System of two coupled maps.
Two types of coupling
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Let us consider a system of two cells each of which is represented by a quadratic map
For example, one may imagine two
biological populations, the number of species in which evolves from year to year in accordance with these equations.
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One way to introduce coupling is to assume that first the species breed (die) inside their own population, and then for some period are allowed to migrate between the populations. The next year the same cycle repeats itself again. The equation looks like
Such coupling tends to equalize the instantaneous states of the cells, and it may be called the dissipative coupling.
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Another way: The species can migrate not participating in the living cycle in their own population. The equation is
Such coupling keeps memory of the state at the previous step of evolution, and it may be referred to naturally as the inertial coupling.
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Finally, it is possible that both types of coupling are present, and this will be the combined coupling:
It appears that there is no necessity to invent some other types of coupling, in some sense this equation serves as a universal model of weakly coupled systems, as follows from the
renormalization group analysis.
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Universal model
Setting
f (x)=1-lx2, where l
is a parameter, we arrive at the equations of the universal model
where n
and g
are parameters of coupling. As follows from numerical computations, relation between these parameters and those participating in the
renormalization group analysis is as follows:
,
Note that
C1 and C2
have sense of coefficients of pure inertial and dissipative coupling, respectively, while is deflection of the control parameter from the critical point. The values
(C, C1, C2)
may be treated as special local coordinates in the parameter space appropriate for
analysis and illustration of the scaling properties.
Parameter space charts,
multistability, scaling properties
Occurence of multistability in a set of coupled maps is due to the following. Let we have first two uncoupled cells, and the control parameter is selected to have stable period-2 orbit in both of them. In the composed system in dependence on initial conditions two cells can oscillate in the same or in the opposite phases.
Both regimes are stable, hence, they continue to exist at finite, at least small arbitrary coupling of the cells. For oscillatory regimes of larger period a number of possible coexisting attractors of the composed system increases.
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Two regimes in-phase and anti-phase in a system of two maps oscillating with period 2 |
Four regimes in a system of two maps oscillating with
period 4. Regime (A) is in-phase, regime (B) is anti-phase, regimes (C) and (D) are mutually symmetric in respect to the exchange of two cells. |
Accounting presence of multistability, we should think of the parameter plane charts of dynamical regimes as consisting of multi-layer partially overlapping surfaces.
Illustration of scaling in the parameter plane of the dissipatively coupled maps for the in-phase and anti-phase sheets.
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Scale change from a picture to the next one is performed by factor d
=4.6692 along the vertical and 2 along the horizontal axis. |
Illustration of scaling in the parameter plane of the inertially coupled maps for the in-phase and anti-phase sheets.
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Scale change from a picture to the next one is performed by factor d
=4.6692 along the vertical and a
=-
2.5029 along the horizontal axis. |
Coupled map lattices
Coupled maps: Two types of coupling
Two coupled maps: Renormalization group analysis
Coupled map lattice: Lattice scaling
From lattice to continuous medium
Medium with different types of coupling
Globally coupled maps
Coupled map lattices and real physical systems
Saratov group
of theoretical nonlinear
dynamics