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Human eye is bored by monotonous landscape. We are depressed by gray skies, unending plain under the wing of an airliner, featureless urban sprawl, geometric canvasses in empty museum halls. Having evolved to meet the challenges of a complex world, we enjoy complexity. Stormy sea entertains us by a spectacle of complexity. It is imitated by entertainers overwhelming us by the information deluge of blinking images in video clips. We also do enjoy, however, the calm of complexity: a mountain landscape, a forest, a meadow, a mosaic.

A general dynamical system is just a set of ordinary differential equations (ODEs) that can be written as a system of first-order equations resolved in respect to the time derivatives:

If the amplitude is allowed to vary in space, a diffusional term is added to the respective amplitude equation. For a bifurcation at zero eigenvalue in a general RDS, spatial dependence can be incorporated in the formal expansion of Sect. 1.3.2 by scaling the spatial derivatives as ∇ = ( when the expansion proceeds to the th order. Starting from a general RDS (1.18), one obtains then the term D∇a with the amplitude diffusivity added to the solvability condition in the respective order, yielding a single () diffusion equation with a polynomial nonlinearity of degree .

A reaction-diffusion model with separated scales suitable for generation of a variety of patterns is the two-component system (1.36) with the diffusivity ratio ∈ ≪ 1. The system includes two variables: a short-range “activator” and a long-range “inhibitor” .

A typical dispersion relation λ(κ) in a spatially extended system just beyond a bifurcation point may have one of the forms shown in Fig. 4.1. In both cases, a narrow band of wavenumbers adjacent to the maximum of the dispersion relation (which may be reached either at κ = 0 or at a finite κ = κ) becomes unstable. Since the dispersion relation should be, generically, parabolic near the maximum, and the leading eigenvalue can be assumed to depend linearly on a chosen bifurcation parameter, say, , the width of the excited band scales as the square root of the deviation from the bifurcation point –. A spectral band of a finite width can be modeled by allowing the amplitude to change on an extended spatial scale, large compared to either κ or any “natural” length scale characteristic to the underlying system.

The complex Ginzburg–Landau (CGL) equation generalizes the normal form (1.70) at the Hopf bifurcation to spatially extended systems.