Effect of noise on the dynamics at the torus-doubling terminal point in a quadratic map under quasiperiodic driving.

Scaling regularities are examined associated with the effect of additive noise on a system driven by an external quasiperiodic force with the golden-mean frequency ratio near the terminal point of the torus-doubling bifurcation curve (TDT point). This point was studied in the context of the problem of the onset of a strange nonchaotic attractor on the basis of renormalization group (RG) analysis [Kuznetsov, Phys. Rev. E 57, 1585 (1998)] and observed in experiments with a quasiperiodically driven resistor-inductor-diode circuit [Bezruchko, Phys. Rev. E 62, 7828 (2000)]. The method implemented in the present paper is based on a generalization of the RG approach of Crutchfield [Phys. Rev. Lett. 46, 933 (1981)] and Shraiman [Phys. Rev. Lett., 46, 935 (1981)], originally developed for the period-doubling transition to chaos in the presence of noise. At the TDT point, a constant determining the rescaling rule for the intensity of noise is found to be gamma = 20.048 637 7 . It means that a decrease of the noise amplitude by this factor ensures the possibility of observing one more level of the fractal-like structure of the dynamics, with increase of the characteristic time scale by [(square root (5) + 1)/2]3. Numeric results demonstrating evidence of the expected scaling are presented, e.g., portraits of the noisy attractors and Lyapunov charts on the parameter plane in different scales.