All Questions
8 questions
4
votes
1
answer
848
views
Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)
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 ...
- 356
6
votes
2
answers
624
views
On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
- 356
6
votes
0
answers
283
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 ...
- 356
9
votes
1
answer
839
views
Conformal changes of metric and geodesics
Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...
- 4,145
6
votes
1
answer
572
views
A vector space associated with a vector field on a symplectic manifold
$\DeclareMathOperator\Div{Div}$Edit: The correct formulation of the vector space $S(X)$ which is defined in this question is the following:$$S(X)=\{Y\in \chi^{\infty}(M)\mid X.\omega(X,Y)=(1/...
- 356
5
votes
1
answer
597
views
A vector field whose flow has constant singular values
$\newcommand{\tr}{\operatorname{tr}}$
$\renewcommand{\div}{\operatorname{div}}$
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi_t$ be its flow.
Does ...
- 6,741
4
votes
0
answers
148
views
Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$
In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
- 91
2
votes
0
answers
479
views
A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
- 356