# Questions tagged [limitcycle]

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32 questions
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### Updated background on Hilbert 16th problem?

What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
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### Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
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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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### Numerical and computational approaches to limit cycle theory

I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach. I would like to ask ...
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### A complex limit cycle not intersecting the real plane

Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow? There is a regular leaf $L$ whose holonomy, along at least one closed curve ...
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### A possible obstruction for existence of limit cycle for analytic vector field on $S^2$

Is there an analytic vector field $X$,on $S^2$ which possess a limit cycle but $X$, satisfy $\nabla_X X =0$ or satisfy $\nabla_X JX= 0$ where $J$ is the standard almost ...
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### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
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### A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$

In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed ...
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### The centralizer of Lienard equation

Consider the lienard vector field $\cases{ x'=y -F(x) \\ y'=-x }$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
Consider $2D$ dynamical systems $X' = F(X)$ where $X=(x,y)$ is a 2-vector, and there is an equilibrium point at the origin. Let $L$ be the set of numbers $x > 0$ such that a limit cycle of the ...
The following question is related to the Seifert conjecture. Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...