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Classifying Fixed Points of 2D Systems

Fixed points of linear two-dimensional differential equations are classified according to their eigenvalues, and represented graphically as local phase portraits. Summary classification of fixed points by the trace and determinants of the matrix. ► MISTAKE: at 11:31 it should be lambda2 ≥ lambda1 ≥ 0 ► Next, we try this classification on a simple example    • Love Dynamics: Coupled Equations of C...   ► Other topics posted regularly Subscribe https://is.gd/RossLabSubscribe​ ► See the introduction to 2D phase portraits    • Phase Portrait Introduction- Pendulum...   ► Apply this in linearizing about fixed points in a nonlinear system    • Nonlinear Systems: Fixed Points, Line...   ► From 'Nonlinear Dynamics and Chaos' (online course). Playlist https://is.gd/NonlinearDynamics ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) http://chaotician.com​ ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes Reference: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 5: Linear Systems Chapters 0:00 - Linear 2D systems 2:09 - Special directions, eigendirections 5:50 - Characteristic equation for eigenvalues 8:42 - Stable and unstable nodes 12:17 - Saddle points 14:49 - Centers and stable and unstable spirals 17:15 - Eigenvalue zero, non-isolated fixed points 18:37 - Degenerate nodes and stars 19:47 - Classification of fixed points, overview 22:19 - Example from a model of gliding flight ► 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 fixed point classification matrix trace matrix determinant complex conjugate eigenvalue pair 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 #NonlinearDynamics #DynamicalSystems #FixedPoints #DifferentialEquations #PlanarSystem #Bifurcation #SaddleNode #Bottleneck #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions