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 fractal attractor and repeller at some set of exceptional points, 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 responsible for scaling in phase space ($α=2.890053...$ and $β=-1.618034...$) and in parameter space ($δ_1=3.134272...$ and $δ_2=1.618034...$). 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 re-injection mechanism) to intermittent chaotic regimes; in the supercritical case it gives rise to a strange nonchaotic attractor.