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Index Theory for Dynamical Systems, Part 1: The Basics
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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.
► For background on 2D dynamical systems, see
► From 'Nonlinear Dynamics and Chaos' (online course).
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
► Follow me on Twitter
► Make your own phase portrait
► Course lecture notes (PDF)
References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 6: Phase Plane
► Courses and Playlists by Dr. Ross
📚Attitude Dynamics and Control
📚Nonlinear Dynamics and Chaos
📚Hamiltonian Dynamics
📚Three-Body Problem Orbital Mechanics
📚Lagrangian and 3D Rigid Body Dynamics
📚Center Manifolds, Normal Forms, and Bifurcations
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
► Next, Poincare-Hopf index theorem for compact manifolds.
► For background on 2D dynamical systems, see
► From 'Nonlinear Dynamics and Chaos' (online course).
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
► Follow me on Twitter
► Make your own phase portrait
► Course lecture notes (PDF)
References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 6: Phase Plane
► Courses and Playlists by Dr. Ross
📚Attitude Dynamics and Control
📚Nonlinear Dynamics and Chaos
📚Hamiltonian Dynamics
📚Three-Body Problem Orbital Mechanics
📚Lagrangian and 3D Rigid Body Dynamics
📚Center Manifolds, Normal Forms, and Bifurcations
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
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