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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A technical lemma in the lecture notes of Yoccoz on interval exchange maps

I'm reading the elegantly written lecture notes "Continued Fraction Algorithms for Interval Exchange Maps" of Yoccoz, available through the link <www.college-de-france.fr/media/jean-...
Xueping's user avatar
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More than one recurrence point (Birkhoff)

Birkhoff's recurrence theorem states that for a compact metric space $X$ and a continuous function $T: X\rightarrow X$, there is a recurrence point $x\in X$; the latter means that for any ...
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Topology of windings on the two-torus

In short my question is: what can we say about the quotient topology induced by the linear flow on a two-torus? I know that an irrational slope leads to a dense winding and hence (if I'm not mistaken) ...
Hapax's user avatar
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Aligning frequencies

Let $\omega_1, \omega_2, \dots, \omega_n$ be frequencies between $1$ and $\log n$. I would like to find an upper bound for a point $t$ that align these frequencies up to a small error $\delta$, that ...
Riobaldo's user avatar
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measurability of a special set

I've been working on some questions in dynamical systems then I faced the following problem: Consider the circle $\mathbb{T}^1:= \frac{\mathbb{R}}{\mathbb{Z}}$. We represent it as a union of disjoint ...
r.ygh's user avatar
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Set of all real numbers $x$ satisfying $\lim_{t\to \infty} \lambda_1(g_t u_x \mathbb Z^2)$ exists

Let $g_t=\text{diag}(e^t,e^{-t})$ and $u_x=\begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix},x \in \mathbb R$. Dani's correspondence establishes the Diophantine approximation properties of $x$ and ...
taylor's user avatar
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Identifying Saddle-node bifurcation of a 3D system of ODEs

I am trying to understand and prove the results shown in the following article. However, I am stuck at a point where it is stated that saddle-node bifurcation of periodic orbits occurs at ...
SHR's user avatar
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Chaotic behaviour of the secant method for $\sin(x)$

For not very serious reasons I was trying to understand the behaviour of the secant method for solving $\sin(x)=0$ starting with $x_0=2$ and $x_1=18$, so $$ x_{n+2}=x_{n+1}-\sin(x_{n+1})\frac{x_{n+1}-...
Neil Strickland's user avatar
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29 views

The relay use of invariant set theory

For a dynamical system, set $A$ is an invariant set with a function $V_1$, whose derivative is semi negative definite on $A$, and the region where the derivative is $0$ is the set $B$, which is also ...
ya g's user avatar
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Simple proof that exactness implies strong mixing

Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
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Continuous version of ergodic with integral

Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
Sean's user avatar
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Proving Hopf bifurcations for 3D system

I am working with a 3D continuous system of ODEs. I have found Hopf bifurcation numerically for a certain value of parameter. However, I want prove it analytically. Is it enough to show that the ...
SHR's user avatar
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Stability of rigid bodies spinning around $z$-axis under gravity

Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
Zhang Yuhan's user avatar
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How should the "measure theoretic" Jacobians of a dynamical map be understood in Lai-Sang Young's "Recurrence Times and Rates of Mixing"

In Young's article: Recurrence Times and Rates of Mixing, she uses multiple times the notation $JF, JF^k, JF^R$ to mean the Jacobian of a dynamical map $F:\Delta\to\Delta$ w.r.t. a given reference ...
Epsilon Away's user avatar
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Regularity and rigidity of stable/unstable distribution for geodesic flow on noncompact negatively curved manifolds

For a volume-preserving $C^\infty$ Anosov flow on a three-dimensional compact Riemannian manifold, it was shown by Hurder & Katok that the Anosov foliations are always of class $C^{1, \alpha}$. ...
Guangqiu Liang's user avatar
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Proving period doubling bifurcation

I am working with a 3D continuous dynamical system. I have plotted the bifurcation diagram and found that period-doubling bifurcation occurs at a certain parameter value. However, I also want to prove ...
SHR's user avatar
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Estimate for the length of a partial orbit for a shift map for which its delta neighbourhood covers an interval

Consider $f:[0,2\pi) \to [0,2\pi )$ given by $f(x) = (x + 1) \bmod 2\pi$ for all $x\in [0,2\pi )$, i.e. a shift map on the unit circle with anti-clockwise shift of $1$. Denote the sequence $\{ x_n \}$ ...
JCM's user avatar
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Identifying bifurcation

[![enter image description here]] 1]1I am trying to analyze the bifurcation of a 3D continuous model. For a certain range of parameter values, the origin is always an unstable point, whereas the ...
SHR's user avatar
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6 votes
1 answer
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Symplectic diffeomorphism of the cylinder moving a point to 0

I am currently reading though part of Zehnder's Lectures on Dynamical Systems. In Chapter VII, I have found myself in the following situation: $Z(1)$ is a subset of standard symplectic space $(\...
user14334's user avatar
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1 answer
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Topological entropy of semi-conjugated dynamical systems

Let $(X,T)$ and $(Y,G)$ be topological dynamical systems. If $(Y,G)$ is a factor of $(X,T)$ it is well known and easy to proof that $h(G)\le h(T)$ , where $h$ denotes the topological entropy. If the ...
Jörg Neunhäuserer's user avatar
3 votes
0 answers
42 views

Perturbation method for time-periodic singular system of ODEs

I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form: $$ \begin{cases} \dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
squille's user avatar
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Under which condition, such that all second-order critical points satisfy $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i\in[n]$?

Consider the following non-convex function $$E(\theta):=-\sum_{i,j}A_{ij}\cos(\theta_i-\theta_j)$$ where $A$ is a symmetric, diagonal-free matrix whose non-diagonal element are $\pm 1$. In other words,...
happyle's user avatar
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How to control the angles of Kuramoto model by controlling its order parameter?

Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $A$ such that all second-order critical points of $E(\theta)$ are in two opposite quadrants, by saying ...
happyle's user avatar
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6 votes
2 answers
561 views

If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?

Let $\{a_k\}_{k\in \mathbb{Z}} \subset \mathbb{R}$ a real sequence and $a\in \mathbb{R}$ such that $$ \lim_{n\to +\infty} \frac{1}{n} \sum_{k=1}^n a_k = a = \lim_{n\to +\infty} \frac{1}{n+1} \sum_{k=0}...
Pac's's user avatar
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2 votes
2 answers
198 views

Devaney chaos and topological entropy

I am searching for dynamical systems on compact spaces which are Devaney chaotic but have topological entropy zero. On the interval such systems do not exist. I think on the Cantor space and on the ...
Jörg Neunhäuserer's user avatar
2 votes
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50 views

Regularization for Newtonian n-body collisions in $\mathbb{R}^3$

In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
user12994's user avatar
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Accessible points of a simply connected domain

We know that if $U$ is an open subset of $\mathbb{\widehat C}$ (extended complex plane), a point $v\in\partial U$ is called accessible from $U$ if there exists a curve $\gamma:[0,1)\to U$ such that $\...
Factorial_zero's user avatar
4 votes
1 answer
348 views

Criteria for extending vector field on sphere to ball

Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file. Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, ...
Zhang Yuhan's user avatar
2 votes
1 answer
134 views

Entire function of finite order with deficient value

There are some sufficient conditions for an entire function to have a finite deficient value e.g., if the order $\rho$ of an entire function $f$ is such that $2<\rho<+\infty$ with all but ...
Factorial_zero's user avatar
1 vote
0 answers
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Multiply connected Fatou component of an entire function

This question may be trivial but still I want to know the answer. Question: Is there any necessary condition (except boundedness of the Fatou component) for the existence of a multiply connected Fatou ...
Factorial_zero's user avatar
3 votes
1 answer
178 views

Is the geodesic flow on a Riemannian manifold conservative?

Let's consider a complete Riemannian manifold $\mathcal{M}$. The geodesic flow of $\mathcal{M}$ is a first-order flow on the tangent bundle $T\mathcal{M}$. My question: Is it conservative? By ...
Gaussioa's user avatar
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1 answer
72 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 ...
NicAG's user avatar
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6 votes
1 answer
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References on semigroup actions

I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994). I would like to ask for references on semigroup actions on ...
Marco Farotti's user avatar
1 vote
1 answer
84 views

Is equal natural density on intervals with matching areas but opposite signs sufficient to use fixed-width part sizes for a simple Riemann sum?

Suppose we have a sequence $\theta_n$ which is dense on $\left(0,2\pi\right)$. Furthermore, if $A=(x,y)\subset(0,\pi)$ and $B=(x+\pi,y+\pi)$ for some $x,y$, and if we define the natural density of a ...
Sky Waterpeace's user avatar
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190 views

Variant of Collatz where one is allowed a series of choices

This question is prompted by thinking about this question and the answer I gave there. Consider the family of functions of the form \begin{equation} f_a(n):=\begin{cases} n/2 & \text{if $n$ ...
JoshuaZ's user avatar
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Any solution of an evolution problem tends to a steady state in $L^2$?

I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
Bogdan's user avatar
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6 votes
2 answers
1k views

5n+1 sequence starting at 7

Consider the following variant of the Collatz function: $f:\mathbb N\rightarrow\mathbb N$ is defined by \begin{equation} f(n):=\begin{cases} n/2 & \text{if $n$ is even}\\ 5n+1 & \...
Riemann's user avatar
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4 votes
0 answers
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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 $X=\mathbb{R}^d$)...
Juno Kim's user avatar
2 votes
0 answers
56 views

Where can I find resources for a paper "Stability analysis of a novel DDE of HIV CD4+ T-cells"?

I am currently working on a the paper [NND]: Question: On page 4, equation 6 introduces a concept related to the infection rate within the context of the HIV model. Unfortunately, the paper does not ...
Furdzik Zbignew's user avatar
1 vote
1 answer
80 views

Norm bound in simultaneous stability to semidefinite program

In the context of robust control, I remember hearing that the two following problems are equivalent. Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\...
wsz_fantasy's user avatar
12 votes
6 answers
1k views

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map $$\begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned},$$ starting ...
Saúl Pilatowsky-Cameo's user avatar
1 vote
0 answers
81 views

Approximating evalutation maps at open sets over invariant measures

Let $G$ be a group acting by homeomorphisms on a compact metrizable space, say $X$; let's denote by $\alpha:G\to\mathrm{Homeo}(X)$ the action, $g\mapsto\alpha_g$, and consider the weak-$^*$ compact ...
J G's user avatar
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0 votes
1 answer
171 views

Hitting times in ergodic dynamical systems

For an ergodic Dynamical System, we know that the return has average equal to 1 over the measure of the set in question. What can we say about the hitting time? Also has finite average? Is it ...
Autovetor's user avatar
2 votes
0 answers
110 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 ...
NicAG's user avatar
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0 votes
1 answer
249 views

Do invariant open sets generate the $\sigma$-algebra of invariant sets?

Let $X$ be a Polish space with Borel $\sigma$-algebra $B(X)$. Let $G$ be a locally compact group. $T:G\times X\to X$ be a continuous action of $G$ on $X$. The $\sigma$-algebra of invariant sets is ...
Cal's user avatar
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-5 votes
1 answer
559 views

Central limit theorem for irrational rotations

Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is $$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$? Birkhoff's ergodic ...
Nikita Sidorov's user avatar
1 vote
0 answers
82 views

Periodic orbits in planar smooth billiard table with large periods

Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period. Formulation of my question: We are considering ...
XYC's user avatar
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2 votes
1 answer
233 views

Does Bernoulli imply exponential mixing?

This question comes from this paper where the authors proved that exponential mixing implies Bernoulli. They also mentioned in the introduction that Bernoulli is the strongest ergodic property and ...
Kousaka_Reina's user avatar
1 vote
1 answer
69 views

Number of ergodic transverse measures for geodesic laminations - bounded by the genus?

Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...
Alejo García Sassi's user avatar
1 vote
0 answers
94 views

Question about ergodic flows and periodicity

Let $X$ be a compact Haussdorf space, let $\mu$ be a Borel measure on $X$ with $\mathrm{supp}(\mu)=X$ and let $(\phi_s)_{s\in\mathbb R}$ be a one-parameter group of homeomorphisms which is continuous ...
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