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Is there a polynomial vector field $$P(x,y,z)\partial_x+Q(x,y,z)\partial_y+R(x,y,z)\partial_z$$ which has a closed orbit $K$ such that $K$ is a non trivial knot?

I am interested in this question since 1999 when I heared the definition of a knot and I read the definition of unknoting and the total curvature of a knot. To what extent can closed orbi …

Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property: For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{-1 …

Inspired by differential equation $$y(1+y'^2)=c$$ which generates the cycloid we consider the following differential equation on a Riemannian manifold: $$f(1+|\nabla f|^2)=c$$ On the other hand th …

It is just the Lotka Volterra system https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations The above link contains materials about this system. I remember I learned about these material fro …

Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\i …

There is no any kind of, generally speaking, "Charactristic curve" for the vector field $z'=f(z)$ when $f$ is a holomorphic function. By charactristic curve I mean any kind of particular curve wh …

It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post Orbits space of real-analytic planar foliations One can …

In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977 "The Products of Manifolds with the f.p.p. Need Not have the f.p.p" who gave an example of two fixed point …

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 $x\in S^2$ t …

This paper contains a conjecture and a partial result about the abelian integral under discussion :"Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals" …

Is there a vector field $X$ on $\operatorname{M}_n(\mathbb{R})$ or $\operatorname{GL}(n,\mathbb{R})$ with the following condition: $$\begin{cases} X\cdot \operatorname{trace}=\operatorname{Det} \\X\c …

We consider the following standard partial order relation on $\mathbb{R}^n$: We say $X=(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots,y_n)=Y$ iff $\sum_{i=1}^k x_i \leq \sum_{i=1}^k y_i,\quad \forall k: 1 …

A physicist colleague asks me the following question. I have no idea to answer him. Your answer is very appreciated. Is there a global first integral on $\mathbb{R}^3$ for the following vector field? …

Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation? If the answer is affirm …