Intermittency Reinjection in the Logistic Map

Abstract

Just below a Period-3 window, the logistic map exhibits intermittency. Then, the third iterate of this map has been widely used to explain the chaotic intermittency concept. Much attention has been paid to describing the behavior around the vanished fixed points, the tangent bifurcation, and the formation of the characteristic channel between the map and the diagonal for type-I intermittency. However, the reinjection mechanism has not been deeply analyzed. In this paper, we studied the reinjection processes for the three fixed points around which intermittency is generated. We calculated the reinjection probability density function, the probability density of the laminar lengths, and the characteristic relation. We found that the reinjection mechanisms have broader behavior than the usually used uniform reinjection. Furthermore, the reinjection processes depend on the fixed point.

Chaotic intermittency with non-differentiable M(x) function

“…Black line: theoretical M (x) function. [36]. Here, we use the results of this paper to verify the theory developed in the previous section.…”

“…Figure 4 shows the map given by equation (30) for µ = 3.8278 < µ c . For the map given by equation ( 30 The second component is given by pre-reinjection points neighboring to the laminar interval lower limit, and they satisfy F (x n−1 ) ∈ (F (−c), c], where x n−1 are the pre-reinjection points [36].…”

“…This paper extends the M function methodology to evaluate discontinuous RPD functions. These RPDs are related to two o more overlapping reinjection mechanisms [1][29] [36]. The paper shows that discontinuous RPD functions correspond to non-differentiable M (x) functions.…”

Review of Chaotic Intermittency

“…Note that a > 0 implies a 3 < −a 2 2 . For this reason, several studies utilize Equation ( 14) as a local map instead of Equations ( 12) or (13).…”

“…Batchelor and Townsend, in the middle of the previous century, used the word intermittency to give an account of the fluctuating velocity in turbulent flows [1]. Subsequently, intermittency has been found in many and varied physical phenomena such as the nonlinear behavior of transient periodic plasma and conducting fluids [2][3][4][5], fluid mechanics and turbulent flows [6][7][8][9][10], Rayleigh-Benard convection [11], electronic digital oscillator [12], logistic map [13], Alfven wave-fronts and derivative nonlinear Schrodinger equation [14,15], premixed combustion [16], Lorenz system [17], coupled oscillators [18][19][20][21], catalytic reactors [22], Ginzburg Landau equation [23], solar cycles [24], spatiotemporal chaos [25,26], thermoacoustic instability [27], control chaos [28], etc. Furthermore, chaotic intermittency is also observed in systems in economics [29,30], medicine, [31,32], neuroscience [33,34], genetics [35], and marine biology [36,37].…”