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Critical behavior associated with quasiperiodicity at the golden-mean frequency ratio

TDT - Torus Doubling Terminal

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Description

This type of critical behavior occurs in the two-parameter analysis of 1D maps quasiperiodically forced with the golden-mean frequency. The critical point is located at the end of the torus-doubling bifurcation curve.

RG equations

Period-3 cycle of the RG equation
g1(x,y) =1- 0.8696y+1.7034y2+…
   x2(-1.0049-0.2989y+0.4635y2+…)
   x4(-0.0719+0.1021y-0.0978y2+…)+…
g2(x,y) =1.2709-1.4085y+1.3329y2+…
   x2(-0.9039-0.3575y-0.1942y2+…)
   x4(0.1149-0.1189y-0.1624y2+…)+…

The orbital scaling factors

Relevant eigenvalues

Codimension
CoDim=2
(For 1D map, and, presumably, for more general systems too)

An example: the forced logistic map

The critical point is located at

Parameter space arrangement
and scaling with factors

Scaling coordinates
l=lc+C1+C2, e=ec+0.3347C2.


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Critical attractor
scaling with factors


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Saratov group
of theoretical nonlinear
dynamics
Хостинг от uCoz