staircase diagrams

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…

staircase diagram for the cosine button-mashing game

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.

staircase diagram with multiple stable and unstable equilibria

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…

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