Index Theory for Dynamical Systems, Part 1: The Basics

Index theory is a powerful global topological method to analyze vector fields, and reveal the existence (or absence) of fixed points and periodic orbits. As in electrostatics, where the vector field along a hypothetical Gaussian surface is used to infer point charges, this method uses the rotation of vectors along a test curve to infer the presence of fixed points. Properties of the index and several examples given.

► Next, Poincare-Hopf index theorem for compact manifolds.
https://youtu.be/CYOzEy0Sptk

► For background on 2D dynamical systems, see
Phase plane introduction https://youtu.be/U4IM7HFzcuY
Classifying 2D fixed points https://youtu.be/7Ewe_tVa5Fs
Linearizing about fixed points https://youtu.be/m0d3sLqPftA
Rabbits versus sheep example https://youtu.be/07V_UNLz0qs
Systems with special structure https://youtu.be/uGUzPZzvPWQ

► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist https://is.gd/NonlinearDynamics

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Subscribe https://is.gd/RossLabSubscribe​

► Follow me on Twitter
https://twitter.com/RossDynamicsLab

► Make your own phase portrait
https://is.gd/phaseplane

► Course lecture notes (PDF)
https://is.gd/NonlinearDynamicsNotes

References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 6: Phase Plane

► Courses and Playlists by Dr. Ross

📚Attitude Dynamics and Control
https://is.gd/SpaceVehicleDynamics

📚Nonlinear Dynamics and Chaos
https://is.gd/NonlinearDynamics

📚Hamiltonian Dynamics
https://is.gd/AdvancedDynamics

📚Three-Body Problem Orbital Mechanics
https://is.gd/SpaceManifolds

📚Lagrangian and 3D Rigid Body Dynamics
https://is.gd/AnalyticalDynamics

📚Center Manifolds, Normal Forms, and Bifurcations
https://is.gd/CenterManifolds
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Видео Index Theory for Dynamical Systems, Part 1: The Basics канала Dr. Shane Ross