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Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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144 views

Fixed-points of a topological circle action

Suppose the circle group $G = S^1$ acts on $X$. If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. ...
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0answers
78 views

Minimal period for a bounded Langton's ant moving on a tessellation

We consider Langton's ant on the 2D plane, but we replace the square lattice by a Voronoi tessellation obtained from a finite set of points (it could be another tessellation, however directions such ...
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0answers
117 views

Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow

Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
2
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0answers
86 views

Particular dynamical system having a bounded orbit

I'm new to dynamical systems, but I've stumbled onto an interesting system and I'm wondering if the orbit is bounded or not for a particular initial condition. The function in question is: $f(x) = \...
1
vote
1answer
312 views

A complex limit cycle not intersecting the real plane

Edit: This is a real coefficient version of the current post. Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow? There is ...
2
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0answers
63 views

A possible obstruction for existence of limit cycle for analytic vector field on $S^2$

Is there an analytic vector field $X$,on $S^2$ which possess a limit cycle but $X $, satisfy $\nabla_X X =0$ or satisfy $\nabla_X JX= 0$ where $J$ is the standard almost ...
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0answers
147 views

Partition the rationals with respect to a multivariate polynomial which sends classes to classes

Let $R$ be a commutative ring and let $f\in R[x_1,x_2,\cdots,x_{n-1}],n\geq 2$ be a polynomial. Definition: We say $f$ is $n$-severable over $R$ if there exists a partition (of set) $$R=\coprod_{i=...
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0answers
47 views

Region of attraction of simple ODE with perturbation

Consider the following simplest example: $$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA. Now consider the two dimensional case: \begin{equation} \begin{aligned} &\dot{x} = x(x-1)(x+1)\\ &...
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0answers
115 views

Stability when linearization fails

The dynamics of the $j$th system: \begin{equation} \begin{split} \dot{\overline r}_j &= h (\overline r_j) \,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...
2
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0answers
107 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 $...
4
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2answers
169 views

Rotation number of composition

Let $f,g:S^1 \to S^1$ be orientation-preserving homeomorphisms. Consider the lift $F,G:\mathbb R \to \mathbb R$. Let $\rho(G)$ and $\rho(F)$ be a rotation numbers. What we can say about rotation ...
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195 views

Canard limit cycle for certain singularly perturbed system(Is there a contradictory situation?)

From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system $$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) ...
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1answer
99 views

How to compute the entropy of a random variable with values in a metric space? [closed]

I have a cloud of points, and I want to compute its 'diversity'. Variance is not appropriate, because a cloud clustering around few points can still have a large variance. To that end, I see the ...
4
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0answers
191 views

Singular foliations of $\mathbb{C}P^2$ that are compatible to Fubini-Study metric

Is there a complete classification of quadratic polynomial vector fields on $\mathbb{C}^2$ whose corresponding singular foliation of $\mathbb{C}P^2$ satisfies the property quoted below? The regular ...
8
votes
1answer
421 views

Existence of at least one compact orbit on the sphere

Good morning, I've came across this question during my researches. It seems apparently very simple, however I Googled a bit and I couldn't find the answer I was looking for. The question is as ...
3
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0answers
67 views

Does the orbital function divided by the volume of a ball decrease?

Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...
2
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0answers
75 views

Ruelle-Perron-Frobenius for continuous time

I'm looking for a proof (or a reference for it) of the following result: Let $L$ be a $n \times n$ $Q$-matrix, $p_0$ probability vector such that $L(p_0) = p_0$ and $V\colon \Omega \rightarrow \...
6
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0answers
210 views

Spectral properties of Non-local Differential operators on real line

I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs. Definition: A ...
12
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1answer
356 views

Entropy of composition

I asked this at math.stackexchange.com, but got no answers. Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (...
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0answers
68 views

Topological transitivity for a self-map of $\mathbb{R}$ with finitely many discontinuities

I started working with a map $f:\mathbb{R} \to \mathbb{R}$ such that it is continuous except on a finite set. I started looking for a definition of topological transitivity and topological mixing in ...
6
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1answer
444 views

A generalization of Gradient vector fields and Curl of vector fields

Let $M$ be a smooth Riemannian manifold. The Riemannian metric enables us to equip the tangent bundle $TM$ with a symplectic structure $\omega$, which is the pullback of the standard symplectic $2$ ...
6
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1answer
172 views

What about of periodic points of $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, $0<x<1$, where $\mu(n)$ is the Möbius function?

Let $\mu(n)$ the Möbius function, we define $F:[0,1]\to[0,1]$ as $$F(x)=\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n.\tag{1}$$ For a function of this kind (I presume that this continuous function has image $[...
6
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1answer
156 views

In a non-compact metric space, topological transitivity need not imply onto

I had asked this question on Mathematics Stack Exchange yesterday but it got no response so I'm asking here. Let $X$ be a compact metric space and $f:X \to X$ be continuous. If $f$ is topologically ...
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0answers
218 views

Connection between integrable systems and group actions

An integrable system can be defined as a symplectic manifold together with the maxiumum possible number of Poisson commuting functions on the manifold which are almost everywhere independent. By the ...
6
votes
1answer
153 views

Possibly new solution to equal-mass three-body problem; refinement required

(This is a repost of this question from 18 months ago on the main Mathematics SE site, as the response there has been underwhelming, and I thought here would be a better authority. As you can probably ...
2
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0answers
56 views

Exponential Convergence Under a Lyapunov-Like Assumption

Consider $V=\mathbb{R}^d\times\mathbb{R}^n$ with coordinate $x^T=[\theta^T,\sigma^T]$. I have an ODE of the form: $\dot{x}=F(x)$, where $F$ is assumed to be sufficiently smooth. Suppose that there's ...
3
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0answers
65 views

Generalisation of Lyapunov time to stochastic dynamical systems

Might there be useful generalisations of the Lyapunov time to stochastic dynamical systems? In particular, I'm interested in methods for calculating confidence intervals around stochastic analogues of ...
7
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0answers
231 views

The graph of Rule 110 and vertices degree

Consider the elementary cellular automaton called Rule 110 (famous for being Turing complete): It induces a map $R: \mathbb{N} \to \mathbb{N}$ such that the binary representation of $R(n)$ is ...
4
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0answers
162 views

How can I catalog these generalized Collatz problems?

The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$: $x + x + 1 \rightarrow x+x+x+1+1;$ $x + x \rightarrow x;$ Whenever a number matches ...
3
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1answer
125 views

Aperiodic tiling of compact space by small number of basic tiles

Suppose we have compact space, like sphere or torus in particular dimension $d$. Is it possible to construct aperiodic tiling in such setting? It seems obvious, answer is yes, because we may just ...
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0answers
92 views

Oja's rule gives unit eigenvectors

Does Oja's rule for normalized Hebbian learning always result in a unit eigenvector which corresponds to the largest eigenvalue? Or are there any specific conditions or assumptions under which this is ...
3
votes
1answer
91 views

Topological universality for Cantor maps

I am afraid this question might be very naïve, but I find it hard to locate a reference that does not answer a slightly different question. Consider the Cantor set $C$ and a continous map $f: C\to C$ ...
6
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1answer
628 views

A problem involving the inverse Collatz map

Let $C$ be the Collatz map on the natural numbers, defined by: $$C(n) := \begin{cases} n/2 & \text{if} \;n \;\text{even} \\ (3n+1)/2 & \text{if} \;n \;\text{odd} \end{cases}$$ The inverse ...
28
votes
1answer
790 views

Vanishing line on Conway's game of life

If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$. ...
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0answers
166 views

Asymptotic formula, polynomial, irrational number and uniformly distribution

Problem 1 Given a irrational number $\alpha$ and two polynomials with positive integer coefficients $P(n),Q(n)$, is it possible to get the asymptotic estimate and reasonable error term for: $$\...
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0answers
45 views

For skew product maps, does ergodicity of the two-point motion imply weak mixing?

Let $\Omega$ be a Polish space equipped with a Borel probability measure $\mathbb{P}$, and let $\theta \colon \Omega \to \Omega$ be a measurably invertible map that is mixing with respect to $\mathbb{...
7
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1answer
417 views

Naturally occuring counting process with a 1/log asymptotics?

Besides prime numbers, is there another physically realizable counting process that exhibits a 1/log density ? The reason I am posting this question is that we are measuring the response of a quantum ...
3
votes
0answers
43 views

Limits of regularized solutions of a singular ODE

Suppose that we have a singular BVP of the form $$\dot{x}=\frac{f(x,t)}{t^n}, \qquad x(0)=x_0$$ where $x\in \mathbb{R}^n$, $n\in \mathbb{N}$. There is of course no more uniqueness and the existence ...
2
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0answers
64 views

Hausdorff dimension of radial limit sets for divergence type subgroups

Let $X$ be a proper $CAT(-1)$ space. Let $\Gamma<Isom(X)$ be a subgroup of divergence type. Is it true that the Hausdorff dimension of the radial limit set of $\Gamma$ in $\partial X$ is equal to ...
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0answers
58 views

Dynamics for sets related to Brownian motion: zero set, fast points

For sets like the Cantor set, we have preserving maps (eg. the shift-maps and conjugates to it) that allows us to study dynamical quantities such as invariant measure and entropy. I am wondering if we ...
2
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1answer
316 views

Does this limit always exist?

Here is the question; it may seem very simple, but it is difficult (at least for me). Let $f(x)$ be a continuous function on $R$ that is strictly increasing, and suppose $g(x)=f(x)-x$ is a periodic ...
3
votes
1answer
94 views

Totally minimal homeomorphisms on connected locally compact noncompact spaces

If $\alpha : X \to X$ is a minimal homeomorphism on a compact Hausdorff space $X$, then if $X$ is connected, $\alpha$ is totally minimal, that is $\alpha^k$ is minimal for every $k \in \mathbb{Z}$. I ...
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2answers
201 views

Is this Riccati equation (“Josephson junction”) always phase-locked at integer rotation numbers?

Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE $$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$ Identifying ...
8
votes
1answer
305 views

Can a harmonic vector field possess a limit cycle?

Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)? Note that the Laplacian of a vector field is defined via natural correspondence ...
3
votes
2answers
295 views

Identification of Invariant Sets for Discrete Dynamical Systems on the Positive Integers

Let $\phi:\mathbb{N}\times \mathbb{N}^+\rightarrow \mathbb{N}^+$ be a dynamical system on the positive integers. Suppose we refer to the orbit of a periodic point of $\phi$ as an invariant set of the ...
4
votes
2answers
167 views

Making images arbitrarily dense under an expanding map

Let $f\colon \mathbb{R}^2 \to \mathbb{R}^2$ be a $C^2$ uniformly expanding diffeomorphism that fixes the origin: that is, $f(0)=0$ and there is $\lambda>1$ such that $d(f(x),f(y)) \geq \lambda d(x,...
16
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0answers
397 views

Trapping lightrays with segment mirrors

Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? I posed this question in several forums before (e.g., here and in an ...
2
votes
2answers
127 views

Planar polynomial vector field for a harmonic pair of polynomials

Has the system of ODEs $$\frac{dx}{dt}=P(x,y)\\ \frac{dy}{dt}=Q(x,y) $$ been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
5
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1answer
107 views

Example of a non-arithmetic Veech surface (other than regular polygon)?

I am reading this paper of Avila and Delecroix of the billiard flow on polygonal surfaces, but I have to get through some basic definitions first. What is a non-arithmetic Veech surface? A Veech ...
3
votes
2answers
150 views

A Really Simple Stochastic Dynamic Billiard

Consider the following stochastic dynamical system. Fix $a > 0$, $b > 0$, $c>0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t),z(t))$ be the position at time $t$ of a point which moves ...