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The Lorenz Equations - Dynamical Systems | Lecture 27

We did it! We made it to 3D systems! In this lecture we do a case study of the celebrated Lorenz equations. This dynamical system is a three-dimensional system that exhibits many of the phenomena we've already seen in one and two dimensions, such as pitchfork bifurcations, Hopf bifurcations, homoclinic connections, and the use of Lyapunov functions. The lecture proceeds by applying the techniques we have developed over the course of this lecture series to understand the Lorenz system, but that will only get us so far. The Lorenz equations are famous for being CHAOTIC, meaning that we need far more advanced techniques to understand them. Here I will review a bit of what happens beyond our analysis, including the transition to chaos. These equations set the stage for the remaining portion of this lecture series where we seek to understand the dynamics high-dimensional systems, primarily through discrete-time mappings which will be the subject of the next video lecture. Simulate the Lorenz equations yourself in Python to see the butterfly attractor at the standard parameter values: https://matplotlib.org/stable/gallery... Learn more about the legendary Ed Lorenz: https://en.wikipedia.org/wiki/Edward_... This course is taught by Jason Bramburger for Concordia University. More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates.