4.0 A better way to understand Differential Equations | Nonlinear Dynamics | Index Theory

Series1: Part 1: https://youtu.be/ySfs8YVMY7Q Part 2: https://youtu.be/I9UEBRya4X0 Part 3: https://youtu.be/6mLCFyEv3Z0 Index theory is a method used to gain global information about a nonlinear differential equation. One powerful insight is that closed orbits (periodic solutions) must have at least one fixed point inside of the curve. In fact, for a periodic orbit, there must be 1 more fixed point of index = +1 than saddle fixed points. Chapters: 0:00 Last time 0:48 Intro 0:58 Motivation 2:04 Index of Closed Orbits 3:53 Index Examples 6:02 Defining Phi and Index 8:18 Proof Index Integer 8:47 Proof Index Constant Under Deformation 10:24 Index of Fixed Points 10:49 Index of Multiple Fixed Points 12:38 Index Closed Orbits = +1 13:00 Examples Closed Orbits 13:53 Ruling Out Closed Orbits 14:13 Next time 14:32 Outro Music: Go Season by Milano -- Patreon: https://www.patreon.com/VirtuallyPassed Instagram: https://www.instagram.com/virtuallypassed/