alphabet

renormalization group method

golden-mean RG analysis

Let us consider a one-dimensional map .
The function
determines evolution operator for one step of discrete time, it is supposed
to have a maximum at the origin. For two steps we obtain ,
and let us introduce instead of *x* a new variable rescaled with factor
. Changing
*x* to*
* in both sides of the equation, we write ,
where .
Now let us take *g*_{1}(*x*) as a new initial function and
produce the same operations. Then, we obtain a renormalized evolution operator
for four steps: ,
where .
Repetition of the procedure yields a recurrent functional equation

.

If the original map *g*_{0}(*x*) depends on parameter and
demonstrates period-doubling cascade, then, at the accumulation point of the
period-doubling, with appropriate selection of rescaling constant a we have
. The limit
function *g*(*x*) will satisfy to the *Feigenbaum-Cvitanović
*equation

,

i.e. it is a *fixed point of the functional equation.* If the maximum
at the origin is quadratic, it will correspond to the Feigenbaum criticality
(type F); powers 4, 6, 8 correspond to other universality
classes (T, S, E).

Let us now search for solution of the functional equation close to the fixed point in a form . At we have

.

This equation has structure of linear operator equation .
If we have some eigenfunction *h*(*x*) and eigenvalue n satisfying
, then a
solution may contain a component that behaves in a course of iterations as .
Those eigenvalues will be of relevance, which are larger in absolute value than
1 and are not associated with infinitesimal variable changes. A number of relevant
eigenvalues corresponds to *codimension* of the given type of criticality.
The eigenvalues themselves define *parameter space scaling factors* in
some special local coordinate system (*scaling coordinats*).

Generalization of the above analysis for two-dimensional mappings ,
consists
in the following. We select appropriate coordinates on the plane (*x*,
*y*) and construct a functional sequence

If it converges (with properly selected and ) to a fixed point, or to a periodic orbit, it will correspond to some criticality type and universality class. Examples are types В, BT, H, FQ, C from our collection.

main science alphabet this page only top

Saratov groupof theoretical nonlinear

dynamics

Хостинг от uCoz