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What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post.. For Mathematical development …

In this question, we are interested in the number of limit cycles which appears in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } …

Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question. Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ( …

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{ …

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 bet …

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 fi …

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 po …

From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system $$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) \e …

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 thi …

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 …

Edit: It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally …

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 fro …

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 questio …

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 m …

Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$. Let $B $ be …