Analysis
Further information: Gradient Pattern AnalysisThe analysis of pattern-forming systems often consists of finding a Partial differential equation model of the system (the Swift-Hohenberg equation is one such model) of the form
where F is generically a nonlinear differential operator, and postulating solutions of the form
where the are complex amplitudes associated to different modes in the solution and the are the wave-vectors associated to a lattice, e.g. a square or hexagonal lattice in two dimensions. There is in general no rigorous justification for this restriction to a lattice.
Symmetry considerations can now be taken into account, and evolution equations obtained for the complex amplitudes governing the solution. This reduction puts the problem into the form of a system of first-order Ordinary differential equation, which can be analysed using standard methods (see dynamical systems). The same formalism can also be used to analyse bifurcations in pattern-forming systems, for example to analyse the formation of convection rolls in a Rayleigh-Bénard experiment as the temperature is increased.
Such analysis predicts many of the quantitative features of such experiments - for example, the ODE reduction predicts hysteresis in convection experiments as patterns of rolls and hexagons compete for stability. The same hysteresis has been observed experimentally.
Read more about this topic: Pattern Formation
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—Karl Marx (1818–1883)
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—Henry Brooks Adams (1838–1918)