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
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► See the introduction to 2D phase portraits
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► Apply this in linearizing about fixed points in a nonlinear system
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► From 'Nonlinear Dynamics and Chaos' (online course).
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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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► Course lecture notes (PDF)
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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
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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

[@user-vq3zo7jq4l]

at 11:31 there is a written mistake, it should be lambda2 > lambda1 > 0.

[@npranavsubhraveti2513]

Thanks for the lecture, was helpful.

[@miaboruto7091]

Can you help me solute questions

[@backbench3rs659]

amazing

[@ProfessorRoss]

Thank you! Cheers!