When Trajectories Converge to a Curve

TL;DR : This is a math post (and part 1 of a theme, here's part 2 ). I was wondering how easy it is to build a dynamical system with a user-defined attracting curve, e.g., a sine wave. You pick a point and if it starts on the curve, then it remains there, otherwise it smoothly gets attracted to the curve as t increases. I don't recall where I first saw this, but I think it's a standard…