From coupled map lattice to continuous medium
Models
Let us consider a coupled map lattice
where f(x)=1-lx2, n is discrete time, m
is spatial index, L is a linear operator that performs averaging over some spatial domain in a symmetric manner.It may be shown that with such structure of the evolution operator, represented as a composition of the local nonlinear and nonlocal linear transformations the coupling between the cells appears to be purely dissipative. Two examples of such coupling are
:
The dissipative coupling tends to equalize the instantaneous states of the interacting cells. As the states of neighboring cells become close in a course of the natural time evolution, description of the lattice as a continuous medium will be natural. Time remains discrete, so this is a coupled map medium. For the future coupling case we have
It is natural to refer to the parameter D as the diffusion coefficient.
Renormalization group analysis
Let the evolution operator over one time step be xn+1=G[xn
]. Let us apply it twice xn+2=G[G[xn]] and perform scale change of the dynamical variable and spatial coordinate by means of an operator Sx(z)=ax(21/2z), where a=-2.5029 is the Feigenbaum constant. Due to realation D=eh2, rescaling of the spatial coordinate by factor 21/2 corresponds to rescaling of the coupling term by 2, that is the eigenvalue associated with the dissipative coupling. In rescaled variables we have xn+2(z)=SG[G[S-1xn]], so, the new evolution operator is G1[x]=SG[G[S-1x]]. Repeating the procedure many times we obtain a sequence of operators
where Gk is an evolution operator over 2k time steps. It appears that at the period-doubling accumulation point of the individual map l=l the operator sequence Gk converges to a universal operator, the fixed point of the renormalization transformation, and it obeys the operator equation
If we depart from the critical parameter value lc a perturbation of the operator fixed point will grow as dk: Gk[x]@G[x]+cdkH[x
], where H is some operator, and µ l-l, d=4.6692...Then, the following properties of universality and scaling should be valid.
Universality. For a medium composed of period-doubling maps with symmetric dissipative coupling the evolution operator over a number of time steps 2
k is asymptotically determined (up to scale changes) by a single parameter µl-l. Scaling. As we have a certain regime of spatio-temporal dynamics at a parameter value close to the critical point l, we have to observe a similar dynamical regime also at l new=l c+(l -l )/d . In the new regime the dynamical variable is rescaled by factor a, the characteristic time scale increases by 2, and the spatial scale by factor 21/2.Computer illustrations of scaling
The first series of pictures relates to the lattice with future coupling
and they represent dynamical variable versus spatial coordinate superimposed for a number of time steps. The first diagram corresponds to l=1.2, so the time period equals 2. The initial conditions are chosen in such way that a domain wall has been formed, whish separates two domains of opposite-phase oscillations in the medium. The width of the wall is determined by the diffusion coefficient, in the present case D=8. Separately, on the right panel, a fragment of the picture is shown, the domain wall itself. The second diagram is obtained at the parameter values rescaled with the rule l
new=lc+(l -l )/d , and with the same diffusion coefficient. Now it is a regime of period 4. In the panel to the right the vertical scale is changed by factor of a and horizontal by factor of 2-1/2 in comparison with the previous diagram. The third picture corresponds to one more level of the scale change. The property of scaling reveals itself as a visually perfect reproduction of the right-hand pictures.
The second series of diagrams relates to supercritical region, and chaotic dynamical regimes take place there. The first picture corresponds to l=1.5437, the second and the third ones to the values rescaled in accordance with the above rule. The diffusion coefficient is D=1. Due to instability intrinsic to the chaotic regimes it can not be expected that the pictures really will look the same, but one can see that they are similar in a more general, statistical sense. In particular, nature of the spatio-temporal dependencies is the same, so the characteristic time and spatial scales behave as expected.
For quantitative check of the scaling one should use appropriate statistical characteristics (spectra, correlation functions) of similar regimes. As an example, we present here spatial spectra in double-logarithmic scale. Curves 1-3 correspond to regimes of the previous figure, and curves 4 and 5 to the next two scaling levels. Observe that the curves look identical up to appropriate shifts along two coordinate axes. The horizontal shift is associated with the spatial coordinate rescaling by 21/2, and the vertical one with rescaling of the dynamical variable.
Universal pattern at the medium edge and finite-size scaling
Let us consider a sufficiently long lattice with a fixed bounadary condition at the left edge xn(0)=0 at the critical parameter value lc=1.401155189:
The upper diagram shows a pattern formating at the edge of the system, and two other panels show fragments of the pattern. The scales on the last two plots differ by factor 2
1/2, in horizontal and by along the vertical direction. Note remarkable correspondence of the yellow and blue panels, which correspond to two levels of scaling. The pattern contais a hierarchy of "tails, associated with different levels of the period doubling, and the tails of each subsequent level are longer by 21/2 times.
In a finite system, for example with boundary conditions xn(0)=xn(m)=0, the tails arise at both edges. If one traces evolution of regimes in dependence of the control parameter, the tails of first levels will have relatively small lengths, but the characteristic length increases by 21/2 times at each subsequent doubling of the temporal period. As the characteristic tail length becomes of order of the system size, it starts to be essential for precise bifurcation values of the parameter, and they will be shifted relatively to the nonperturbed ones that correspond to an individual map or to a spatially uniform state of the unbounded medium.
For period-doubling bifurcations in a finite system the following relation holds:
where D
is a constant responsible for a shift of the scaling center for the tails from the true edge and dependent on concrete
boundary conditions. The next figure shows a chart of dynamical regimes in the parameter plane of the finite-size coupled map
lattice and diagrams illustrating evolution of patterns on a way to chaos. Note that chaos appears as irregular oscillations in the
middle part of the system. Initially it is obviously finite-dimensional, and then larger number of spatial modes enter the play
(compare two top tiagrams).
Saratov group