Introduction to the Modeling and Analysis of Complex Systems

12.2. PHASE SPACE VISUALIZATION 211For example, a one-dimensional binary CA model with L = 10 has 2 101 = 1024 possibleconfigurations, a two-dimensional binary CA model with L = 10 has 2 102 ≈ 1.27 × 10 30possible configurations. The latter is large, but it isn’t so huge compared to the size of therule spaces you saw in the exercises above.Exercise 12.3 Calculate the number of all possible configurations of a twodimensional,three-state CA model with L = 100.12.2 Phase Space VisualizationIf the phase space of a CA model is not too large, you can visualize it using the techniquewe discussed in Section 5.4. Such visualizations are helpful for understanding the overalldynamics of the system, especially by measuring the number of separate basins of attraction,their sizes, and the properties of the attractors. For example, if you see only onelarge basin of attraction, the system doesn’t depend on initial conditions and will alwaysfall into the same attractor. Or if you see multiple basins of attraction of about comparablesize, the system’s behavior is sensitive to the initial conditions. The attractors may bemade of a single state or multiple states forming a cycle, which determines whether thesystem eventually becomes static or remains dynamic (cyclic) indefinitely.Let’s work on an example. Consider a one-dimensional binary CA model with neighborhoodradius r = 2. We assume the space is made of nine cells with periodic boundaryconditions (i.e., the space is a ring made of nine cells). In this setting, the size of its phasespace is just 2 9 = 512, so this is still easy to visualize.In order to enumerate all possible configurations, it is convenient to define functionsthat map a specific configuration of the CA to a unique configuration ID number, and viceversa. Here are examples of such functions:Code 12.1:def config(x):return [1 if x & 2**i > 0 else 0 for i in range(L - 1, -1, -1)]def cf_number(cf):return sum(cf[L - 1 - i] * 2**i for i in range(L))Here L and i are the size of the space and the spatial position of a cell, respectively.These functions use a typical binary notation of an integer as a way to create mapping