Linked Questions

8 votes
0 answers
497 views

A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
Ali Taghavi's user avatar
4 votes
0 answers
486 views

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow. Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
Ali Taghavi's user avatar
2 votes
1 answer
463 views

A curvature description for center condition for quadratic vector field

We consider the quadratic vector field $V$ $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end {cases}\;\;\;\;(V)$$ where $P,Q \in \mathbb{R}[x,y]$ are polynomials of degree $2$ with $P(0,0)=Q(0,0)=...
Ali Taghavi's user avatar
6 votes
0 answers
454 views

An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)

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 ...
Ali Taghavi's user avatar
2 votes
0 answers
419 views

Lifting a quadratic system to a non-vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non-vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
Ali Taghavi's user avatar
4 votes
0 answers
325 views

The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$: $\tilde{D}_{X}(f)=...
Ali Taghavi's user avatar
5 votes
0 answers
305 views

Is the closed orbit of the Van der Pol equation a stable periodic orbit?

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 ...
Ali Taghavi's user avatar
6 votes
0 answers
282 views

A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
Ali Taghavi's user avatar
5 votes
0 answers
278 views

The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$. We consider the following polynomial vector field on ...
Ali Taghavi's user avatar
3 votes
0 answers
356 views

(Some possible obstructions to ) Limit cycles as closed geodesics(3)

First we explain our Motivation: Motivation: First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ ...
Ali Taghavi's user avatar
2 votes
1 answer
210 views

A complex limit cycle not intersecting the real plane(2)

Inspired by this question and the counter example provided in its answer we ask: Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the ...
Ali Taghavi's user avatar
3 votes
0 answers
201 views

Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
Ali Taghavi's user avatar
3 votes
0 answers
161 views

Flat Riemannian metrics adapted to quadratic vector fields with center

Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$ Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
Ali Taghavi's user avatar
0 votes
0 answers
159 views

A closed leaf with two different index with respect to two different Riemannian metrics

Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question: Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, ...
Ali Taghavi's user avatar
1 vote
0 answers
157 views

Indecomposability of image transformations (pure algebra). Open questions

W-transformations -- definitions We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
Włodzimierz Holsztyński's user avatar

15 30 50 per page