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

[@undisclosedmusic4969]

"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!

[@hoseinzahedifar1562]

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👌🏻

[@neyhmor]

It is not easy. He just makes it look so easy (and I mean it as a compliment)

[@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.

[@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?

[@Music-qm4pw]

Very good!!!!

[@CISMD]

A video on equilibria of equivariant systems would be cool

[@seslocrit9365]

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

[@juliangomez8162]

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

[@Applefarmery]

Will you also cover the center manifold theorem?

[@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.

[@SSNewberry]

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

[@dharshan272]

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?

[@MrHaggyy]

Take a look at the proof for this question. Hartman Grobman is derived from the Jacobian and eigenvalues of a system and works in theory if there is no 0 realpart or repeated eigenvalue. In practice, you need quite stable or unstable eigenvalues. The more you zoom in the smaller your realparts can be, but the area your linearisation holds get`s smaller too.

[@animeshjain8932]

blue diagram at 6:07 looks sus (just joking don't take too seriously)

[@JuanFernandez-hd7de]

Great video, as always. Specially the last part where the equation for the stable manifold Is derived. How would you derive the unstable on? It seens much more cumbersome....

[@mohammadabdollahzadeh268]

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