Non Euclidean Phase Spaces (e.g. Invariant Spheres), Lyapunov's 2nd Method (Non Hyperbolic) Examples

The nonlinear pendulum model has a phase portrait (phase plane) that is periodic with infinitely many equilibrium points. Therefore it makes more sense to consider its phase space to be a cylinder which can be obtained as a topological quotient space where we "glue" and identify the vertical lines at pi and -pi together. The flow on the cylinder now just has two equilibrium points, which matches physical reality. Some three-dimensional systems of ordinary differential equations, along with their three-dimensional flows have invariant spheres (two dimensional surface examples of invariant manifolds). This induces a vector field and flow on the individual invariant spheres that will always have at least one equilibrium point (this is the "Hairy Ball Theorem"). They are also very beautiful flows. Stability theorems are reviewed, including Lypaunov's Second Method. Then Lyapunov's Second Method is applied to a one-dimensional example of a non-hyperbolic equilibrium point, a two-dimensional example of a non-hyperbolic equilibrium point, and a three-dimensional example of a non-hyperbolic equilibrium point. Content about constructing a local Lyapunov function for an equilibrium point of a competing species model is included.

Bill Kinney's Differential Equations and Linear Algebra Course, Lecture 36A.

(a.k.a. Differential Equations with Linear Algebra, Lecture 36A, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 36A).

#lyapunovstability #phasespace #nonlinearity

(0:00) These lectures are the pinnacle of what we’ve done
(1:38) Best (most “natural”) phase space for the pendulum model
(5:06) Phase cylinder (undamped and damped case)
(8:30) 3-Dimensional System with Invariant Spheres
(9:45) Conserved quantity (conserved function)
(11:57) Curve of equilibria (curve of equilibrium points)
(15:29) Flow on the unit sphere
(18:04) Mathematica visuals
(20:56) Stability and eigenvalues (Lyapunov’s First Method)
(21:55) Lyapunov’s 2nd Method
(24:39) Example: dy/dt = -y^3 (equilibrium at y = 0)
(28:04) Example (2-dimensional): Polar coordinates example from Lecture 33A
(29:01) Example (3-dimensional): origin has a local Lyapunov function
(31:59) Trapping Region on Mathematica

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[CristalMediumBlue]

Amazing video! The bifurcations that occur as the radius increases are very interesting!

[georgesadler]

Professor Kinney, thank you for anther outstanding summary and analysis of the Nonlinear Pendulum and its phase portrait. The use of Mathematica helps with the visualization of the Spherical and Cylindrical dimensions of these Linear/Nonlinear Systems. Lyapunov's 2nd Method (Non-Hyperbolic) examples and problems increases my understanding of what a Lyapunov Function is and how to determine if its stable or unstable.