Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,397
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Closed orbit for vector field $f(\bar{z})$ where $f$ is holomorphic function
Edit : According to the comments of Michael Renardy and Christian Remling I revise the question as follows:
Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such ...
- 312
39
votes
2
answers
3k
views
3D Billiards problem inside a torus
I have been trying to simulate the behavior of a light particle being reflected inside of a torus (essentially a 3D billiards problem). I have found that after a few thousand bounces, it converges on "...
- 589
2
votes
1
answer
179
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Riemannian metric adapted to singular $1$-dimensional foliation
Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$, with the following properties:
1) The origin is an isolated singularity for $X$ and its linear ...
- 312
4
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0
answers
177
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Explicit symbolic codings
The short version of my question is that I need examples of explicit continuous symbolic codings of invertible dynamical systems.
Here's a longer version. Suppose $(\Omega,\mu,T)$ is an invertible ...
- 2,085
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votes
1
answer
55
views
The number of limit cycles of a quadratic vector field with a unique singularity
Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?
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0
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343
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Is there a relationship between the Jacobian at a point and the curvature at that point?
The Jacobian $J$ For a dynamical system $\dot{\textrm{x}}=F(\textrm{x})$ determines the dynamics in the tangent plane at a given point. Intuitively speaking the Jacobian evaluated at a point should ...
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1
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491
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A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?
Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll ...
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2
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386
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Orbits of rational functions
This is a generalization of Integrality of iterates of rational functions.
The question is: given an infinite sequence of rational numbers $a_0, a_1, \dotsc, a_n, \dotsc,$ is it always an orbit of $...
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answer
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Integrality of iterates of rational functions
Let $f(x)$ be a rational function which is a ratio of two integral polynomials, and $n \in \mathbb Z$. Then the sequence of iterates $n, f(n), f(f(n)), f(f(f(n)), ...$ will be an infinite sequence of ...
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2
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269
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Reference needed: $C^r$ convergence of Euler's method
Let $U\subset R^n$ be open, $F\colon U\to \mathbb{R}^n$ a $C^\infty$ vector field, and $x(t)$ the solution of
$$x’ = F(x)$$
with initial condition $x(0) = y$, which we assume defined at least for $t\...
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Why is a hyperbolic basic set of dimension 2 either an attractor or a repeller?
I'm currently trying to understand the Birman-Williams Template Theorem, proved in the paper "Knotted periodic orbits II: Fibered knots, Low Dimensional Topology". Unfortunately, there doesn't seem to ...
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1
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830
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Hilbert 16th problem via hyperbolic geometry
More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...
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The notion of shadowing property is independent of choice of metric or not? [closed]
Can anyone tell me that the notion of shadowing property is independent of choice of metric or not?
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Self-map of a set for which the sizes of fibers of iterates are given by polynomials
I am interested in functions $f\colon X\to X$ (where $X$ is some countable set) such that for every $x \in X$ there exists a polynomial $P_x$ such that $\#(f^k)^{-1}(x)=P_x(k)$ for all $k \geq 1$.
...
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1
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Are all Torus Links in fact Lorenz links or not?
I'm currently trying to work through the material on Lorenz knots in the literature and there seems to be conflicting information.
On p. 66, in the Birman-Williams' paper Knotted Periodic Orbits in ...
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answer
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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 $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\...
- 312
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votes
4
answers
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Iteration cycles of Z_n weights in path graphs: Why cycles of length 182 for a 6-node path?
Assign to the $n$ nodes of a path graph vertex weights
forming a permutation of $(0,\ldots,n{-}1)$.
Now iterate the following update repeatedly:
Each node sums the weights of its neighbors, and that ...
- 149k
3
votes
1
answer
193
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An explicit formula for a flat metric compatible to certain polynomial vector field with center
Let $X$ be the following vector field on the plane:
$$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$
The vector field $ (X)$ has a non isochronous center at the origin.The ...
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transforming a Ricatti equation into a generalised Ricatti equation [closed]
C̶o̶n̶s̶i̶d̶e̶r̶ ̶a̶ ̶R̶i̶c̶a̶t̶t̶i̶ ̶e̶q̶u̶a̶t̶i̶o̶n̶ ̶o̶f̶ ̶t̶h̶e̶ ̶f̶o̶r̶m̶
$$ y' + y^2 = S(x), \qquad \qquad \qquad (1)$$
w̶h̶e̶r̶e̶ ̶$̶S̶(̶x̶)̶$̶ ̶i̶s̶ ̶a̶ ̶m̶e̶r̶o̶m̶o̶r̶p̶h̶i̶c̶ ...
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575
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A Rokhlin lemma with a prescribed height function?
Let $T$ be a ergodic automorphism of a non-atomic Lebesgue probability space $(X, \mathcal{A}, \mu)$.
The celebrated Rokhlin tower lemma says that given an integer $n>0$ and $0 < \epsilon < ...
- 2,411
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votes
1
answer
65
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Boundedness of particle motion with time-varying force
Consider the differential equation
$$ m \ddot{x} + k \dot{x} = - W_t x $$
where
$m$ and $k$ are nonnegative.
$x_t \in \mathbb{R}^n$
$W_t$ is a matrix that satisfies $$ \alpha I \succeq W_t \...
- 23
2
votes
1
answer
87
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Herman's paper on the smoothness of conjugacy for irrational circle rotation in english
Can someone help me and tell me where to find the Herman's paper in english (or proof of his theorem) Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations?
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1
answer
154
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Crossed products and unitaries implementing $\mathbb{Z}_n$-actions
I'm working through Li's and Barlak's Cartan Subalgebras and the UCT Problem but I'm stuck at one of the simpler proofs of the paper. On page 9 they deal with masas (maximal abelian subalgebras) of a ...
- 741
4
votes
1
answer
259
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Ergodicity of the Form Factor in Random Matrix Theory
This question is motivated by recent works in quantum gravity, particularly in the analysis of the Sachdev-Ye-Kitaev (SYK) model. The SYK model is a one-dimensional quantum mechanical model which, in ...
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322
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Long wavelength instability: Linear Vs nonlinear phenomenon
I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by:
$\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}...
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3
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1
answer
152
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Is the space of harmonic functions invariant under the derivational operator associated with a geodesible flow?
Assume that $V$ is a vector field on a
Riemannian manifold $(M,g)$ with natural volume form $\Omega$ arising from $g$.
Assume that the solution curves of $V$ are parametrized geodesics of the ...
- 312
3
votes
0
answers
83
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Holder potentials over time-one map of Anosov flow
I have two quick questions that I've been having a hard time finding an answer to. Let $f^{t}: M \rightarrow M$ be a transitive Anosov flow on a closed Riemannian manifold $M$.
(1) Let $\varphi: M \...
- 179
3
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0
answers
135
views
Two semi stable limit cycles with disjoint interior
What is a precise example of a quadratic vector field on the plane with at least one semi stable limit cycles?
Furthermore, is there a quadratic polynomial vector field on the plane with two ...
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0
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55
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Which planar smooth foliations are not smooth equivalent to a foliation arising from level sets of a harmonic function?
Is there an smooth foliation of the plane which is not smoothly equivalent to a foliation $dH=0$ where H is a harmonic function without critical values?
If the answer is negative then we conclude ...
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0
answers
234
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A cubic system with two nested limit cycles with opposite orientations(2)
The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...
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516
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Limit cycles as closed geodesics(2)
Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
- 312
2
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1
answer
463
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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)=...
- 312
2
votes
0
answers
205
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A sum with integer parts
Let $ \mathcal{A} $ be a set of reals such that $ \sum_{a \in \mathcal{A} } \frac{1}{a} = \infty $ and $ \sum_{a \in \mathcal{A} } \frac{1}{a^2} < \infty $. For instance, $ \mathcal{A} = \mathbb{N}^...
- 549
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161
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Is a non vanishing holomorphic vector field necessarily a geodesible vector field?
Motivated by the "The obvious Fact" part of this answer,, we ask the following question:
First we recall a definition, which is used in the above link:
Definition: A non vanishing vector ...
- 312
1
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1
answer
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Finding the "orthogonal" map of a given 1d map
Let $f:\mathbb{C}\to\mathbb{C}$ be meromorphic or even entire. Let $z:\mathbb{C}\to\mathbb{C}$ be such that it holds
$$z(t+1) = f(z(t)).$$
If $f$ is entire and we choose carefully $z$ can be ...
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votes
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answers
161
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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 ...
- 312
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2
answers
583
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Intermediate results for Langton's ant highway conjecture
This paper states the following theorem about Langton's ant:
The set of cells that are visited infinitely often by the ant (for a given initial configuration) has no corners. A corner of a set is a ...
- 1,793
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votes
1
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205
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Non conformally geodesible vector field
What is an example of a smooth vector field $V$ on an open set of the plane which is a geodesible vector field but there is no a conformal metric $g$ such that $V$ is geodesible ...
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votes
0
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486
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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:$$\begin{cases} x'=P(...
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1
vote
2
answers
185
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Transformation which sends asymptotic lines to principal lines over a surface
Suppose $M$ is an $C^\infty$ surface in $\mathbb{R}^3$ and let
$$
K_-=\{p\in M:\ K(p)<0\}\neq \emptyset,
$$
where $K$ denotes Gaussian curvature. Consider the following statement:
Let $p\in ...
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votes
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answers
468
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Diffeomorphisms of $\mathbf R^n$
Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies.
Question: Is there an example to ...
- 1,772
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1
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How to deduce the following inequality by use of the Shannon-McMillan-Breiman theorem?
My question is how to deduce the red inequality from the Shannon-McMillan-Breiman theorem?
First I state some lemmas which will be used in the QUESTION:
the proceeding lemmas are in the setting where ...
32
votes
2
answers
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A Collatz-like problem on prime numbers
Consider the function $f$ on the prime numbers defined by $$ f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle $$(3,7,5,11,23,...
3
votes
0
answers
119
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Irregularly Intertwined Linear Recursions: Other References?
I was wondering if anyone had run across the following notion of intertwined linear recursions. I'm looking for references, or even a standard name. (I know one source, which is the genesis of this ...
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4
votes
1
answer
353
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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 $...
- 312
4
votes
1
answer
713
views
A certain generalization of the Poincare Bendixson theorem
Assume that we have a $n-1$ dimensional integrable distribution $D $ on $\mathbb{R}^n \setminus \{0\}$ which generates a foliation $\mathcal{F}$. We fix an orientation for $D$.(For $n=2$ we ...
- 312
4
votes
1
answer
222
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Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus
Again a very simple question. I currently hold two contradictory ideas in my head
1) A hamiltonian diffeomorphism of a torus necessarily has fixed points
2) most hamiltonian actions on a torus in an ...
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5
votes
2
answers
606
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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, ...
- 103
4
votes
1
answer
146
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Equicontinuity and orbits of compact open sets
Let $X$ be a compact zero-dimensional space, let $S \subseteq \mathrm{Homeo}(X)$ and let $U$ be a compact open subset of $X$. Suppose that $s^{-1} \in S$ for all $s \in S$, and that $S$ restricts to ...
- 4,678
10
votes
2
answers
437
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When is every orbit closure uniquely ergodic?
Given a topological dynamical system $(X,T)$ (so that $T$ is a homeomorphism of the compact metric space $X$) and a point $x\in X$ we call the set ${\mathcal O}(x):=\overline{\{T^nx:n\in\mathbb Z\}}$ ...
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