Introduction to the Modeling and Analysis of Complex Systems

5.2. PHASE SPACE VISUALIZATION OF CONTINUOUS-STATE DISCRETE-TIME... 63For discrete-time systems with continuous-state variables (i.e., state variables thattake real values), drawing a phase space can be done very easily using straightforwardcomputer simulations, just like we did in Fig. 4.3. To reveal a large-scale structure of thephase space, however, you will probably need to draw many simulation results startingfrom different initial states. This can be achieved by modifying the initialization functionso that it can receive specific initial values for state variables, and then you can put thesimulation code into for loops that sweep the parameter values for a given range. Forexample:Code 5.1: phasespace-drawing.pyfrom pylab import *def initialize(x0, y0): ###global x, y, xresult, yresultx = x0 ###y = y0 ###xresult = [x]yresult = [y]def observe():global x, y, xresult, yresultxresult.append(x)yresult.append(y)def update():global x, y, xresult, yresultnextx = 0.5 * x + ynexty = -0.5 * x + yx, y = nextx, nextyfor x0 in arange(-2, 2, .5): ###for y0 in arange(-2, 2, .5): ###initialize(x0, y0) ###for t in xrange(30):update()observe()plot(xresult, yresult, ’b’) ###