Introduction to the Modeling and Analysis of Complex Systems

14.3. LINEAR STABILITY ANALYSIS OF CONTINUOUS FIELD MODELS 27514.3 Linear Stability Analysis of Continuous Field ModelsWe can apply the linear stability analysis to continuous field models. This allows us toanalytically obtain the conditions for which a homogeneous equilibrium state of a spatialsystem loses its stability and thereby the system spontaneously forms non-homogeneousspatial patterns. Note again that the homogeneous equilibrium state discussed here is nolonger a single point, but it is a straight line (or a flat plane) that covers the entire spatialdomain.Consider the dynamics of a nonlinear continuous field model(∂f∂t = F f, ∂f )∂x , ∂2 f∂x , . . . (14.35)2around its homogeneous equilibrium state f eq , which satisfies0 = F (f eq , 0, 0, . . .) . (14.36)The basic approach of linear stability analysis is exactly the same as before. Namely,we will represent the system’s state as a sum of the equilibrium state and a small perturbation,and then we will determine whether this small perturbation added to the equilibriumwill grow or shrink over time. Using ∆f to represent the small perturbation, we apply thefollowing replacementf(x, t) ⇒ f eq + ∆f(x, t) (14.37)to Eq. (14.35), to obtain the following new continuous field model:(∂f eq + ∆f= F f eq + ∆f, ∂(f )eq + ∆f), ∂2 (f eq + ∆f), . . .∂t∂x ∂x(2∂∆f= F f eq + ∆f, ∂∆f )∂t∂x , ∂2 ∆f∂x , . . . 2(14.38)(14.39)Now, look at the equation above. The key difference between this equation and theprevious examples of the non-spatial models (e.g., Eq. (7.67)) is that the right hand side ofEq. (14.39) contains spatial derivatives. Without them, F would be just a nonlinear scalaror vector function of f, so we could use its Jacobian matrix to obtain a linear approximationof it. But we can’t do so because of those ∂’s! We need something different to eliminatethose nuisances.