Introduction to the Modeling and Analysis of Complex Systems

164 CHAPTER 9. CHAOSfigure()ax = gca(projection=’3d’)ax.plot(xresult, yresult, zresult, ’b’)show()This code produces two outputs: one is the time series plots of x, y, and z (Fig. 9.7), andthe other is the trajectory of the system’s state in a 3-D phase space (Fig. 9.8). As you cansee in Fig. 9.7, the behavior of the system is highly unpredictable, but there is definitelysome regularity in it too. x and y tend to stay on either the positive or negative side,while showing some oscillations with growing amplitudes. When the oscillation becomestoo big, they are thrown to the other side. This continues indefinitely, with occasionalswitching of sides at unpredictable moments. In the meantime, z remains positive all thetime, with similar oscillatory patterns.Plotting these three variables together in a 3-D phase space reveals what is called theLorenz attractor (Fig. 9.8). It is probably the best-known example of strange attractors,i.e., attractors that appear in phase spaces of chaotic systems.Just like any other attractors, strange attractors are sets of states to which nearbytrajectories are attracted. But what makes them really “strange” is that, even if they looklike a bulky object, their “volume” is zero relative to that of the phase space, and thusthey have a fractal dimension, i.e., a dimension of an object that is not integer-valued.For example, the Lorenz attractor’s fractal dimension is known to be about 2.06, i.e., itis pretty close to a 2-D object but not quite. In fact, any chaotic system has a strangeattractor with fractal dimension in its phase space. For example, if you carefully look atthe intricate patterns in the chaotic regime of Fig. 8.10, you will see fractal patterns theretoo.Exercise 9.5 Draw trajectories of the states of the Lorenz equations in a 3-Dphase space for several different values of r while other parameters are kept constant.See how the dynamics of the Lorenz equations change as you vary r.Exercise 9.6 Obtain the equilibrium points of the Lorenz equations as a functionof r, while keeping s = 10 and b = 3. Then conduct a bifurcation analysis on eachequilibrium point to find the critical thresholds of r at which a bifurcation occurs.