In this video (which happens to be my first ever 1080p video!), I discuss linear stability analysis, in which we consider small perturbations about the fixed point, and then analyze the local behavior of the differential equation/dynamical system around the fixed point.
The derivation involves a simple linear expansion of the differential equation function around the fixed point. When the x derivative of the ODE function f(x) is positive, we have an unstable fixed point. When it's negative, we have a stable fixed point. I didn't discuss a special (but more rare case) in the video, but when f'(xf) is zero, we usually have a 'half-stable' fixed point (stable from one direction, unstable from the other).
Questions/requests? Let me know in the comments below!
Prereqs: The first 2 videos in my Nonlinear Dynamics/Dynamical Systems playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4UkZFbndqWWZMVHc
Special thanks to my Patrons: Jacob Soares