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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
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
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.