Linked Questions
17 questions linked to/from Limit cycles as closed geodesics (in negatively or positively curved space)
172
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Proposals for polymath projects
Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
44
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5
answers
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Finding a 1-form adapted to a smooth flow
Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
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11
votes
2
answers
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Elliptic operators corresponds to non vanishing vector fields
Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ...
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9
votes
1
answer
838
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Conformal changes of metric and geodesics
Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...
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4
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1
answer
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A cubic system with two nested limit cycles with opposite orientations
What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...
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3
votes
1
answer
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A non vanishing vector field compatible to a Riemannian metric
Assume that $(M, g)$ is a connected Riemannian manifold which is either open or is compact with zero Euler characteristic.
Is there a non vanishing vector field $X$ on $M$ such ...
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6
votes
0
answers
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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 ...
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7
votes
0
answers
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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 ...
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4
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0
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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(...
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1
vote
1
answer
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Ergodicity of geodesic flow in negative curvatutre as a possible obstruction for consideration of limit cycles as closed geodesics(4)
Does the ergodicity of geodesic flow of compact surfaces with negative curvature stile hold for non compact case?
Is not the ergocity theorems of geodesic flow an obstruction to have a ...
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6
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0
answers
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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 ...
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3
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0
answers
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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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5
votes
0
answers
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Unit eigenvalue of the linearized Poincare return map
Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...
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3
votes
0
answers
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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 ...
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3
votes
1
answer
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Obstructions for a foliation to be transformed to a Frenet foliation
Assume that we have a $1$ dimensional foliation of $\mathbb{R}^2$. Is there a global diffeomorphism of the plane which maps all leaves of the foliation to curves with non zero curvature?
One can ...
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3
votes
0
answers
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Flat Riemannian metrics adapted to quadratic vector fields with center
Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$
Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
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2
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0
answers
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An isocline geodesic characterization of $2$ dimensional flat metrics
Lets we have a Riemannian metric on an open subset of the plane which satisfies the following local property.
Local Property: For every point $x$ and every foliation by geodesics around $x$, ...
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