dynamical systems come in two flavors — continuous and discrete time. we’ll be covering lots of interesting examples in both settings. but for the moment, it’s worth noting that 1-d discrete-time systems ( or “maps” ) provide a microcosm of what dynamical systems entails.

illustrated below is a **staircase diagram** of the *logistic map*, defined by ** x(n+1) = f(x(n))**, where f is a quadratic of the form

**. as you change the value of the parameter**

*f(x) = a x ( 1 – x )***and the initial condition**

*a>0***, then poltting**

*x(0)***versus**

*x(n+1)***is illuminating. you can see stable and unstable equilibria, periodic orbits of various types, and everything changing. how to make sense of it all? that’s what dynamical systems covers.**

*x(n)*