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 f(x) = a x ( 1 – x ). as you change the value of the parameter a>0 and the initial condition x(0), then poltting x(n+1) versus x(n) 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.