Introduction to the Modeling and Analysis of Complex Systems

9.4. CHAOS IN CONTINUOUS-TIME MODELS 167Compare your result with numerical simulation results obtained in the previousexercise.By the way, I said before that any chaotic system has two dynamical processes:stretching and folding. Where do these processes occur in the Lorenz attractor? It isnot so straightforward to fully understand its structure, but the stretching occurs wherethe trajectory is circling within one of the two “wings” of the attractor. The spirals seenon those wings are unstable ones going outward, so the distance between initially closestates are expanded as they circle around the spiral focus. In the meantime, the foldingoccurs at the center of the attractor, where two “sheets” of trajectories meet. Those twosheets actually never cross each other, but they keep themselves separated from eachother, forming a “wafer sandwich” made of two thin layers, whose right half goes on tocircle in the right wing while the left half goes on to the left wing. In this way, the “dough”is split into two, each of which is stretched, and then the two stretched doughs are stackedon top of each other to form a new dough that is made of two layers again. As this processcontinues, the final result, the Lorenz attractor, acquires infinitely many, recursivelyformed layers in it, which give it the name of a “strange” attractor with a fractal dimension.Exercise 9.7 Plot the Lorenz attractor in several different perspectives (the easiestchoice would be to project it onto the x-y, y-z, and x-z planes) and observe itsstructure. Interpret its shape and the flows of trajectories from a stretching-andfoldingviewpoint.I would like to bring up one more important mathematical fact before we close thischapter:In order for continuous-time dynamical systems to be chaotic, the dimensions of thesystem’s phase space must be at least 3-D. In contrast, discrete-time dynamical systemscan be chaotic regardless of their dimensions.The Lorenz equations involved three variables, so it was an example of continuous-timechaotic systems with minimal dimensionality.This fact is derived from the Poincaré-Bendixson theorem in mathematics, which statesthat no strange attractor can arise in continuous 2-D phase space. An intuitive explanationof this is that, in a 2-D phase space, every existing trajectory works as a “wall” that you