Abstract: I will speak of the “garden-variety” differential equation x’’ +.1 x’ + sin(x)=cos(t), which models a forced damped pendulum. It has phenomenally complicated dynamics: it has an “attracting mode”, which has the Wada property: every point on the boundary of one basin is on the boundary of all infinitely many others. I will also discuss motions “controlled" by symbolic dynamics.

The lecture is intended to illustrate interesting dynamics, but slide to illustrate what can now, with the help of computers, be taught even in undergraduate courses. It also illustrates that “chaos” is the flip side of “controllability”.