Torus fractalization and intermittency.

Abstract

The bifurcation transition is studied for the onset of intermittency analogous to the Pomeau-Manneville mechanism of type I, but generalized for the presence of a quasiperiodic external force. The analysis is concentrated on the torus-fractalization (TF) critical point that occurs at some critical amplitude of driving. (At smaller amplitudes the bifurcation corresponds to a collision and subsequent disappearance of two smooth invariant curves, and at larger amplitudes it is a touch of attractor and repeller at some fractal set without coincidence.) For the TF critical point, renormalization group (RG) analysis is developed. For the golden mean rotation number a nontrivial fixed-point solution of the RG equation is found in a class of fractional-linear functions with coefficients depending on the phase variable. Universal constants are computed that are responsible for scaling in phase space (alpha=2.890 053... and beta= -1.618 034...) and in parameter space (delta(1)=3.134 272... and delta(2)=1.618 034...). An analogy with the Harper equation is outlined, which reveals important peculiarities of the transition. For amplitudes of driving less than the critical value the transition leads (in the presence of an appropriate reinjection mechanism) to intermittent chaotic regimes; in the supercritical case it gives rise to a strange nonchaotic attractor.