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 https://youtu.be/07V_UNLz0qs
► *MISTAKE* at 3:05, it should be f(x* + u, y* + v)
► For background, see
classifying 2D fixed points https://youtu.be/7Ewe_tVa5Fs
phase portrait introduction https://youtu.be/U4IM7HFzcuY
geometric analysis in 1D https://youtu.be/Mcqrn9V7_YI
► 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
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► Make your own phase portrait
https://is.gd/phaseplane
► For more about hyperbolic vs. non-hyperbolic fixed points in N-dimensional systems
https://youtu.be/5d0UhnBm16g
► 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
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