Introduction to the Modeling and Analysis of Complex Systems

270 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSIShand side. But this non-autonomy can be easily eliminated by replacing x with a newstate variable y that satisfies∂y= 0,∂t(14.4)y(x, 0) = x. (14.5)In the following, we will continue to use x instead of y, just to make the discussion easierand more intuitive.To find an equilibrium state of this system, we need to solve the following:0 = D∇ 2 c eq + sin x (14.6)= D d2 c eq+ sin x (14.7)dx2 This is a simple ordinary differential equation, because there are no time or additionalspatial dimensions in it. You can easily solve it by hand to obtain the solutionc eq (x) = sin xD + C 1x + C 2 , (14.8)where C 1 and C 2 are the constants of integration. Any state that satisfies this formularemains unchanged over time. Figure 14.1 shows such an example with D = C 1 = C 2 = 1.Exercise 14.1 Obtain the equilibrium states of the following continuous fieldmodel in a 1-D space:∂c/∂t = D∇ 2 c + 1 − x 2 (14.9)As we see above, equilibrium states of a continuous field model can be spatially heterogeneous.But it is often the case that researchers are more interested in homogeneousequilibrium states, i.e., spatially “flat” states that can remain stationary over time. This isbecause, by studying the stability of homogeneous equilibrium states, one may be ableto understand whether a spatially distributed system can remain homogeneous or selforganizeto form non-homogeneous patterns spontaneously.Calculating homogeneous equilibrium states is much easier than calculating generalequilibrium states. You just need to substitute the system state functions with constants,