Let us consider a system of two coupled maps

where g(x)=1�1.5276x²+0.1048x⁴+... is a function representing the fixed point of the Feigenbaum�Cvitanovic equation g(x)=ag(g(x/ a)), a=�2.5029... is Feigenbaum's constant, e is small parameter, j (x, y) � is function of coupling that obeys a condition j (x, x)=0. Near the diagonal on the plane (x,y) we have

.

Two-fold iteration of the above equations yields

and after renormalization of scale u® u/a, v® v/a we obtain

These are equations of the same form as the original ones, but with a new function of coupling

Let us turn now to the eigenvalue problem

Analysis and numerical solution reveal two relevant eigenfunctions:

The first one corresponds to the inertial, and the second one to the dissipative coupling. Beside of them there is an eigenvalue d =4.6692..., that is present in the Feigenbaum theory and is associated with deflections of the control parameter in an individual map from the critical point. So, the renormalized evolution operator over large number of iterations near the critical point is represented as

This implies the property of universality: for any two weakly coupled period doubling maps the evolution operator near the diagonal is expressed by this universal relation and is determened by three parameters (C, C₁, C₂). Next, the property of scaling holds: At the point of the parameter space (Cd ^�1, C₁a^�1, C₂2^�1) dynamical regimes arise of the same kind as at the point C, C₁, C₂), but with doubled time scale, and initial conditions for dynamical variables are obtained by rescaling with factor a=- 2.5029�