(1 + ǫ)yn − ay n(ǫ, a > 0), respectively [2]. The characteristic relation of type-I intermittency is 〈l〉 ∝ ǫ, where 〈l〉 is the average laminar length and ǫ is the channel width between the diagonal and the local Poincaré map. Those of type-II and III intermittencies are 〈l〉 ∝ ln(1/ǫ) where 1+ ǫ 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 〈l〉 ∝ 1/ǫ. 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 − ln ǫ and ǫ (0 ≤ ν ≤ 1/2). When the lower bounds of the reinjection (LBR) are below and above the tangent point the critical exponent is always −1/2 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, the characteristic relations are − ln ǫ, ǫ, and ǫ 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 ǫ (1/2 ≤ ν ≤ 1) dependent on the RPD. When RPDs are uniform, of the form x around the tangent point, and fixed very close to the tangent point, the characteristic relations are ǫ, ǫ, and ǫ, respectively [6]. In this report we discuss the characteristic relations of type-III intermittency analytically, and obtain ǫ characteristic relation experimentally in an electronic circuit that consists of inductor, resistor, and diode with uniform RPD. 2 Since the local Poincaré map of type-III intermittency can be described to be yn+2 = (1 + 2ǫ)yn + byn , which is the same as that of type-II intermittency [3], it is enough to discuss the characteristic relations of type-II intermittency according to the RPD without loss of generality. For the given local Poincaré map of type-II intermittency, if we set a gate such that |yin| ≤ c on deviations in the laminar region, the laminar length l(yin , c) for the reinjection at yin is obtained in the long laminar length approximation l(yin , c) = 2 ln [