If we search a sulution of the operator renormapization group equation G_k+1[x]=SG_k[G_k[S^-1x]] as G_k[x]=G[x]+H_k[x], where ||H||<<1 and G is a fixed point of the operator equation, we can find that the operators H obey the iterative relation

It may be concluded that in a one-dimensional lattice constructed of cells with dominating symmetric dissipative (diffusive) coupling the following properties of universality and scaling must be valid.

Universality. Evolution operator ofer 2^k time steps in asymptotic of large k is determined (up to scale changes) by four parameters. They are the control parameter of individual cells l and three parameters of coupling: anti-symmetric inertial a, anti-symmetric dissipative b, symmetric inertial g.

Scaling. Being given a definite regime of spatio-temporal dynamics at some parameter set (l, a, b, g) we have to observe similar dynamics at l_new=l_c+(l–l_с)/d, a_new=a/n₁, b_new=b/n₂, g_new=g/n₃ with appropriate initial conditions (rescaling by a). In the last case a characteristic time scale grows by 2 times, and a spatial scale by 2^1/2 раза.

Let us consider a model of coupled map lattice

with boundary conditions x(0)=0, x(M)=0. Here the simmetric dissipative coupling, or diffusion, is represented by operator L and the anti-symmatric inertial coupling is characterized by parametera. The top figure to the right is the chart of regimes in the plane of coupling parameter a and control parameter l. Color areas designate states that are approximately uniform in space in central region of the medium. Colors code time periods (inicated by numbers 1, 2, 4, 8). The gray area corresponds to appearance of non-uniform states, travelling regular or chaotic waves. Note self-similarity of the chart: near the critical point a=0, l=l_с the parameter plane arrangement repeats itself in smaller scales, and the scaling factors along the vertical and horizontal axes are d=4.6692 and n₂=1.7698, respectively.

Let us consider now a lattice with evolution operator that is factorizable as in the case of the diffusion coupling, but asymmetric

with boundary conditions x(0)=x*, x(m)=x*, where x* - is the fixed point of an individual map. Beside the pure diffusion we have here anti-symmetric dissipative coupling characterized by parameter b. The top picture to the left is the chart of regimes in the parameter plane (b,l). Color areas designate states that are approximately uniform in space in central region of the medium. Colors code time periods (inicated by numbers 1, 2, 4, 8). The gray area corresponds to appearance of non-uniform states, travelling regular or chaotic waves. Near the critical point b=0, l=l_с the chart of regimes manifests scaling property, with factors d=4.6692 and n₂=1.4142 along vertical and horizontal axes, respectively.

Let us consider a coupled map lattice

,