Introduction to the Modeling and Analysis of Complex Systems

74 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISx maxx 3x 2x 1x tx minx minx 0x t-1x 1x 2x maxFigure 5.9: Drawing a cobweb plot (6).5.4 Graph-Based Phase Space Visualization of Discrete-State Discrete-Time ModelsThe cobweb plot approach discussed above works only for one-dimensional systems,because we can’t embed such plots for any higher dimensional systems in a 3-D physicalspace. However, this dimensional restriction vanishes if the system’s states are discreteand finite. For such a system, you can always enumerate all possible state transitionsand create the entire phase space of the system as a state-transition graph, which can bevisualized reasonably well even within a 2-D visualization space.Here is an example. Let’s consider the following second-order (i.e., two-dimensional)difference equation:x t = x t−1 x t−2 mod 6 (5.18)The “mod 6” at the end of the equation means that its right hand side is always aremainder of the division of x t−1 x t−2 by 6. This means that the possible state of x islimited only to 0, 1, 2, 3, 4, or 5, i.e., the state of the system (x, y) (where y is the previous