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Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field w …

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) } …

Let $\mathbb{D}=\{z\in \mathbb{R}^2\mid |z|<1\}$ Is it true to say that every homeomorphism of $\mathbb{D}$ is conjugate to a self homeomrphism of the disk extendable to a homeomorphism of $\bar{\math …

According to existing informative and interesting answers one gets that the local dynamical behavior around fixed points or around periodic orbits may generates some obstructions for being conjuga …

Inspired by these two posts on knots orbits of polynomial vector fields on $\mathbb{R}^3$(A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit) and (Are total curvature and the …

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 …

I would appreciate if you introduce me a reference (paper or book) who address the concept of hyperbolic dynamics but with emphasis on absence of periodic orbits. a possible tit …

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 …

I had already asked this question in MSE then I ask here at MO. Assume that $A\in M_n(\mathbb{R})$ is a non singular matrix. Is the flow of the linear vector field $X'=AX$ a geodesible flow on $\ma …

Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field on $M$. Are the following 3 conditions equivalent? 1)For every fixed $t$ the diffeomorphism $\phi …

Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is den …

Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question. The complex Lie group $H=\math …

By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1 …

I would like to apply the known version of the conjectural formula (11) page 10 of the paper Number theory and dynamical Lefschetz trace formula. Disclaimer: I do not have a complete unde …