Effect of noise on the Torus Doubling Terminal criticality

Let us consider a period-doubling system driven by a quasiperiodic external force. An appropriate model is the logistic map with an additional term corresponding to the external driving

The ratio of the frequencies of the driving and of the rhythm of the discrete time is taken to be equal to the golden-mean irrational. On the parameter plane (l, e) there is a curve of the so-called torus-doubling bifurcation, which starts at the point l=0.75, e=0 and follows with increase of e up to the final point, the critical point TDT (Torus Doubling Terminal), placed at l_c=1.158096856726, e_c=0.360248020507.

The renormalization group (РГ) analysis of the dynamics at the TDT point was developed in [Kuznetsov, Pikovsky and Feudel, 1988]. The critical behavior at the TDT point and various dynamical regimes characteristic to its vicinity was observed in an electronic experiment with LR-diode circuit under quasiperiodic driving. [Bezruchko, Kuznetsov and Seleznev, 2000]. As in real physical systems presence of noise is inevitable, it is of interest to consider its effect on the dynamics at the TDT point. Let us turn to a model with added stochastic term:

Here x_n is a sequence of statistically independent random variables with zero mean and fixed variance s, k is a parameter of intensity of noise, which is supposed to be small.

Below we present Lyapunov charts on the parameter plane (l, e) without noise (the first diagram) and in presence of noise at k=0.03 (the second diagram). In the first picture we indicate the critical point TDT and present inscriptions explaining nature of the dynamical regimes: smooth torus attractors T1 and T2, strange nonchaotic attractor SNA, chaos. For each pixel at certain l and e we compute the Lyapunov exponent and code the pixel with a color - from dark to light for values from minus infinity to zero, and black for positive values.

In presence of noise, the picture of main areas on the Lyapunov chart remains visible, although fine details are smeared out.

The following picture shows phase portraits of the attractors at the critical values of the parameters l and e on iteration diagrams x_n+1 versus x_n. The first diagram corresponds to the pure dynamics (no noise), the second and the third to the noise intensity parameter values indicated on the plots k.

Observe how successively disappear with increase of noise first the subtle and tham more rough details of the attractor.

As certain scaling regularities take place near the TDT point, the effect of noise also must reveal some scaling properties.

Renormalization group analysis of the problem with noise leads to a conclusion that for observation of each next level of the small-scale structure associated with the point TDT (that corresponds to increase of the characteristic time scale by factor W³=4.236068...) the noise amplitude must be reduced by the factor g=20.048637712.

For more accurate formulation of the scaling regularities, it is appropriate to introduce a special local coordinate system on the parameter plane. In accordance with the numerical computations, for our model it may be defined by the following relations:

Let us assume that near the TDT in presence of the noise of intensity k we observe some regime at the parameter values l and e, which correspond to local coordinates c₁ and c₂. Then, at the parameter values corresponding to c₁/d₁ and c₁/d₂, where d₁=10.5029835, and d₂=5.1881181, and the noise level e/g the statistically similar regime will take place with characteristic time scale larger by the factor W³=[(5^1/2+1)/2]³. In this situation the distribution law for the variable x near the extremum x=0 will be similar to the original, with the characteristic scale in less by factor x в a=3.96376656. For small deflections from the criticality and small noise, the distribution, as supposed, is of universal nature and does not depend on details of the distribution for the external noise and of its correlation properties (if the correlation function decays sufficiently fast).

Let us consider several computer illustrations of the suggested scaling law.

We start with the plots depicting the "noisy" attractor in coordinates u, x. In the second plot the noise intensity is reduced by the factor g in comparison with the first one. The parameter values l and e correspond to the critical point TDT. On each diagram we select a fragment reproduced in the inset in the middle; there the magnification factor for the second attractor is larger by the factor a along the axis x and by the factor W₃ along the axis u. A nice correspondence of the superimposed "red" and "green" objects illustrates the scaling property for the noisy system.

Let us turn now to the diagrams indicating dependence of the Lyapunov exponent on the noise intensity at the TDT point. As seen from the picture, presence of the noise leads to decrease of the Lyapunov exponent, i.e. has stabilizing effect (in contrast to the Feigenbaum transition when the noise gives rise to increase of the Lyapunov exponent). The inset shows a fragment of the plot with rescaling in horizontal direction by factor g, and in vertical direction by factor в W³. Similarity of both these pictures reveals the expected scaling property.

One more way to demonstrate scaling consists in representation of the Lyapunov exponent versus the noise intensity in the double logarithmic scale. As seen from the plot, the dots follow in average a straight line with the slope log_gW³=0.4815.

Another series of the illustrations presents the Lyapunov charts on the parameter plane relating to a small vicinity of the TDT critical point depicted in "scaling coordinates". The gray tones from dark to light code a level of the Lyapunov exponent from minus infinity to zero, and black corresponds to positive values of the Lyapunov exponent. With passage to each the next picture, the horizontal and vertical coordinates are rescaled by factors d₁ and d₂, respectively, and the level of the external noise is decreased by factor g. Beside that, we redefine a gray tone coding rule to account the change of the characteristic time scale by factor W³.

• Effect of noise on the Feigenbaum transition
• Effect of noise on the golden-mean quasiperiodicity at the chaos threshold
• Effect of noise on the period-tripling
• Effect of noise on the torus doubling terminal criticality