All Questions
9 questions
8
votes
1
answer
365
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Can a harmonic vector field possess a limit cycle?
Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)?
Note that the Laplacian of a vector field is defined via natural correspondence ...
- 356
7
votes
1
answer
841
views
Hilbert 16th problem via hyperbolic geometry
More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...
- 356
7
votes
0
answers
521
views
Limit cycles as closed geodesics(2)
Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
- 356
6
votes
0
answers
283
views
A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory
The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
- 356
6
votes
0
answers
537
views
Counting limit cycles via curvature in Riemannian geometry
In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...
- 356
4
votes
0
answers
495
views
Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)
This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.
Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
- 356
3
votes
0
answers
210
views
Jacobi equation and conjugate points on solution curves of the Van der Pol vector field
Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
- 356
3
votes
0
answers
360
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(Some possible obstructions to ) Limit cycles as closed geodesics(3)
First we explain our Motivation:
Motivation:
First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ ...
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1
vote
0
answers
61
views
Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?
Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
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