Nonlinear Systems: Fixed Points, Linearization, & Stability

► *MISTAKE* at 3:05, it should be f(x* + u, y* + v)

► For background, see

► From 'Nonlinear Dynamics and Chaos' (online course).

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)

► Follow me on Twitter

► Make your own phase portrait

► For more about hyperbolic vs. non-hyperbolic fixed points in N-dimensional systems

► Course lecture notes (PDF)

Reference: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 6: Phase Plane

► Courses and Playlists by Dr. Ross

📚Attitude Dynamics and Control

📚Nonlinear Dynamics and Chaos

📚Hamiltonian Dynamics

📚Three-Body Problem Orbital Mechanics

📚Lagrangian and 3D Rigid Body Dynamics

📚Center Manifolds, Normal Forms, and Bifurcations

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

[lionelinx]

Very good series, has helped me a lot!

[advancedappliedandpuremath]

Hi Sir hope you are having great time. I have a question regarding fixed point. In context of a function fixed point is defined where f(x) = x or standard notion is T(x)=x. But in context of dynamical system when we use the word fixed point it seems that it doesn't relate to definition T(x) = x but rather we say that fixed point is where the derivative is zero. So my question is that does the word fixed point meaning differs in both contexts or is there any analogy.

[江百川]

I really like all series of your video. But there is a writing mistake at 3:05, and maybe it should be: f(x^* +u, y^* + v). I just want to remind other learners.

[frimpongemmanuel]

what if there were trig function on the right hand side (e.g cos(x) or sin(y) ), do we have to include it ??

[wentaowu]

I wonder if they could plug the advertisements in the beginning and end of each video. I am focusing on your lecture, suddenly an advertisement. It is scary and annoying.