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).

References

http://dare.uva.nl/document/15299 Cars H.Hommes. Adaptive learning and roads to chaos. (1990). .. The dynamics of the expected prices in the model is described by a one-dimensional nonlinear difference equation \(x_{n+1}=f(x_n)\). Chiarella (1988) approximated this model by the well known logistic map \(x_{n+1}=\mu x_n (1-x_n)\)..

http://csc.ucdavis.edu/~cmg/papers/EoMfaDS.pdf James P. Crutchfield and Bruce S. McNamara. Equations of Motion from a Data Series. (1987–2013) .. The first example shows the effect of extrinsic noise on the model entropy and the optimum model. We consider the stochastic logistic map \(x_{n+1} = r x_n(1−x_n) + ξ_n\) where the nonlinearity parameter r is 3.7, so that the deterministic behavior is chaotic. ..

http://www.sciencedirect.com/science/article/pii/0375960181901079 G. Mayer-Kress, H. Haken. Intermittent behavior of the logistic system. Physics Letters A. Volume 82, Issue 4, 23 March 1981, Pages 151–155. In the discrete logistic model a transition to chaotic behavior via intermittency occurs in a neighborhood of periodic bands. Intermittent behavior is also induced if a stable periodic orbit is perturbed by low-level external noise, whereas alterations due to computer digitalisation produce remarkable periodicities. We compare our numerical results with the predictions of Pomeau and Manneville for the Lorenz system.

Tufillaro, Nicholas; Tyler Abbott, Jeremiah Reilly An experimental approach to nonlinear dynamics and chaos. 1992. NY. Addison-Wesley New York. ISBN 0-201-55441-0.

Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0-7382-0453-6. Nonlinear Dynamics and Chaos. NY, 2000. Perseus Publishing. ISBN 0-7382-0453-6.

Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850840-9.

http://www.springerlink.com/content/d73u677140l3j826/ Bjerkl\"ov K. Strange Nonlinear Attractors in the quasi-periodic family. Communications in Mathematical Physics. {\bf 286}. 2009. P.137--161

http://arxiv.org/pdf/1306.3626.pdf Wang Liang. Describe Prime number gaps pattern by Logistic mapping. 2013.

http://jphyslet.journaldephysique.org/articles/jphyslet/abs/1979/23/jphyslet_1979__40_23_609_0/jphyslet_1979__40_23_609_0.html Y. Pomeau et P. Manneville. Stability and fluctuations of a spatially periodic convective flow. J. Physique Lett. 40, 609-612 (1979) DOI: 10.1051/jphyslet:019790040023060900 Using a model equation we analyze the onset of convection for a roll instability. The wavelength increase follows from a variational approach. For certain wavelengths, phase diffusion turns out to be unstable against compression (or dilatation) and torsion of the rolls. Thermal phase fluctuations induce a tiny brownian motion of the structure which should be observable in carefully designed experiments such as Rayleigh-Bénard in liquid helium or Carr-Helfrich in nematic liquid crystals.

http://www.sciencedirect.com/science/article/pii/0167278980900135 Paul Manneville, Yves Pomeau. Different ways to turbulence in dissipative dynamical systems. Physica D: Nonlinear Phenomena Volume 1, Issue 2, June 1980, Pages 219–226. The Lorenz model is studied in details for σ = 10, and 145 < r < 170. Between r = 145 and r = 148.4 the Lore nz attractor disaggregates itself into a limit cycle through a cascade of bifurcation with successive undoubling of periods. At r = 166.07 this limit cycle looses its stability through “intermittency”, giving rise to a second aperiodic attractor. We give a semi-quantitative interpretation of these processes and discuss their relation with the different transitions to turbulence observed experimentally.

http://pra.aps.org/abstract/PRA/v26/i4/p2117_1 Carson Jeffries and Jose Perez. Observation of a Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator. Phys. Rev. A 26, 2117–2122 (1982) For a driven nonlinear semiconductor oscillator which shows a period-doubling pitchfork bifurcation route to chaos, we report an additional route to chaos: the Pomeau-Manneville intermittency route, characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. This occurs near a tangent bifurcation as the system driving parameter is reduced by ε from the threshold value for a periodic window. Data are presented for the dependence of the average laminar length ⟨l⟩ on ε, and also on additive random noise voltage. The results are in reasonable agreement with the intermittency theory of Hirsch, Huberman, and Scalapino. The distribution P(l) is also reported.