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

TDT - Torus Doubling Terminal alphabet

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. show enlarged figure show enlarged figure Critical attractor scaling with factors show enlarged figure

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