Pomeau-Manneville

Pomeau-Manneville or Pomeau-Manneville scenario is concept of transition of a dynamic system from smooth and easy-toobserve behaviour the the chaotic fluctuations that can be treated in terms of the probability theory and stochastic models.

J.-P. Eckmann suggests the following qualification of the Pomeau–Manneville scenario and its original description

"This scenario (Pomeau and Manneville, 1980; Manneville and Pomeau, 1980) has been - correctly - termed  Transition to turbulence through intermittency.  Its mathematical status is soemwhat less satisfactory than that of the tro other scenarios presented here..."

Simple model
The simple dynamics, that can be interpreted in terms of chaos, refers to the process \(F\) such that

(1)\(~ ~ ~ F(z+1)=T(F(z))= c~ F(z)~ \Big(1-F(z)\Big)\)

where \(c\) is constant (often it is assumed that \(3\!\le\! c\! \le\! 4)\), and \(T\) can be interpreted as the Transfer function. The opinions about applicability of Pomeau–Manneville to simple mathematical models are pretty different.

Pawel Elutin and anonymous reviewers express enthusiasm, believing, that sequence \(F\) by (1) realises the Pomeau-Manneville and should have applications in theory of economics, turbulent flows and other stochastic processes.

Ю, contrary, expresses skepticism, expecting, that neither simple descriptions with models similar to equation (1), nor the holomorphic solution \(F\) of (1) have application in any science or technology.

In this article, the links about Pomeau-Manneville are collected. The goal is to formulate the concept in a way, that satisfies the TORI axioms

Keywords
Logistic sequence, Logistic operator, Transfer function, Философия Ю (in Russian).