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 Charles Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations 2d ODE vector field topology cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex dynamics point vortices pendulum Newton's Second Law Conservation of Energy topology #NonlinearDynamics #DynamicalSystems #VectorFields #topology #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #Bifurcation #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #DynamicalSystems #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion