Introduction to the Modeling and Analysis of Complex Systems

68 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSIS32101233 2 1 0 1 2 3Figure 5.3: Phase space drawn with Code 5.3.5.3 Cobweb Plots for One-Dimensional Iterative MapsOne possible way to solve the overcrowded phase space of a discrete-time system is tocreate two phase spaces, one for time t − 1 and another for t, and then draw trajectoriesof the system’s state in a meta-phase space that is obtained by placing those two phasespaces orthogonally to each other. In this way, you would potentially disentangle thetangled trajectories to make them visually understandable.However, this seemingly brilliant idea has one fundamental problem. It works onlyfor one-dimensional systems, because two- or higher dimensional systems require fourormore dimensions to visualize the meta-phase space, which can’t be visualized in thethree-dimensional physical world in which we are confined.This meta-phase space idea is still effective and powerful for visualizing the dynamicsof one-dimensional iterative maps. The resulting visualization is called a cobweb plot,which plays an important role as an intuitive analytical tool to understand the nonlineardynamics of one-dimensional systems.Here is how to manually draw a cobweb plot of a one-dimensional iterative map, x t =f(x t−1 ), with the range of x t being [x min , x max ]. Get a piece of paper and a pen, and do the