Phase Portrait Introduction- Pendulum Example

In the geometric or graphical study of two-dimensional nonlinear ODEs, our goal is to determine all the qualitatively different system behaviors, that is, find the phase portrait. The pendulum example introduces the concept.
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► Teacher Bio Dr. Shane Ross, professor, Virginia Tech
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► Related Courses and Series Playlists by Dr. Ross
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Reference: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 6: Phase Plane
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 #DifferentialEquations #Bifurcation #SaddleNode #Bottleneck #Circle #CuspCatastrophe #CatastropheTheory #CuspBifurcation #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

[@christianwesthof9136]

Thanks for that. That was helpful to see the pendulum moving as the dot on the phase space traced through the orbit.

[@ProfessorRoss]

Thank you. I love to be able to provide helpful visualizations whenever possible.

[@wentaowu3070]

These are really great lectures on nonlinear analysis.

[@SujanDahal]

Thank you ❤ From Nepal

[@triton62674]

Very cool stuff great insights!

[@fardeenrazif9377]

Thanks for the lecture

[@ProfessorRoss]

So nice of you

[@ragin_cajuns]

Awesome explanation professor.

[@ProfessorRoss]

Glad you liked it

[@danielprieto3563]

In the phase diagram picture, what are those lines above and below the closed loops, is it suggest that if the pendulum moves to fast the ball will break off and fly of to infinity?

[@ProfessorRoss]

Daniel, good question. The lines above and below correspond to the case where the pendulum keeps turning in one direction, over and over, like a windmill. There's no going off to infinity since the angle variable is cyclic. More at 5:52

[@juanfa98]

jefe