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14 votes
1 answer
2k views

The perturbation of non-Hamiltonian algebraic vector fields

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) } \...
0 votes
1 answer
77 views

Same occupation measure $\Rightarrow$ same trajectory

Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ The occupation ...
2 votes
0 answers
136 views

Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures "Global Conjecture: There is a dense set $D$ of dynamics such that any element of ...
1 vote
0 answers
64 views

Physical measure of a dynamical system in terms of its density

Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ In ergodic theory, the occupation measure is $$\mu_{x, T}(...
0 votes
0 answers
58 views

Role of basins of attraction in the Morse decomposition

Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by $$\dot{x}=F(x(t))$$ An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
0 votes
0 answers
303 views

Proof that a first integral is not a constant function

Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions $$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$ such that all of them are differentiable and ...
5 votes
1 answer
205 views

Bias of DS literature to polynomial ODEs

In the literature on continuous time dynamical system, we generally deal with an open set $U \subset \mathbb{R}^n$ and a vector field $F: U \rightarrow \mathbb{R}^n$ and define a DS by the ODE $$\...
0 votes
0 answers
70 views

Example of DS with a dense trajectory in the whole state space

Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure) $$\dot{\mathbf{...
7 votes
2 answers
2k views

The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
1 vote
0 answers
70 views

What *piecewise* smooth curves/surfaces/hypersurfaces give rise to forward-invariant regions of dynamical systems?

Consider a set $\mathcal{B}\subset \mathbb{R}^n$ that is homeomorphic to a closed n-dimensional ball, and denote its boundary by $\mathcal{H}$. Assume that $\mathcal{H}$ is a "piecewise smooth&...
9 votes
1 answer
481 views

Existence of a vector field with a finite number of limit cycles.

The following question is related to the Seifert conjecture. Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
2 votes
1 answer
296 views

Isochronization of quadratic vector fields with center

What is a classification of all quadratic vector fields $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}\qquad (V)$$ with a center at origin such that $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\...
3 votes
0 answers
74 views

A foliation version of S.Husseini counter example in fixed point theory

In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977 "The Products of Manifolds with the f.p.p. Need Not have the f.p.p" who gave an example of two ...
0 votes
0 answers
72 views

$\mathbb{R}^n$-flow, cross-section and Whitney theorem

For a $\mathbb{R}$-flow (X, $\Phi_{\mathbb{R}}$), the (local) cross-section is well defined (recall that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]}(x)=\{x\...
1 vote
0 answers
61 views

Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?

Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties? The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
2 votes
1 answer
123 views

Keeping track of limit cycles via certain second order differential operator

Inspired by the two posts which are linked bellow we ask the following question: Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
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 ...
2 votes
0 answers
150 views

Global solution of second order ODE defined on riemannian manifold

Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
3 votes
0 answers
194 views

If the sum of everywhere linearly independent vector fields are periodic, are the component vector fields periodic?

I feel like the above must be true but embarrassingly cannot seem to prove it. Take linearly independent, commuting vector fields $X$ and $Y$ on a manifold and corresponding flows $\Phi^t_X$, $\Phi^...
5 votes
2 answers
647 views

Flow of a nowhere vanishing complete vector field

Let X be a nowhere vanishing complete vector field on a manifold M, $\gamma: \mathbb{R} \to M$ be its flow with $\gamma(0)=p \in M$ and suppose it is not periodic. If $\gamma(\mathbb{R})$ is closed, ...
2 votes
0 answers
108 views

Does a smooth dynamical system always come with a metric

Warning: My education in formal mathematics is very weak so I apologize for any confusions/errors in the following, please don't hesitate to correct me. Question: Consider a smooth dynamical system $...
3 votes
0 answers
165 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 ...
3 votes
1 answer
195 views

An explicit formula for a flat metric compatible to certain polynomial vector field with center

Let $X$ be the following vector field on the plane: $$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$ The vector field $ (X)$ has a non isochronous center at the origin.The ...
12 votes
3 answers
2k views

Vector field with holomorphic flow

Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms. I know one reference ...
4 votes
1 answer
541 views

A vector field on the tangent bundle which is not equivalent to any second order ODE

A second order differential equation on a manifold $M$ is a vector field $X$ on $TM$ which is not only a section of the vector bundle $T(T(M)) \to TM $ with the obvious structure, ...
2 votes
0 answers
191 views

Geometric properties of solutions of Hamiltonian system

Context : We are interested in the following dynamic with state $(q,\varphi)$ $$ \dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi) $$ ($\varepsilon >0$ ...
5 votes
2 answers
258 views

Noninvariance for a specific nonlinear oscillator

Consider the nonlinear system \begin{align*} \frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} x_2(t) \\ -4x_1(t) + x_1^2(t) \end{pmatrix}, \end{align*} which admits ...
1 vote
0 answers
462 views

Partial feedback linearization (Control theory)

I'm trying to understand a theorem about partial feedback linearization from the paper "On the largest feedback linearizable subsystem" by R. Marino (published in: Systems & Control Letters, ...
6 votes
0 answers
201 views

The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow $\...
6 votes
1 answer
508 views

Estimating the flow when we know the vector field

Suppose we have a $C^k$ vector field $v$ and let $\phi_t$ be the corresponding flow. I have estimates on $v$ and its derivatives: $|v| < C_0$, $|Dv| < C_1$, $|D^2v| < C_2$, ... $|D^kv| < ...
1 vote
1 answer
901 views

How to deduce the existence of stationary points from fixed points of evolution maps?

This is probably a very elementary question. Nevertheless I decided to post it on MO. Consider a smooth manifold $M$ and a smooth complete vector field $v:M\rightarrow TM$. Consider an autonomous ...