2.0 A better way to understand Differential Equations | Nonlinear Dynamics | 2D Linear Diff Eqns

[@americanbeing5998]

This is best video I've ever seen regarding non-linear DE. Thanks

[@virtually_passed]

Glad you enjoyed it ☺️

[@ReasonableSwampMonster]

Using these in tandem with Strogatz’s book and lecture notes they’ve been super helpful to clear up what is otherwise opaque. Something about seeing the concepts animated and in colour makes them seem far less daunting than a book or chalkboard lecture lol

[@virtually_passed]

Really glad you liked it :)

[@user-pb8yw8cw3s]

Strogatz's book is amazing, and so this video !

[@Somendra_Kharola]

Learnt so much! Thank you! The narration, the explanation, the structure, the articulation, the animation were all brilliant!

[@virtually_passed]

Glad you enjoyed it!

[@juanpina6638]

I used to get bored with differential equations until I saw this video series

[@abdurrahmanlabib916]

Greaat video! Eagerly waiting for part 3:D

[@virtually_passed]

Part 3 is finally done: kzbin.info/www/bejne/bJ6vdHmvettmkJI

[@TungNguyen-vb4on]

great thanks from Vietnamese student !

[@agrajyadav2951]

This is similar quality as that of a 3b1b video. Great job and thank you!

[@virtually_passed]

3b1b is a hero of mine :)

[@YualChiek]

Beautiful! I'm glad you highlighted how such a high number of different stable flow formations can be characterized with surprising uniformity. The spiral formation is even elliptical in shape. I can't help but think that the vortex theories of gravitation held by the likes of Descartes and Leibniz would have been significantly more difficult to dismiss since a corpuscular flow suitable for celestial mechanics might then have been describable.

[@mnada72]

That is absolutely great explanation. Thank you

[@virtually_passed]

Thanks!

[@leleogere]

Awesome animations congrats!

[@virtually_passed]

Thanks!

[@spranav9878]

Keep Going man Great content Your channel will grow soon

[@virtually_passed]

Thanks!

[@kiiometric]

Amazing video!

[@virtually_passed]

Part 2 is finally done! Working on Part 3 now...

[@virtually_passed]

Part 1: kzbin.info/www/bejne/r4TJpGuPi7KMbbM Part 2: kzbin.info/www/bejne/f2q4dnWIrsZnjpI Part 3: kzbin.info/www/bejne/bJ6vdHmvettmkJI

[@oliverhees4076]

Very interesting video! The end of my ODEs class last semester touched upon using slope fields to analyze the behavior of systems, and I'm doing a nonlinear dynamics course next semester, so this is all very relevant to what I'm learning. Very well-explained.

[@virtually_passed]

Hey Oliver, thanks for the comment. Indeed if you're studying nonlinear dynamics next semester then this will be the first the first video in this series will be the first thing you will learn. If you want additional materials to learn from I'd recommend the nonlinear dynamics and chaos book by Steven Strogatz :)

[@glenm99]

Beautiful video. I love it.

[@virtually_passed]

Thanks :)

[@RSLT]

Wow, what a great video...hats off to you...

[@virtually_passed]

Thanks so much

[@zaynbashtash]

Awesome

[@elvanaud]

This is fantastic ! Do you have any further ressources that I could look up if I want to know more about this ? Especially the link between differential equations and eigen-stuff

[@virtually_passed]

Hello, great question! Yes I do. There are two books that I used for this mini-series. One is "Nonlinear Dynamics and Chaos" by Steven Strogatz. The other is "Nonlinear Differential Equations and Dynamical Systems" by Ferdinant Verhulst. In my opinion the book by Strogatz is much easier for beginners. Good luck!

[@Ghostalitta]

Great series and very intuitive! What is the name of the app that you are using for drafting/hand written derivations?

[@virtually_passed]

Thanks! I use Microsoft one note and use my hp envy x360 touch screen to write. I record the screen using filmora pro. Although, there are many other options. Camtasia studio and Photoshop also work but are more expensive

[@Ghostalitta]

@@virtually_passed thanks 🙏

[@shebo96]

hello, great video how did you draw the vector field from the differential equation

[@virtually_passed]

I used a python library called manim. :)

[@shebo96]

@@virtually_passed for example if i wanna draw it by hand how do i do the math, do i need only the eighen vectors and values

[@virtually_passed]

@@shebo96 for a linear system (like the ones shown in this video) then yes, you only need the eigenvalues and eigenvectors to know what the entire vector field will look like.

[@shebo96]

@@virtually_passed thanks for replying. How do you figure the vector field from the eighenvector and eigenvalue. For example at point (10,9) how do you evaluate the vector there from the given eighenvectors (1,1) (1,0) and eigenvalues 2 and 4

[@Beb001-w2x]

have a question. the stability and instability of the fixed points... do what exactly to the system as a whole... does the solution to the nonlinear ODE.. blow out , resulting in no solution to the equation. Stability.. reduction in oscillations in the system, so it decays to the solutions of the equation..

[@virtually_passed]

Good question. The stability of fixed points only gives you very local information, it tells you absolutely nothing about the system as a whole. Even though fixed point analysis is easy to do and quite useful, it's almost always never enough to analyze a nonlinear differential equation

[@Beb001-w2x]

@@virtually_passed so is the approach to solving the system behavior to examine it numerically....

[@blacklistnr1]

@3:10 "guess solution" it's the reason I strongly disliked math teachers in school. if you "happen to guess correctly", I also "happen to guess correctly" the results directly and better not ask me how because I'll say I guessed like you do. If you have list of probable guesses tell it to me; If it's a heuristic that physical systems tend to have these sort of solutions just say so; If you only know how to solve this type of problem, again it's fine but just say it. If your prepared problem solution involves guessing and pulling tricks out of a hat, you're teaching voodoo divine intuition, not math where all steps are logically linked.

[@virtually_passed]

Hey! Thanks for the comment. You raise a good point! You're right that the guess isn't 'completely arbitrary'! Let me give you some motivation behind this guess. Consider the 1D version of the differential equation (where 'a', 'x' and 't' are scalars): dx/dt = a x We can solve this equation directly without any guesses needed. Dividing by x and integrating both sides yields: ∫1/x dx = ∫a dt ln(x) = a t + c x = e^(at+c) x = e^(at)*e^c Define e^c = v x = e^(at)*v As you can see the solution to the 1D version is in the form of our guess for the 2D problem. In my view, you can consider this 'good motivation' for the guess shown at 3:10. ------------------------- However, if you're deadset on solving this set of linear differential equations without any 'lucky' guesses, then this is also still possible! Let's say you have the following set of equations. 1) dx/dt = ax + by 2) dy/dt = cx + dy Differentiate equation 1) wrt time. d^2x/dt^2 = a*dx/dt + b*dy/dt substitute out dy/dt using equation 2) d^2x/dt^2 = a*dx/dt + b*(cx+dy) Now you can solve for y purely as a function of x, dx/dt and dx^2/dt^2. And as a result, you can uncouple the differential equations and solve them directly. Hope that helps :)

[@blacklistnr1]

@@virtually_passed Thanks for the reply! This really ties together the video. It was a really interesting series, btw. Phase space diagrams are really cool!

[@willie333b]

Is there a saddle spiral?

[@virtually_passed]

That's not possible. If you solve for the eigenvalues you'll find you need to solve a quadratic equation. By playing around with it you'll see it's impossible to get complex conjugates and different real values at the same time.

[@MrRyanroberson1]

instead of linking part 1 / 2 / 3 in each video, you might have a better time linking a playlist of them all

[@virtually_passed]

Thanks for the suggestion!