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This question is inspired by the concept of "Differential Inclusion". The ring of entire holomorphic functions is denoted by $Hol(\mathbb{C})$. Is there a complete classification of all $f\in Hol(\ …

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 …

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation? $$\begin{cases}\dot{x}=y-x^{3}\\\dot y= …

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 …

We consider the planar polynomial vector field $$(*) \;\;\;\begin{cases} \dot x= P(x,y) \newline \dot y =Q(x,y)\end{cases}$$ We replace the real variables $x,y$ with complex variables $x:=x_{1}+ …

Is there a non vanishing real analytic vector field $X$ on $S^3$ such that $X$ has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case? …

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$ Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such tha …

A second order differential equation on a manifold $M$ is a vector field $X$ on $TM$ which is not only a section of the vector bundle $T(T(M)) \to TM $ with the obvious structure, b …

Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?

This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation). Linear vector fields are always complete vector field so they do not satis …

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? …

What is a classification of all quadratic vector fields $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}\qquad (V)$$ with a center at origin such that $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\righ …

The rescalling $(V')$ of $(V)$ as described in the question has always an isochronous center when $(V)$ is a quadratic system with center. The reason is that $d\theta(V')=1$ where $d\theta=(\frac{1} …

Consider the vector field $$X=(y-10x)\partial_x-x\partial_y$$ it is not a gradient vector field with respect to the standard Riemannian metric of $\mathbb{R}^2$ but it is a gradient …

Is there a non vanishing vector field on $S^{3}$ with an infinite family $T_{\lambda}$ of invariant torus such that each $T_{\lambda}$ has an structure of a Kronecker foliation but for e …