Cascade of Attractor-Merging Crises to the Critical Golden Torus and Universal Expansion-Rate Spectra

Two chaotic attractors with rotation numbers Pm = Fm-r/Fm, Fm being the moth Fibonacci number, and pm+! just before their attractor-merging crisis converge to the critical golden torus as m -H)(). A critical scaling is shown to hold for those chaotic attractors for large m; leading to a universal spectrum of the local expansion rates of nearby orbits. The two-frequency dynamical systems exhibit a great variety of bifurcations in the two-dimensional space of parameters which control the ratio of the two frequen cies and the strength of the nonlinearity, resulting in fascinating chaotic attractors.l) A typical example of the two-frequency systems is the sine-circle map,l),2) which is the map on a circle [0,1] onto itself and expressed as