Give a talk on the long time behavior of dissipative systems.

The professor, the screen is yours.

We're looking forward to your talk.

Okay, thank you.

I can start. Yes.

Yes, please.

Okay.

So thank you very much for the invitation and presentation.

So I think you see in the abstract of the talk, I think, have start immediately with a very simple elementary

example to explain more or less what the phenomenon I'm going to describe here.

So let us consider just a particle that is moving on the line under constant force.

Okay. So, and the visit is some of the dissipation.

Okay.

So if we simply suppose you will dissipation with under some friction, we have a friction.

So the differential equation is like that.

So what I mean in the equation goes to infinity.

Behavior is very different. So because if you do not have dissipation, if you conserve energy, that is acceleration is constant, it goes to infinity, it is growing velocity.

Otherwise, it's not like that.

And actually, this particle essentially is a model of a boat in the water. If you have a boat in the water, if you apply some force, it goes with a constant velocity.

Stronger force, greater velocity. If you do not, if water does not move and you do not apply any force, it just stay.

It stops. If we apply force and go with some velocities, we stop to apply force. It stops very fast.

So it looks like it follows not a Newtonian mechanics. It follows Aristotle mechanics rather than Newton one.

At least in the visual. Perhaps Aristotle invented his mechanics, his physics, watching the ships in the sea, more or less.

And it was plausible.

Indeed, let us look on the phase portrait on this absolute elementary system.

So we take a phase portrait.

Take an axis, here is going to be Q, another axis is Q dot.

And then the trajectory phase portrait is as follows.

We have a very particular solution, this one with constant force. I'm sorry, it's not quite good to draw in the screen.

Something like that is constant and all other solutions tend very fast, exponentially fast to this one.

This is a constant, not constant solution, this is a horizontal line solution.

And all other solutions go quite fast to this one. Look more or less like that. I'm sorry, this goes in one direction, of course.

Like that. They look essentially like that.

This is a picture. And this is what you see. You see only this solution because the rate with which they go to this particular one with constant velocity is exponentially higher than the velocity, than the rate of your particle on this one.

So long time behavior is this one.

And actually what you see.

And moreover, if you look a little bit on this phase portrait, intuitively, you can imagine if you take not a constant force, but close to constant,

put here something like a BVQ, qualitative picture should be the same. So this invariant, so your system has invariants of many fold and that is projected very well on the velocities on the configuration space.

So for any point Q, we have a prescribed velocity somehow here.

It is not any more constant if you just a little bit, little bit perturbed this B, the solution will be similar, but of course it's the invariant guy will be not constant, but just changes a little bit.

But the structure is the same. Everything goes to this invariant. It cannot disappear because it is what people call normally stable. So it's invariant sub manifold such that everything tends to it and moreover it tends much faster than the dynamics on the sub manifold.

It's like the definition, but I do not want to.

If your dissipation is small and B really depends, if B is constant, so even for small dissipation, we have the same picture, just this invariant sub manifold move more or less the velocity grows when dissipation decreases in velocity.

So if the dissipation is small and B starts to be reasonable change, it depends on who knows more while the little bit more while then the structure tends completely, change completely some vertices appears and we do not have the prescribed velocity.

So dynamics really can be seen only in the phase space q, q dot and not not the velocity is not function of q only.

Even in the long, even in the long time.

And what I'm going to explain today is that it is actually universal phenomenon, at least in the case of

mechanical systems. It is just a mechanical system with isotropic dissipation, subisotropic dissipation like that in any dimension, actually in any manifold.

And moreover, we should control what is the threshold dissipation. So if we have some dynamical system with kinetic and potential energy.

And in terms of this energy, we have a well defined and easily computable, computable tensor, curvature tensor that control where the head is like essentially like that everything tends to the dynamics with the prescribed velocity that depends only on the point.