Effect of noise on the period-tripling

In the quadratic map zn+1=l-zn2 let us regard z as a complex variable and l as a complex parameter. Then, beside of the period-doubling bifurcation cascade (along the real axis of parameter l), one can get so well the bifurcation cascades of period-tripling and others. Golberg, Sinai, and Khanin have developed an RG analysis analogous to the Feigenbaum theory, appropriate to the period-tripling. The picture to the right shows a chart on the complex parameter plane of the quadratic map, where a fractal Mandelbrot set may be seen. It is formed by points on the parameter plane, which correspond to the dynamics starting from the origin, which is restricted in a finite part of the complex plane z. In the diagram a critical point of Feigenbaum is marked F, and the critical point of Golberg - Sinai - Khanin (GSK). To come to this point one has to follow a road on the complex plane through the "leaves" of the Mandelbrot cactus of tripled period. In the top semi-plane the point GSK is placed at

lGSK = 0.0236411685 + 0.7836606508i,

and in the bottom semi-plane at the conjugate value l. [A.I. Golberg, Y.G. Sinai, K.M. Khanin. Universal properties of the period-tripling sequence. Russ. Math. Surveys, 38, 1983, 187.]

Near the critical point GSK a structure of "leaves" of the Mandelbrot set possesses a property of scale invariance in respect to rescaling of g-gGSK with the complex factor d=4.600225588.98122473i. The below diagram illustrates this statement on the Lyapunov chart, where values of the Lyapunov exponent are coded by colors from dark to light from minus infinity to zero, and by black for positive values. A uniform gray tone corresponds to escape of the orbits to infinity. On each successive diagram, the coding rule is redefined to take into account triple decrease of the characteristic scale for the Lyapunov exponent.

Let us consider now a stochastic map

where xn is a sequence of statistically independent complex random variables with zero mean and fixed variance s. As seen from the following picture (built for e=0.001), in presence of noise the scale invariance of the Lyapunov chart violates.

Renormalization group (RG) analysis of the problem with noise leads to conclusion that a universal constant exists shown how much has one to decrease the noise intensity to make observable one more level of the period-tripling: g=12.2066409 [Isaeva, Kuznetsov, Osbaldestin, 2004] (Content of the RG analysis see here .)

Let us assume that at some complex value of the parameter l close to the critical point GSK at a sufficiently small noise intensity e some certain regime of behavior of the model is observed. Then, with the deflection from the critical point (l-lGSK)/d, where d=4.600225588.98122473i is complex scaling factor, and at the noise level e/g a statistically similar regime will occur with characteristic time scale triple as large. The probability distribution for the variable z near the origin will be similar to the original one, with rescaling of the coordinate z by factor a=2.09691989+2.35827964i.

Let us consider two computer illustrations of the scaling property.

The first figure shows Lyapunov charts analogous to the previous diagram, but now each next picture corresponds to the noise intensity decreased by factor g. As seen, similarity of the pictures is recovered.

The next figure shows portraits of "noisy attractors" of the complex quadratic map at parameter values, which correspond in the noiseless case to the so-called superstable orbits of period 3, 9 and 27, i.e. to those containing a point z=0, respectively, at

and for the noise intensity parameters indicated on the plots. As observed, with appropriate rescaling of the complex coordinate (see the selected fragments on the diagrams), the portraits look similar.

Saratov group
of theoretical nonlinear
dynamics
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