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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) } …

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 …

Added:8/15/2024 What about holomorphic or real analytic version? Please see the comment discussions on this post. Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We def …

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 …

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 …

We consider the Van der Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$ It is well known that this equation has a unique limit …

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 …

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 …

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book …

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 …

What is a precise example of a quadratic vector field on the plane with at least one semi stable limit cycles? Furthermore, is there a quadratic polynomial vector field on the plane with two se …

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 …