i think the best way to begin learning about discrete-time dynamical systems is with the following well-known example. take a calculator; set it to radians; type in a random number; then hit the cosine button over & over. what happens is convvergence to a stable equilibrium. the way the covergence happens is through over- then under-shooting. why? let’s take a look…

for discrete-time 1-d dynamics, the best way to visualize is via a staircase diagram, in which you plot *x_{n+1}* versus *x_n*, where *x_{n+1} = f(x_n)*. this allows you to investigate what happens for arbitrary functions *f*.

if you look closely, you can see the differences between stable and unstable equilibria. of course, more interesting things can happen, but we’ll talk about that later…