visualizing 1-d flows

visualization is so central to dynamical systems. one great place to get started is with 1-d continuous-time autonomous systems: ODEs of the form dx/dt=f(x). there are a lot of ways to illustrate such systems. one is to plot the solution x(t) for an initial condition x(0). that’s ok, but it’s even better to look at the flow, where you look at all initial conditions at once and let them evolve. but it does not end there. i like to supress time altogether, and plot x versus dx/dt. this makes it possible to visualize the dynamics as that of a vector field on the x-axis. you can see all of these perspectives illustrated below: