SCALING PROPERTIES AND RENORMALIZATION INVARIANTS FOR THE HOMOCLINIC QUASI-PERIODICITY

Some bifurcation scenarios near the ''gluing'' homoclinic bifurcation are considered. The attracting trajectories, related to irrational rotation numbers, do not look like motions on a 2-torus; however, qualitatively they have much in common. The unusual metric properties of bifurcation sequences are studied with the help of renormalization, which shows that the prevalence of superexponential scaling both in the subcritical (pre-chaotic) and supercritical parameter domains is due to the existence of singular fixed points of the corresponding renormalization transformations; an approximate analysis near these points yields quantitative characteristics of scaling laws. On the boundary between subcritical and supercritical regions the scaling is exponential; the transformations, however, possess an additional invariant, which characterizes the mismatch of two branches of the return map near the saddle-point of the initial continuous system and influences both local (for a given rotation number) and global quantitative characteristics of the bifurcation diagram.