Dimension Three

In this chapter, we introduce four vector fields to illustrate selected highlights from three-dimensional dynamics and bifurcations. In the first example, a periodic orbit of a vector field in IR3 yields its stability to another periodic orbit of approximately twice the period. In the second example, as a periodic orbit becomes unstable, an invariant torus appears nearby. Using an appropriate Poincaré map, these two local bifurcations are, respectively, the counterparts of the period-doubling and Poincaré-Andronov Hopf bifurcations of a fixed point of a planar map from Chapter 15. The third example illustrates an important source of chaotic dynamics other than successive period-doubling bifurcations, a special type of homoclinic orbit—Sil’nikov orbit or saddle-focus. The final example features the Lorenz equations, the strange attractor mast, photographed.