Introduction to the Modeling and Analysis of Complex Systems

Chapter 14Continuous Field Models II: Analysis14.1 Finding Equilibrium StatesOne nice thing about PDE-based continuous field models is that, unlike CA models, everythingis still written in smooth differential equations so we may be able to conductsystematic mathematical analysis to investigate their dynamics (especially their stabilityor instability) using the same techniques as those we learned for non-spatial dynamicalsystems in Chapter 7.The first step is, as always, to find the system’s equilibrium states. Note that this is nolonger about equilibrium “points,” because the system’s state now has spatial extensions.In this case, the equilibrium state of an autonomous continuous field model(∂f∂t = F f, ∂f )∂x , ∂2 f∂x , . . . (14.1)2is given as a static spatial function f eq (x), which satisfies(0 = F f eq , ∂f )eq∂x , ∂2 f eq∂x , . . . . (14.2)2For example, let’s obtain the equilibrium state of a diffusion equation in a 1-D spacewith a simple sinusoidal source/sink term:∂c∂t = D∇2 c + sin x (−π ≤ x ≤ π) (14.3)The source/sink term sin x means that the “stuff” is being produced where 0 < x ≤ π,while it is being drained where −π ≤ x < 0. Mathematically speaking, this is still a nonautonomoussystem because the independent variable x appears explicitly on the right269