Linked Questions
11 questions linked to/from Hilbert 16th problem and dynamical Lefschetz trace formula
172
votes
36
answers
35k
views
Proposals for polymath projects
Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
44
votes
5
answers
6k
views
Finding a 1-form adapted to a smooth flow
Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
- 114k
21
votes
2
answers
2k
views
Applications of number theory in dynamical systems
I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics.
...
- 760
4
votes
1
answer
366
views
A cubic system with two nested limit cycles with opposite orientations
What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...
- 356
2
votes
2
answers
584
views
A non-vanishing vector field on $S^3$ whose flow does not preserve any transversal foliation
Is there a non-vanishing vector field $X$ on $S^3$ which does not admit a transversal $2$-dimensional foliation? if the answer is negative, is there a non-vanishing vector field $X$ on $S^3$ which ...
- 356
5
votes
1
answer
164
views
A non-geodesible foliation of $S^3$ or $S^2\times S^1$
Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?
If the answer is ...
- 356
8
votes
0
answers
508
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 $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
- 356
2
votes
0
answers
624
views
Why Poincare sphere compactification and not torus compactification
The Poincare compactification is a method to carry a polynomial vector field on the plane to an analytic vector field on $S^{2}$ via analytic embedding $$(x,y)\to (\frac{x}{\sqrt{1+x^{2}+y^{2}}},\...
- 356
2
votes
0
answers
427
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 ...
- 356
2
votes
0
answers
136
views
Irrational closed orbits of vector fields on $S^2$ (Limit cycles and trace formula)
Motivations: We first introduce our motivations: We wish to find an operator-theoretical interpretation for the number of limit cycles of a polynomial vector field on the plane. We quote the ...
- 356
0
votes
0
answers
76
views
A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field
Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...
- 356