The Core of Dynamical Systems

[@graf_paper]

Omg, the PDF is wonderful. Sooo appreciate you putting that together!!

[@Khashayarissi-ob4yj]

Beautiful episode. Thank you.

[@benjamingoldstein1111]

Good video again! This channel deserves its fast growth!

[@dibeos]

Thanks for the encouragement, Benjamin :D

[@agnelomascarenhas8990]

This is a beautiful presentation. Thanks!

[@ЗакировМарат-в5щ]

Thank you. But I think that state space of simple pendilum is a cylinder not torus because velocity (x with dot) could have any value. Also I want you to solve some interesting and yet practical task. For example find optimal route in mountain region using differential geometry.

[@TacoDude314]

The torus at 5:40 describing the state of the pendulum is just wrong. 1. (Minor point) The state space diagram is wrong according to the pendulum diagram. The stable point is at x=0 mod 2pi. 2. The square cut out at 5:20 and the torus at 5:40 clearly contain points above/below the separatrix. Those points will flow outside of the square. 3. Once you identify 0 and 2pi then the rotational orbits are periodic on the (infinite) state space cylinder. There is no reason to (attempt to) exclude them 4. There is clearly nothing periodic about the state space in the x' direction so there is no reason to turn it into a torus. There is no reason to identify points representing the pendulum swinging spinning "fast enough" clockwise to spinning "fast enough" counterclockwise, which is what you're doing by cutting out an arbitrary velocity range and gluing it into a torus. It seems like you're trying to link to unrelated concepts: state space/dynamics of a pendulum and quasiperiodic orbits on a torus.

[@tensorfeld295]

Nice animations. Do you know the book 'Dynamics: Geometry of Behaviour' by Abraham and Shaw?

[@dibeos]

No, I don't but I will search about it! Is there a specific chapter that you would like us to make a video about?

[@tensorfeld295]

@@dibeos I mentioned it, because it has a lot of pictures in it. Maybe something about seperatrices, generic properties and bifurcations? Other topics I like, not necessarily connected to the book: * Hamiltonian Systems and KAM-Theorem * 16th Hilbert-Problem and Limit Cycles * all about manifolds ... ^^

[@dibeos]

@ yeah, there are a lot of things missing in the pdf. The goal was not to make a complete course on Dynamical Systems but just show its main “core”. Yeah, but these that you mentioned are very interesting things for us to talk about in future videos about DS. Thanks for letting us know 😎

[@yanntal954]

0:09 Vortex formation, magnetic levitation and planetary devastation 😅

[@otterlyso]

The PDF has a lot more content!

[@ethandennis368]

Hey, big fan of the videos - love your mission to make things accessible. In this video I am not sure I agree with how you introduced tori. They absolutely are a useful concept in many systems but not in the way you showed with the pendulum. Maybe I misunderstood and also lied to another commenter but it seems you just saw that the phase space was a cylinder so you just truncated it and turned this subspace into a torus for fun. I will be handwavy here and overly intuitive, but this is my field so if you want me to get into things like symplectic geometry and stuff I can - would be happy to talk to you guys as well if you have interest in continuing in this direction at all. The tori that are interesting and heavily studied is in invariant tori which do not contain any compact invariant submanifolds (like the quasi-periodic orbits you went on to mention). These are the interesting structures which describe our bounded motion and organize phase space. In the case of the pendulum, we have an 'integrable' system, which pretty much just means that the whole space is filled with an infinite number of tori (we can use actions and angles as coordinates), but in a 2D hamiltonian system they can only be 1-tori (i.e., periodic orbits, which we say are 'Lagrangian' tori in this case). Maybe I am sounding pedantic here, but in these systems there is a natural connection between positions and momenta (see: symplectic forms) in such a way that makes it so that the structure of the pendulum phase space is naturally and ideally this cylinder. If you truncate it, the best course of action (pun?) would then be to undo that and still just say we just locally have a cylinder but just locally now. The reason invariant tori are interesting is as you alluded to - if you are preserving some structures in the flow (keeping some constant of motion - maybe even time dependent in some cases depending on specifics - we will just say energy even if its an oversimplification) nearby trajectories dont really have the freedom to diffuse throughout phase space or converge to a set of lower measure. Thinking about the pendulum, we dont expect a 2D set of trajectories to converge to the equilibrium like we would in the dissipative case because level sets of energy which are in this set would need to converge towards each other for that to be the case. The result is that equilibria in these nonlinear systems must admit oscillatory or hyperbolic character but would never be like asymptotically stable. Handwaving all the harder theory away to generalize this further, the result is that these invariant tori are the only bounded motion in these physical systems. Their dynamics are then a combination of these oscillatory motions. Remarkably, these tori seem to exist in perturbed systems in much more useful situations (see KAM theory, although its claims are like 10000x times more conservative than what we find empirically), and tori are often just the most useful real structure that can be found in a system's dynamics. These tori even can exist amidst chaotic dynamics, and the interplay between how that is possible is interesting. We see these tori in applications like quantum theory, plasma/fluid dynamics, nuclear reactors, etc.. My favorite though is that every single real world orbital motion in space resembles motion on some higher dimensional tori (not rigorously true, but its very very close). We claim things are periodic orbits for convenience but they are simply not.

[@finite1731]

At 2:41, there's no square on the dt and d^2x/dt^2, if consistant, shouldn't have (t) to its right because all other other x's also have a time dependence which isn't shown explicitly.

[@aymenzaki6786]

i wonder if you do animations about lagrangien manifolds and Hamiltonien systems from vladimir Arnold book mathematical methods of classical mechanics thanks

[@dibeos]

Thanks for letting us know what you want! We will add to our list of ideas ;)

[@RF-fi2pt]

Comparing the same graphics at 3:05 and 4:30, we see is wrong at 3:05 put the π (and multiples) at rest point. The correct π is at Sela points and 0 at rest points ,as your graphics at 4:30 show.

[@Xen-p6p]

Would love to see a video about the pantograph equation and the deformed exponential function which is the solution to: f'(x)=f(ax) , f(0)=1 f(x)=sum(0,infinity)[a^(n(n-1)/2)*x^n/n!] (Converges for |a| less than or equal to 1, however, I think there may be an analytical continuation to all values of a just haven't seen anyone try yet) It's similar to the differential equation for the exponential; f(x)=af(x) , but unlike it, it stretches the function on the x-axis as opposed to stretching on the y-axis like the taking the derivative of e^(ax) does.

[@physicslover9227]

Respectful Authorities can you please make video connecting Geometrical flows and dynamical systems with General Relativity.My motivation is to study Raychaudhari equation as a geometrical flow equation...Thankyou so much for your valuable vidoes :)...

[@physicslover9227]

Sir/Maam any response for making videos ? Please let me know :)...

[@RAyLV17]

Great video! I always wanted to learn the concepts in Dynamical Systems (such as Lie Groups, Symplectic Methods) 5:15 When you mapped state-space onto the Torus, you did not include region beyond the periodic region (anything above the seperatix), why is that? (is it still capturing the entire state-space of the pendulum?)

[@ethandennis368]

It seems they are just mapping the subspace inside the separatrices, but this is not the only possible way to do it. I do not believe this to be interesting in the way tori often are in dynamics (see my other comment I am about to leave). Pretty much the phase space of the system is S1 x R (since we have a pendulum angle and its momentum that could get as big as you want) which is an (inifinite) cylinder. However, because energy is conserved and the position variable is periodic, you will naturally require some momentum bound since if you came around in position space and your momentum was not the same, the energy along this trajectory would not be preserved. So it seems they are just sort of choosing a cut off and identifying the ends of this truncated cylinder. If you chose your momentum subspace to be bigger than this, the separatrix is still there, and the trajectories outside of it will also be separatrices on the torus. I dont believe theres a whole lot of reason to do this, since the ends you are identifying are not like a continuous part of the same family. I would guess symmetry leads to some ways to represent this on like klein bottle and fix that but again, why lol.

[@dibeos]

Thank you! The mapping focuses on the periodic and quasi-periodic regions because these correspond to bounded motion. The separatrix represents the boundary between bounded and unbounded regions, and beyond it, the pendulum’s dynamics (like full rotations) are not captured on the torus since it’s not a bounded manifold. This keeps the visualization specific to the periodic state-space

[@ayoubelazzouzi5600]

some contexts that we can apply this studies? with an added value

[@cosmicwakes6443]

Alexander Grothendieck sought to understand the underlying principle of the dichotomy between the discrete and continuous through his project of anabelian geometry, so in that spirit, could dynamical systems also pay a role in bridging the divide between number theory and geometry, that is, the continuous and discrete? I know that dynamical systems has illuminated many areas of number theory through the branch of mathematics know as arithmetic dynamics, including Mordell's conjecture. It would seem that dynamics could have far reaching consequences for mathematics as a whole.

[@GrowthInMath]

I wish you will make a video in italian

[@dibeos]

Vorremmo anche farlo in italiano, ma in inglese abbiamo più visualizzazioni. Il nostro obiettivo è far crescere questo canale fino a un certo numero di iscritti e poi assumere qualcuno che ci aiuti con le animazioni. Poi possiamo iniziare un canale tutto in italiano! Questo è il nostro obiettivo!

[@Neilcourtwalker]

You are not presenting the yellow press. Stop with that nonsens of changing the presenter every second sentence. This is so offputting and makes the videos complete bullshit.