The Hartman-Grobman Theorem, Structural Stability of Linearization, and Stable/Unstable Manifolds

This video explores a central result in dynamical systems: The Hartman-Grobman theorem. This theorem establishes when a fixed point of a nonlinear system will resemble its linearization. In particular, hyperbolic fixed points, where every eigenvalue has a non-zero real part, will be "structurally stable" so that the nonlinear system resembles its linearization in a neighborhood of the fixed point. We also discuss the existence of stable and unstable manifolds.

@eigensteve on Twitter

This video was produced at the University of Washington

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0:00 Hartman-Grobman and hyperbolic fixed points
10:36 Stable and unstable manifolds
13:50 Example of stable manifold

[undisclosedmusic]

"It requires technical language but let me draw you a picture to show you what I mean" are the words of every great teacher in the world.

[pongballchannel]

These videos are PHENOMENAL! This is exactly what I needed for my research, thank you so much 🙏

[nikkatalnikov]

Hartman-Grobman theorem... my respect, professor!

[MrHaggyy]

This is excellent stuff and really helps to illustrate stability. Add some numbers and margins to it and you can engineer quite neat systems that your controller will like.
So please do more lectures about these pictures and how to increase or break stability.

[hoseinzahedifar]

Great lecture... Especially the last seconds about the term y = \phi(x) which might be what and how to be calculated...Thank you very much...
I've never known about this interesting theory...

[guuspalmero]

Best explanation, so easy but also so practical👌🏻

[CISMD]

A video on equilibria of equivariant systems would be cool

[seslocrit]

The axis (in the last example) can be considered as position vs. velocity.

[Applefarmery]

Will you also cover the center manifold theorem?

[juliangomez]

Steve, please tell us about what is a manifold, and the "manifold theorem".

[bonbonpony]

13:44 That's what I'm interested in the most: how can we find equations for those stable manifolds and closed orbits. I see an example of that at the end of the video, but I don't quite follow where everything came from.
14:21 So what was the nonlinear differential equation that this system originated from? If x' = dx/dt, then substituting it to the 2nd equation gives me d²x/dt² + dx/dt = x² which is nonlinear all well, but then what should I do with the first equation? The form of the 2nd-order non-linear equation that I got from the 2nd equation doesn't seem to depend on the first equation, so I clearly must be doing something wrong here :q How should I involve the 1st equation into play?
15:44 But a trajectory doesn't necessarily have to be a function of one coordinate in terms of the other. What then? Would a parametrization work equally well?

[shocklab]

Hi Steve, these are truly wonderful lectures. A very naive question: A stable node is also, as far as I'm aware, structurally stable. In two dimensions it has two real eigenvalues, therefore it would seem to fulfil the definition you give for hyperbolicity, but it does not have a stable and an unstable manifold - ie. it's not a saddle. Does hyperbolicity hold independent of the relative signs of the real parts of the eigenvalues? The same question holds for any eigenvalues of the Jacobian for which there also exist imaginary parts.

[dharshan]

I think there was a mistake in calling it the stable manifold. The points on the manifold should approach a point. Here the manifold you drew would be referred to as an unstable manifold as the emerge from a point. At least, that's all the references I've been reading says.

[mmaroti]

If I understood it correctly, then a spiral in fixed point, e.g. with eigen value -1+i, would also be preserved locally to the non-linear case, right?

[SSNewberry]

The proof is not hard but requires some mind-expanding.

[mohammadabdollahzadeh]

Hi dear Dr, please make a tutorial about how to design adaptive dynamic pid controller in matlab
Im looking forward to hearing from you
Sincerely
Mohammad

[animeshjain]

blue diagram at 6:07 looks sus




(just joking don't take too seriously)