Introduction to the Modeling and Analysis of Complex Systems

150 CHAPTER 8. BIFURCATIONSdef initialize():global x, resultx = 0.1result = []def observe():global x, resultresult.append(x)def update():global x, resultx = x + r - x**2def plot_asymptotic_states():initialize()for t in xrange(100): # first 100 steps are discardedupdate()for t in xrange(100): # second 100 steps are collectedupdate()observe()plot([r] * 100, result, ’b.’, alpha = 0.3)for r in arange(0, 2, 0.01):plot_asymptotic_states()xlabel(’r’)ylabel(’x’)show()In this code, r is gradually varied from 0 to 2 at intervals of 0.01. For each value of r, themodel (Eq. (8.37)) is simulated for 200 steps, and only the second half of the state valuesare recorded in result. Once the simulation is finished, the states stored in resultare plotted at a particular value of r in the plot (note that the expression [r] * 100 inPython produces a list of one hundred r’s). The alpha option is used to make the markerstransparent so that the densities of markers are also visible.The result is shown in Fig. 8.10. This bifurcation diagram still shows how the system’sstate depends on the parameter, but what is plotted here are no longer analytically ob-