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.
Highlights
Intermittency is a route by which a dynamical system evolves from regular behavior to chaos
We studied the intermittency reinjection processes for the logistic map
To describe the reinjection mechanisms, we evaluated by means of the M function methodology the following statistical variables: the reinjection probability density function, φ(x), the laminar length, l(x, c), the probability density of the laminar lengths, ψ(l), and the characteristic relation, L(ε)
Summary
Intermittency is a route by which a dynamical system evolves from regular behavior to chaos. Thirty years later, chaotic intermittency was classified into three types named I, II, and III [16] This categorization follows the periodic orbit stability loss using the system Floquet multipliers or the fixed-point loss of stability in a map taking the map’s eigenvalues. The local map determines the intermittency type, and the reinjection process drives the trajectories’ return from the chaotic behavior to the laminar zone. A more general scheme, called the M function methodology, was introduced [3] It has accurately worked for maps showing type-I, -II, -III, and -V intermittencies with and without noise [28–31]. We studied the reinjection processes in type-I intermittency for the third iterate of the logistic map. The main objective is to show that the reinjection process is more complex than described in previous studies and to introduce a work methodology to evaluate the main statistical variables of chaotic intermittency
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