Topics in Dynamical Systems: Fixed Points, Linearization, Invariant Manifolds, Bifurcations & Chaos

This video provides a high-level overview of dynamical systems, which describe the changing world around us. Topics include nonlinear dynamics, linearization at fixed points, eigenvalues and eigenvectors, bifurcations, invariant manifolds, and chaos!!
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This video was produced at the University of Washington
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0:00 Introduction
4:35 Linearization at a Fixed Point
9:37 Why We Linearize: Eigenvalues and Eigenvectors
14:46 Nonlinear Example: The Duffing Equation
19:59 Stable and Unstable Manifolds
21:12 Bifurcations
25:20 Discrete-Time Dynamics: Population Dynamics
27:02 Integrating Dynamical System Trajectories
29:07 Chaos and Mixing

[@gretarmark]

Nice, exactly what I'm doing at my job right now. Linearization.

[@lexinwonderland5741]

Fantastic video, I look forward to seeing more in the series!

[@jamesjohn2537]

This is good excellent explanation and overview of the subject. I can't wait to reached there at that level, this is really cool stuff!.

[@Pedritox0953]

Great video!

[@gabrielh5105]

Thank you for your videos, professor Brunton

Amazing stuff! Thanks

[@tiledesta5825]

You are always smart. Dr. I wish long life!!

[@hoseinzahedifar1562]

Amazing... thank you very much.

[@AleeEnt863]

a big thanks!

[@lightnlies]

Can you do a lecture series on Proper generalized decomposition methods? I would love that!

[@MrHaggyy]

Awesome video, ^^ as usuall. If you mention literatur in the video, it would be nice if you add them to the description as well. Doesn't have to be links or any sophisticated citation. Just a quick reference you can copy to google and maybe even some worthy mentiones to start digging into the topic.

[@bricematuszczak7929]

Thank you for all the content you post online!
Regarding bifurcation theory, is there a chapter in any of your books addressing this?
Thank you in advance

[@user-wz7ch8vf2p]

Steve, thank you for your videos. While studying non-linear systems I found the theme manifolds difficult and couldn't do it any better than 'ok that's something like a subspace, so nothing gets out of subspace'. Should you make a clarifying video on the topic that would be great

[@zoheirtir]

I agree with you, hope Steve spend some time to this topic

[@pak7524]

Next time
I will hope you to explain "Numerical Weather Prediction".

[@StEvUgnIn]

Hi Steve, could you please a video series on clustering algorithms?

[@mohammadsajjadi1433]

Dear Brunton, thank you so much for your videos. My major, like you, is Control Engineering. I have a question. What can we say about the equilibrium points and limit cycles of a nonlinear system having 5 negative Lyapunov exponents and 1 zero Lyapunov exponents?

[@ASKUJapan]

Hello Dr Brunton, have you have ever thought of tree growth as a dynamical system?

[@sibik2473]

Sir thank you so much for your videos can you also teach us PINNS and Physics I formed deep learning

[@user-uf2uc3ce2r]

20:17 Should the green manifold be stable and the red manifold be unstable instead of what the picture says? I mean if we were to look where the arrows are pointing at.

[@hoseinzahedifar1562]

I think the green manifold is unstable because any point starting on the manifold goes to the points specified at the ends of the red manifold.

[@rodbhar6522]

yeah i think he mixed those up

[@ndmath]

I think the text contents should be switched.

[@mesdouifatiha1579]

Please Steve, could you do a video on lyapunov exponents with MATLAB code.

[@science_engineering]

Mixing is a sufficient condition that a dynamical is chaotic. Mixing is NOT a property of chaos...

[@olegageev9712]

Nothing new though. Some particular topics.