Introduction to the Modeling and Analysis of Complex Systems

46 CHAPTER 4. DISCRETE-TIME MODELS I: MODELINGgrowth with constant removals due to fishing, and growth of credit card balanceswith constant monthly payments (both with negative b). Change the values of aand b, and see if the system’s behaviors are the same as those of the simpleexponential growth/decay model.4.4 Simulating Discrete-Time Models with Multiple VariablesNow we are making a first step to complex systems simulation. Let’s increase the numberof variables from one to two. Consider the following difference equations:x t = 0.5x t−1 + y t−1 (4.14)y t = −0.5x t−1 + y t−1 (4.15)x 0 = 1, y 0 = 1 (4.16)These equations can also be written using vectors and a matrix as follows:( xy)t=( ( )0.5 1 x−0.5 1)yt−1(4.17)Try implementing the simulation code of the equation above. This may seem fairlystraightforward, requiring only minor changes to what we had before. Namely, you justneed to simulate two difference equations simultaneously. Your new code may look likethis:Code 4.11: oscillation-wrong.pyfrom pylab import *def initialize():global x, y, xresult, yresultx = 1.y = 1.xresult = [x]yresult = [y]def observe():