## 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…

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…

## equilibria: stable & unstable

dynamical systems begins with the notion of equilibria — states that remain constant over time. In the simplest cases, one distinguishes between stable and unstable equilibria, though to specify what exactly these mean requires care (and math).

one simple example of stable and unstable equilibria is found in a rigid-rod pendulum, rotating about a pivot point (with or, as here, without, friction). letting the system evolve from a variety of initial conditions reveals a stable equilibrium (bottom) and an unstable equilibrium (top).

there’s a lot to discuss about this system. skipping ahead (for the sake of seeing something wonderful), let’s consider what happens when you give this system a shake, vibrating the pivot point up & down. the simulation indicates that past a criticla threshold of amplitude/frequency, shaking the pendulum converts the unstable equilibrium to a stable equilibrium.

that’s swell. but does the mathematics prove that this is the expected behavior? well, that’s one of the reasons why you take a dynamical systems course.