Analysis of Nonlinear Systems, Part 1 (Nullclines and Linearization), and a Long and Lame Joke

Differential Equations (with DE Tools Printed Access Card) 4th Edition: https://amzn.to/3a6E3J2 Differential Equations Lectures Playlist: https://www.youtube.com/watch?v=8ZgHotcDMa8&list=PLmU0FIlJY-MkNz306pfzQE2D9tNHU6SQp. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP. Check out my blog: https://infinityisreallybig.com/ (0:09) Intro to the series. (0:37) Dr. Kinney's Long and Lame Jokes to come in the first 3 videos. (1:53) Note that the problems take a while. (2:20) Example: dx/dt = xy - 4x, dy/dt = y - x^2. Note: it's nonlinear. (4:00) Find 3 equilibrium points. (6:54) Draw equilibrium points. (8:27) Define and draw nullclines. (14:02) Determine the directions of the vector field in the various regions the nullclines break the plane up into. (18:10) Linearize near the equilibrium points (a more important application of linearization than those applications encountered in Calculus). Linearizing near the origin amounts to ignoring nonlinear terms in the original system (create an associated linear system). (29:10) Linearization near the other equilibria with the Jacobian matrix, determining the nature of the equilbria with the trace and determinant of the Jacobian matrix (this trick only works if all eigenvalues have nonzero real part). Mention the idea of a separatrix. (34:36) Long and Lame Joke of the Day. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.