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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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 ...
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Do not take one more exploration when the best outcome is bad. Decision theory and statistical learning

Imagine a two-stage dynamic problem with information update. At stage $0$, we are in the state of $d$ success out of $n$ trials. We need to decide we try or not now. If we try, we will go to two ...
David Smith'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 $\{ x_n \}$ ...
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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 ...
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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 $(\...
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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
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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,...
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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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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}...
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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
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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
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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
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1 answer
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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
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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
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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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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 ...
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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
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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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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 $n$ ...
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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{\...
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2 answers
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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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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
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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
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1 answer
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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
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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
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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 ...
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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
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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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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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1 answer
558 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
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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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1 answer
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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
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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 ...
Lau's user avatar
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1 vote
1 answer
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Does every proximal dynamical system have zero topological entropy?

A dynamical system is proximal if $$\:\forall (x,y) \in X \times X, \: \liminf_{n \rightarrow \infty} d(f^{n}(x),f^{n}(y)) = 0 $$ (where $X$ is a compact metric space with metric $d$). Is it true that ...
Matej Moravik's user avatar
2 votes
0 answers
133 views

Unipotent closure in classical groups

Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...
Mathew's user avatar
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8 votes
4 answers
743 views

Ergodic theory applied to number theory

I am interested in the links between Ergodic Theory and Number Theory. Can anyone give some references for papers to read in this field? Any open problems? Or ideas where it may be applicable in NT?
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0 answers
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Sufficient conditions for chain recurrent set equal to set of non wandering points

Given a generic diffeomorphism, I know that the set of nonwandering points is contained in the chain recurrent set, but the converse is not always true. Is there some sufficient conditions under which ...
giangian's user avatar
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3 votes
1 answer
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Rate of convergence for Markov chain in random environment

Let $(\Omega,\mathfrak{F},\mathbb{P})$ be a probability space and $\sigma:\Omega\to\Omega$ be an ergodic, invertible and measure preserving transformation. Consider a family of column stochastic ...
JayP's user avatar
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Is the global solution to this ODE bounded?

Consider $$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$ where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
tony's user avatar
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2 votes
1 answer
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When does uniqueness of a stable equilibrium imply it is globally stable?

Given a gradient dynamical system $$\dot x=-\nabla f(x),$$ my question is: (1) If there exists only one equilibrium $x^*$ which is stable (if necessary, this can be changed to stable asymptotically ...
tony's user avatar
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Closed subgroups in Ratner's orbit closure theorem on unipotent flows

Let $G$ be a semisimple (real or $p$-adic) Lie group and $\Gamma$ a discrete and cocompact subgroup of $G$, as in the setting of Ratner's theorems on unipotent flows (see for example here \url{https://...
Zhang's user avatar
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3 votes
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Stability of indefinitely damped mechanical system with diagonal stiffness

I'm trying to find conditions for the asymptotic stability of the following linear system, \begin{equation} \mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0 \end{equation} given the ...
Shivang Rawat's user avatar
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0 answers
24 views

Coefficients in the series expansion of a central manifold are all zero

I have a system of 4 ODEs, which linearized around the origin gives $$ \begin{align} &\dot{q_1}=a\, q_1\\ &\dot{q_2}=b\,q_2\\ &\dot{q_3}=0\\ &\dot{q_4}=c\,q_4 \end{align} $$ with $a$, $...
F.Mor's user avatar
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3 votes
1 answer
132 views

Topological amenability of actions - forgetting topology

Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there ...
Alcides Buss's user avatar
3 votes
1 answer
158 views

Chain components and posets

Let $(X,f)$ be a topological dynamical system ($f$ continuous, $X$ compact, metric with distance $d$). Let $C\subseteq X^2$ indicate the chain recurrence relation: $$xCy\iff \forall \epsilon>0\ \...
Alessandro Della Corte's user avatar
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1 answer
148 views

Help in understanding the singular system of linear forms and non escape of mass

I am having some trouble in understanding certain portions of the following paper by KKLM https://link.springer.com/article/10.1007/s11854-017-0033-4 So in proposition 3.1, they proved the estimate ...
User1723's user avatar
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