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Nonlinear Systems: Fixed Points, Linearization, & Stability

The linearization technique developed for 1D systems is extended to 2D. We approximate the phase portrait near a fixed point by linearizing the vector field near it. Two worked examples are given. A word of caution when dealing with borderline cases (centers, degenerate nodes, stars, or non-isolated fixed points). Next, we consider in-depth a 2D population model of rabbits vs sheep    • Phase Plane Analysis: Worked Example ...   ► MISTAKE at 3:05, it should be f(x* + u, y* + v) ► For background, see classifying 2D fixed points    • Classifying Fixed Points of 2D Systems   phase portrait introduction    • Phase Portrait Introduction- Pendulum...   geometric analysis in 1D    • Graphical Analysis of 1D Nonlinear ODEs   ► 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   / rossdynamicslab   ► Make your own phase portrait https://is.gd/phaseplane ► For more about hyperbolic vs. non-hyperbolic fixed points in N-dimensional systems    • Hyperbolic vs Non-Hyperbolic Fixed Po...   ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes Reference: 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 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 #NonlinearDynamics #DynamicalSystems #FixedPoint #DifferentialEquations #Bifurcation #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions