Introduction to the Modeling and Analysis of Complex Systems

14.3. LINEAR STABILITY ANALYSIS OF CONTINUOUS FIELD MODELS 2770 1-D space1f(x)space discretizationDh = 1 / n n = 5 n = 10n → ∞-1 1 0 0 01 11 -2 1 0 00 1 -2 1 0 0 0 0 0 00 0 1 -2 1 0 0 0 0 00 1 -2 1 0 = l = lf(x+Dh)+f(x–Dh)–2f(x)lim0 0 0 1 -2 1 0 0 0 00 0 0 0 1 -2 1 0 0 0Dh 20 0 1 -2 1Dh 2Dh → 0 Dh 20 0 0 0 0 1 -2 1 0 00 0 0 1 -1-1 1 0 0 0 0 0 0 0 01 -2 1 0 0 0 0 0 0 00 0 0 0 0 0 1 -2 1 00 0 0 0 0 0 0 1 -2 10 0 0 0 0 0 0 0 1 -1= lmatrixeigenvectormatrixeigenvectorlinear operator ( 2 )eigenfunctionFigure 14.2: Relationship between matrices/eigenvectors and linear operators/eigenfunctions.∆• For L = ∂∂x :f(x) = Ce λx (14.41)• For L = ∂2∂x 2 :f(x) = C 1 e √ λx + C 2 e −√ λx(14.42)Here, λ is the eigenvalue and C, C 1 , and C 2 are the constants of integration. You canconfirm that these eigenfunctions satisfy Eq. (14.40) by applying L to it. So, if we use suchan eigenfunction of the spatial derivative remaining in Eq. (14.39) as a small perturbation∆f, the equation could become very simple and amenable to mathematical analysis.There is one potential problem, though. The eigenfunctions given above are all exponentialfunctions of x. This means that, for x with large magnitudes, these eigenfunctionscould explode literally exponentially! This is definitely not a good property for a perturbationthat is supposed to be “small.” What should we do? There are at least two solutionsfor this problem. One way is to limit the scope of the analysis to a finite domain of x (andλ, too) so that the eigenfunction remains finite without explosion. The other way is to