Let us have a complex analytical map corresponding to a critical point of the accumulation of the period-tripling bifurcations, with added random force:
where
xn is a sequence of statistically independent complex random variables with zero mean and fixed variance s.
Let us apply this map three times. In assumption that the noise intensity parameter e is small, in the first order in this parameter we get:
or, after scale change x-->x/a,
with the complex scaling constant of Golberg, Sinai and Khanin
The expression with the coefficient e may be interpreted as a new random variable designated as U1(zn)x'n, where the factor U1(zn)x'n is determined from a condition the value x'n to have the same variance s as the original random sequence x. As the numbers xn, xn+1, xn+2 are statistically independent, the average squared values of the terms of the sum accumulate. So, the equation for the three-fold iteration is reduced to the same form as the original, namely,
but with new functions
The procedure performed may be applied many times that yields a sequence of functions gk, Uk, satisfying a chain of recurrence functional equations. This sequence gk converges to a definite limit, the fixed point of the Golberg - Sinai - Khanin equation
Its solution has been found numerically with high precision via a polynomial approximation of g(z), see here.
A solution for Uk may be searched for in a form |Uk(z)|2µgk(z), which gives rise to an eigenvalue problem
We emphasize that here we have a real function of complex argument (z), and a real eigenvalue g.
In accordance with results of numerical computations, g=12.2066409. A plot of the corresponding eigenfunction is shown in the figure.
