Linearizing Nonlinear Differential Equations Near a Fixed Point

This video describes how to analyze fully nonlinear differential equations by analyzing the linearized dynamics near a fixed point. Most of our powerful solution techniques for ODEs are only valid for linear systems, so this is an important strategy for studying nonlinear systems.

This is a hugely important step towards analyzing nonlinear systems with linear techniques.

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This video was produced at the University of Washington

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0:00 Overview
3:37 Fixed points of nonlinear systems
5:32 Zooming in to small neighborhood of fixed point
7:03 Solving for linearization with Taylor series
12:10 Computing Jacobian matrix of partial derivatives
15:10 Example of linearizing nonlinear system

[stephanel]

There is a missing factor of 1/n! in the Taylor series. Luckily here it is of no consequence as it only affects the higher order terms that are dropped. Great series of lectures!

[Joao-ujkm]

Video looks so elegant with the colors and background! Great lecture, beautiful work!

[JeonghunKang-oxsk]

What a wonderful explaination! Thanks for saving my life.

[shayislam]

This is so interesting and helps me so much with my research. Thank you very much, Dr. Brunton. Keep'em coming.

[jamesmosher]

I’ve never thought about DEs in this manner with the fixed points, etc. interesting.

To me, what’s even more interesting are BVP on irregular domains. Like how the solution to the Helmholtz equation on a rectangle is the 2D fourier series, but, if you go to a rectangle with one quadrant missing, the eigenfunctions are nearly impossible to to represent in a “clean” fashion.

[prodbyryshy]

i wish i could remember everything from my ODE class! i sort of turned towards more computational and statistical work, but pdes are beautiful math

[rediculousman]

I'm pretty rusty on this, hence why I'm watching these to try and refresh my memory (10 years out of uni).
I always liked to think about the local stability by imagining the state space as an n dimensional space with gravity. If you choose a point and drop a marble, you can watch which direction it rolls. If it falls into a low point and stops, it's stable. If it rolls away forever it's unstable. There are also points where the marble can roll away and then get stopped somewhere else.
If you want to develop a controller, you have to figure out what force vectors you need to apply to keep the marble fixed in the point that you dropped it.

In real life, the state spaces can be massive, so you can just choose a small sample that you can stay within, so that allows you to approximate it linearly.

[sinarezaei]

you make me love math
thanks for your lectures
❤❤❤

[GeoffryGifari]

On the illustration drawn near the beginning of the video we see two fixed points, and it seems like our dynamical system flows from one fixed point into the other. Is this always the case? can we have multiple fixed points but the phase portrait only flows around their own fixed points and never crossing into each other?

[julioosorio]

Excellent explanation. greetings from Peru.

[dungtrananh]

Hello sir, according to my knowledge, the linearity around the equilibrium point of a nonlinear system is only true within a small range (in vicinity) around this equilibrium point. Could you please help me with a method to quantify the vicinity around any equilibrium point of a system?

[individuoenigmatico]

Yes, x0 is a fixed point of a differential equation if and only if x(t)=x0 for all t is a solution of the differential equation.

Of course in our differential equation x'=f(x), x0 is a fixed point if and only if f(x0)=0.

[mdshahporan]

Dear sir,
I hope you will answer my question. If we linearize a non-linear system near the equilibrium point, then we are limited only a very small region of our whole system. My question is what if I want to solve or operate at any other location except the equilibrium point? And since the linear version of non-linear system explains a very small region, I think this is not so meaningful if we are interested in our whole non-linear system. In that situation, how do we explain or solve the system?

[MLDawn]

What if there are no fixed points?! Does it mean linearization is not an option? Then what should we do?

[BalaramPradhan-if]

Plz make videos on how to draw this graph in mathematica or matlab

[YuriGorokhov]

I feel like taking the Taylor series in powers of delta x could use a bit more elaboration, quite a jump from simple Taylor series expansion. Especially confusing by the overuse of variables that are variations of x in this lecture… x, x bar, delta x :)

[cleisonarmandomanriqueagui]

in what book can i find this theory ? i can not find it

[mariarahelvarnhagen]

Classrooms Are Using dx For dz All The Time And Ignoring dt/dt In Maxwell's Equations

[lyaeusv]

if only i can summon a girlfriend like you summon your delta x

[firosiam]

Please try to keep ur vedios a bit shorter like I feel interested to watch the series and I see these long vedios and I dnt thnk I have the time to watch every vedios and catch up to the current vedio