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!

What a wonderful explaination! Thanks for saving my life.

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

Very useful. Thank you professor!

Thanks prof. Big fan.

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

I love you! Thank you for this video!

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

Great video. Thank you

Excellent explanation. greetings from Peru.

ti's getting interesting thanks

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.

@kodfkdleepd2876

Жыл бұрын

Differential equations are equivalent to vector fields and so studying vector fields provides different perspective. Specifically closed integral paths are precisely periodic solutions of the differential equation. When you use irregular domains you are excluding these integral paths as solutions. The boundaries are then not "natural" in the sense that they interrupt the natural flow of these integral paths forcing more complex solutions.

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?

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.

great Sir

Perfect

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?

thank youuu

Nice

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

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.

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

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

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 :)

@rajinfootonchuriquen

Жыл бұрын

For a vector function, you take the iteration fo the jacobian with is the tensor product between the gradient vector and the function vector, for terms of O(x^2), you are dealing with tensor of rank bigger than 2.

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?

@zaynbashtash

Жыл бұрын

Hello sir, did you find any resources on this topic?

@mdshahporan9069

Жыл бұрын

@@zaynbashtash Not yet

@cedricvillani8502

Жыл бұрын

Control Theory 🎃 See NASA

@darkside3ng

Жыл бұрын

After linearizing the system, its operation can tested by simulation or analytic methods, so you can assess how far from the equilibrium point you can go without losing control characteristics. If one point is not sufficient to properly control the system, you can choose other points to cover the entire range of operation of your control system and use a gain scheduling approach to change the model. Try to search for "What Is Gain Scheduling? | Control Systems in Practice" from Matlab channel, it is a good starting point.

@mdshahporan9069

Жыл бұрын

@@darkside3ng Thank you so much sir for giving me the instruction. I was looking for this answer in different books also, but didn’t get a satisfactory solution.

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

@mariarahelvarnhagen2729

Жыл бұрын

Greedy Instructor Perspective

He is a left handy right ?

How does this man write backwards

@marekw4353

Жыл бұрын

The image is mirrored. In one of his lectures he mentiones he is left-handed

@wuyizhou

Жыл бұрын

@@marekw4353 that makes so much sense, thanks!

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

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

@rajinfootonchuriquen

Жыл бұрын

Bro he is giving education for free. Don't be entitled

@tascker0

Жыл бұрын

Feel intersted? What a low frustration tolerance.

What's going on in here