Introduction to the Modeling and Analysis of Complex Systems

66 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISax = gca(projection=’3d’)for x0 in arange(-2, 2, 1):for y0 in arange(-2, 2, 1):for z0 in arange(-2, 2, 1):initialize(x0, y0, z0)for t in xrange(30):update()observe()ax.plot(xresult, yresult, zresult, ’b’)show()Note that it is generally not a good idea to draw many trajectories in a 3-D phase space,because the visualization would become very crowded and difficult to see. Drawing asmall number of characteristic trajectories is more useful.In general, you should keep in mind that phase space visualization of discrete-timemodels may not always give images that are easily visible to human eye. This is becausethe state of a discrete-time system can jump around in the phase space, and thus thetrajectories can cross over each other (this will not happen for continuous-time models).Here is an example. Replace the content of the update function in Code 5.1 with thefollowing:Code 5.3: phasespace-drawing-bad.pynextx = - 0.5 * x - 0.7 * ynexty = x - 0.5 * yAs a result, you get Fig. 5.3.While this may be aesthetically pleasing, it doesn’t help our understanding of the systemvery much because there are just too many trajectories overlaid in the diagram. Inthis sense, the straightforward phase space visualization may not always be helpful foranalyzing discrete-time dynamical systems. In the following sections, we will discuss afew possible workarounds for this problem.