Introduction to the Modeling and Analysis of Complex Systems

278 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSISfurther investigate those eigenfunctions to see if they can take a form that doesn’t showexponential divergence. Is it possible?For Eq. (14.41), a purely imaginary λ could make f(x) non-divergent for x → ±∞, butthen f(x) itself would also show complex values. This wouldn’t be suitable as a candidateof perturbations to be added to real-valued system states. But for Eq. (14.42), there aresuch eigenfunctions that don’t explode exponentially and yet remain real. Try λ < 0 (i.e,√λ = ai) with complex conjugates C1 and C 2 (i.e., C 1 = c + bi, C 2 = c − bi), and you willobtainf(x) = (c + bi)e iax + (c − bi)e −iax (14.43)= c(e iax + e −iax ) + bi(e iax − e −iax ) (14.44)= c(cos ax + i sin ax + cos(−ax) + i sin(−ax))+ bi(cos ax + i sin ax − cos(−ax) − i sin(−ax)) (14.45)= 2c cos ax − 2b sin ax (14.46)= A(sin φ cos ax − cos φ sin ax) (14.47)= A sin(φ − ax), (14.48)where φ = arctan(c/b) and A = 2c/ sin φ = 2b/ cos φ. This is just a normal, real-valuedsine wave that will remain within the range [−A, A] for any x! We can definitely use suchsine wave-shaped perturbations for ∆f to eliminate the second-order spatial derivatives.Now that we have a basic set of tools for our analysis, we should do the same trickas we did before: Represent the initial condition of the system as a linear combination ofeigen-(vector or function), and then study the dynamics of each eigen-(vector or function)component separately. The sine waves derived above are particularly suitable for thispurpose, because, as some of you might know, the waves with different frequencies (a inthe above) are independent from each other, so they constitute a perfect set of bases torepresent any initial condition.Let’s have a walk-through of a particular example to see how the whole process oflinear stability analysis works on a continuous field model. Consider our favorite Keller-Segel model:∂a∂t = µ∇2 a − χ∇ · (a∇c) (14.49)∂c∂t = D∇2 c + fa − kc (14.50)The first thing we need to do is to find the model’s homogeneous equilibrium state which