Ali Taghavi ,

For comparison, look at what the idea of ​​calculating the oscillation period of a metaphysical pendulum led to:

(8) (PDF) Chaotic dynamics of an electron (researchgate.net)

https://www.researchgate.net/publication/325226826_Chaotic_dynamics_of_an_electron

Ali Taghavi, Vladimir Sergeevich Anashin

Dear Ali, let me digress further from the topic you set, turning to limit cycles in 3- and 4-dimensional space. However, it is possible that having dealt with multidimensional cases, the problematic issues with limit cycles on the plane will be clarified. In my head, I have a plan to construct a dynamic system in 4-dimensional space whose limit cycles lie on a 3-sphere, and a plan to construct a dynamic system in 3-dimensional space whose limit cycles lie on a classical torus. Moreover, such a toric inertial manifold can be transformed into a 2-sphere with pole-cut outs, the geometry of which is used in the article from the previous post. If you and (or) Vladimir Sergeevich Anashin are interested, we could develop this topic in a joint article.

Sincerely, Igor.

Dear Igor, thank you for your kind invitation. Generally, the theme is interesting for me though I am afraid that my personal contribution (if any) may not be significant. Also, please note that in August and September I will be busy in other projects. Anyway, please keep me in due course, I am staying tuned.

Best wishes,

Vladimir

Igor Bayak Vladimir Sergeevich Anashin

While limit cycles originaly defined in dimension 2 (may be initiate by Poincare, "Cycle Limit" but is very interesting to consider a higher dimensional analysis.

In this line I would appreciate if you give comment on Remark at the bottom of page 2 and point 3 in page 3 of the following note

Article Some Questions around The Hilbert 16th Problem

BTW your comments on the following question for Hamiltonian with finite number of limit cycle are very appreciated

https://mathoverflow.net/questions/210996/an-algebraic-hamiltonian-vector-field-with-a-finite-number-of-periodic-orbits-2?noredirect=1&lq=1

Dear Igor I will start reading your paper tomorrow

Thanks again for sharing and your proposal

Best regards

Ali

That's what I meant.

A dynamical system on a plane with a limit cycle on a unit circle:

I=y∂x-x∂y

1=x∂x+y∂y

ρ2=x2+y2

Γ=I+log(ρ)1

Similarly, a dynamical system in a 4-dimensional space with an inertial manifold in the form of a unit sphere is obtained:

I=x2∂x1-x1∂x2+x4∂x3-x3∂x4

1=x1∂x1+x2∂x2+x3∂x3+x4∂x4

Γ=I+log(ρ)1

At the same time, a dynamical system in 3-dimensional space with an inertial manifold in the form of a torus can be constructed from limit cycles Γ coinciding with Villarceau circles and hyperbolic dynamical systems

J=y∂x+x∂y

on planes orthogonal to Villarceau circles and dividing them in half, which are annulled on isotropic lines intersecting Villarceau circles.

I have not yet thought through the transformation of a toroidal inertial manifold into a sphere with gouged poles, but even the presented inertial manifold requires more rigorous formalization. It would also be nice to think about visualizing this dynamical system.

You can also think about dynamical systems with inertial manifolds nested inside each other like matryoshka dolls.

Ali Taghavi

Ali, you are asking for comments, but I need to learn more before commenting. I hope that in the process of working together I will learn a lot more about Hamiltonian and polynomial dynamical systems. In the meantime, I don't know anything, I'm just guessing.

Our dynamical system may be more attractive from the point of view of physics if we accept

Γ=I∕ρ+log(ρ)1∕ρ

Then, using the current lines of the vector field 1∕p and I∕ρ as the coordinates (r=log(ρ),φ) of the manifold, we obtain from the dynamical system on the plane and in 4-dimensional space dynamical systems on cylindrical manifolds ℝ1×S1 and ℝ3×S1, respectively. At the same time dφ∕dr=1∕r and therefore, a material point with a gravitational potential 1∕r consists of limit cycles that close on the 3-sphere.

Ali Taghavi

It seems that dividing by p is not required. Check out the post explaining this issue

https://www.researchgate.net/post/What_is_an_action

I appreciate your interest in my question. But i do not see relation between my original question and the MO link I provided with the materials of your paper. your paper is interesting but please explain the relation

Ali, you are right, these are two different things and I have given this link in vain. But you have awakened my interest in limit cycles. However, if you do not want me to deviate from the topic you have set, then I will shut up.

Ali Taghavi, I went into the subject a little bit. Could you look at the text from page 6 to page 8. Is there any connection with Hilbert's 16th problem there?

Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS

Ali Taghavi

Okay, let me ask you straight out - is this a polynomial dynamical systems?

Σn ∂ρi ∕ ∂φi = Σn 1 ∕ (ρi-1)

where

∂φi = (y-yi)∂x-(x-xi)∂y

∂ρi = (x-xi)∂x+(y-yi)∂y

ρi = ((x-xi)2+(y-yi)2)½

Igor Bayak Vladimir Sergeevich Anashin ladimir Sergeevich Anashin

Now I read your two papers on "ALGEBRA" of linear vector fields. They are interesting and raise some new questions

For an arbitrary Riemannian manifold we classify all finite dimensional vector space of vector fields which are closed under the "CONNECTION" operation (as you did in your interesting paper) such that the connection operator is indeed an associative algebra operator. It was the case for linear vector field you did.(In your paper in Journal of geometry and symmetry in physics".

I try to read again your two papers then we will have more discussion.

BTW I am interested in this question since many years ago

https://mathoverflow.net/questions/382577/does-p-xp-yq-xq-y-0-implies-non-existence-of-limit-cycle-for-p-partial-x

Your comments are very appreciated

Ali Taghavi Thank you for your interest in my writings. For my part, I will try to understand yours. But a little later.