In application to all situations of the quasiperiodic motion with the frequency ratio
given by the golden mean, the main idea of the RG analysis consists in examination
of evolution operators for time intervals determined by successive Fibonacci numbers:
where xn is a sequence of statistically
independent random values with a zero mean and a fixed mean-square value
s, k
is a parameter of intensity of noise supposed to be small,
Applying successively the maps corresponding to
Let us redefine the stochastic term.
Let us have an orbit launched at some moment from a point
(xi, yi).
Consider an ensemble of random numbers
{xi,
xi+Fk+1}
with zero mean and variation s2
and compose a sum with coefficient represented by functions of
xi and yi.
As the terms are statistically independent, the sum
may be represented again as a random number
with zero mean and variation s2,
multiplied by a function of xi and yi:
Let us introduce
and rewrite the equation in the form analogous to the original one,
with redefined random variable and functions
To obtain a closed system of functional equations,
we square both parts of the equation and perform
averaging over the ensemble of the noise realizations. As
we come to a relation
where a prime designates derivative in respect to the first argument.
Following the main idea of the RG approach, let us perform a scale change
where a is the scaling constant
found earlier for the TDT critical point.
Then, in terms of the redefined functions
the above equations imply
These relations define the RG transformation for the set of functions
{gk, fk, Yk,
Fk}.
The routine may be repeated again and again to obtain the functions for larger and larger
k, i.e. to determine the renormalized evolution
operators for time intervals corresponding to iteration numbers Fk.
As known, at the critical point TDT the functional sequence
gk, fk
converges to a stationary solution of period 3,
It implies that a solution of the recursive linear functional equations
{Yk,
Fk}
asymptotically will be determined
by an eigenvector associated with the maximal
eigenvalue W
for the following eigenproblem
where the right-hand part linear operators
are expressed as
A numerical solution yields
W=401.94787411...
Now, in linear approximation in respect to the noise amplitude the stochastic map corresponding
to F3k+q and F3k+q+1+1 steps of evolution at the
TDT critical point may be written in the renormalized variables as
where
Taking into account additional perturbations of the evolution
operator caused by parameter shifts from the critical point,
one can derive from this the scaling law formulation given in the main text.
where the terms of the first order in the noise intensity e are taken into account.