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Consider $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,y)=\mathrm{e}^{x}\cos y$ then $\nabla (f)$ is nothing but $\mathrm{e}^{\bar{z}} :\mathbb{C} \rightarrow \mathbb{C}$, with image neither co …

For every positive real number $x$ we define $$E(x)= \int_0^{\infty} x^t/t!\,\mathrm dt$$ where $t!=\Gamma(t+1)$. This is motivated by classical exponential function. Is this function well defined (t …

Updated 1/25/2023 I just added a related post below: Jacobi fields, Conjugate points and limit cycle theory EDIT: Here is a related post which concern quadratic vector fields rather than Van de …

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 …

There is sequence of continuous functions $f_{n}$ on the unit interval $[0,1]$ which converges to a function $f$ such that $f$ is discontinuous at rational points of $(0,1)$, a dense subset …

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point. Let $p: \mathbb{R}^n \to \mathbb{R}$ be a polynomial function. We define the Ham …

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet …

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map. Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?

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 …

A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions. Motivated by conversations on this questions we ask: …

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 …

Assume that $(M,g)$ is a Riemannian manifold. A vector field $X$ on $M$ is called a harmonic vector field if the corresponding $1$-form $\alpha$ with $\alpha(Y)= \langle X,Y \rangle_g …

Let $P$ be a homogenous polynomial with real coefficients in several variable(at least three variable) Is the following statement true: For every $\epsilon$ there is a $\delta$ such that for ev …

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 …