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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4 votes
2 answers
163 views

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 $$\...
1 vote
0 answers
1k views

Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero

Using the following definition of Dirichlet L-function $$ L(1,\chi)=\begin{cases} \dfrac{2\pi h}{w\sqrt{m}} & \textit{if}\ \chi(-1)=-1 \\\\ \dfrac{2 h \log{|\epsilon|}}{w\sqrt{m}} & \textit{...
2 votes
0 answers
384 views

Semigroup transformation (symmetry?) and hamiltionan dynamic. Noether Theorem generalization?

In reasoning about symmetries of dynamical systems usually there is an Legrangian $ L(p,q) $ and symmetry transformation $s' = f(s)$ where $s = p$ or $q$. If $f(s)$ represent continuous symmetry of ...
11 votes
1 answer
2k views

Entropy of first return map and suspension flows

There are some well know formulas of Abramov about derived systems. Firstly let $(X,\mu,f)$ be a probability preserving system and $A\subset X$ is measurable such that $\bigcup_{n\ge0}f^nA=X$. Let $\...
-1 votes
0 answers
51 views

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, ...
1 vote
1 answer
133 views

Multidimensional intersection property

Consider the multidimensional annulus $\{(p,\theta)\} = \mathbb R^n\times\mathbb T^n$ endowed by the $1$-form $\omega=p\,d\theta$. A diffeomorphism $A$ of this annulus onlo itself is said to be exact ...
1 vote
1 answer
78 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 $$\...
7 votes
2 answers
369 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to L^{...
0 votes
0 answers
14 views

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 ...
0 votes
1 answer
95 views

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 $\...
1 vote
0 answers
93 views

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 ...
2 votes
2 answers
203 views

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 ...
1 vote
1 answer
181 views

Repelling invariant manifold of a discrete dynamical system

Given a $C^\infty$ map $Q: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following properties $Q$ fixes the $x_1$-axis, i.e. $Q(x_1,0,\dotsc,0) = (x_1,0,\dotsc,0)$. For $x_1$ in a neighborhood of $...
2 votes
0 answers
56 views

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 ...
2 votes
1 answer
59 views

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 ...
2 votes
0 answers
24 views

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 $C^{1, \alpha}$. ...
0 votes
0 answers
39 views

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 $\{ x_n \}$ ...
-1 votes
0 answers
39 views

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 ...
10 votes
1 answer
306 views

Fixpoints of $m\longmapsto \mathrm{rad}(\phi(m^2))$ under iteration

Given a strictly positive integer $m$ let $\alpha(m)=\mathrm{rad}(m\phi(m))$ be the radical (product of all distinct prime divisors) of the product of $m$ and of Euler's totient function $\phi(m)=m\...
9 votes
1 answer
2k 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 ...
5 votes
5 answers
2k views

Fractal questions: Weierstraß-Mandelbrot

Coming from a specific field in algebraic geometry, I am a total noob in Fractal Theory and I'd like to learn it a bit. I hope I am tolerated for my maybe-trivial questions. I just read about the ...
5 votes
1 answer
376 views

Positiveness of the largest Lyapunov exponent

Let $\alpha\in \mathbb{R} / \mathbb{Q}$, let $A(x)$ be the $2$-by-$2$ matrix $$ A(x)=\begin{pmatrix} \dfrac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\dfrac{1}{{\lambda}} \\ \dfrac{...
6 votes
1 answer
136 views

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 $(\...
6 votes
1 answer
93 views

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 ...
1 vote
0 answers
57 views

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,...
3 votes
0 answers
41 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(...
6 votes
1 answer
407 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 the LHS ...
0 votes
0 answers
22 views

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 ...
2 votes
2 answers
196 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 ...
6 votes
2 answers
559 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}...
2 votes
0 answers
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, ...
2 votes
1 answer
292 views

(Exponential) Mixing property for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$, ...
4 votes
1 answer
337 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, ...
14 votes
1 answer
745 views

On the iterated automorphism groups of the cyclic groups

Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
3 votes
1 answer
292 views

Equidistribution of the orbit $\{\text{diag}(t^a,t^{-a})\Lambda \}_{t>0}$ for a.e. $\Lambda\in \text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$

$\DeclareMathOperator\diag{diag}\DeclareMathOperator\SL{SL}$It is well-known that geodesic flow $g_t=\{\diag(e^t,e^{-t}) \}_{t>0}$ acts ergodically (actually mixing) on $\SL(2,\mathbb R)$ (Howe–...
14 votes
1 answer
1k 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) } \...
2 votes
1 answer
133 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 ...
1 vote
0 answers
57 views

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 ...
20 votes
2 answers
1k views

Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

I saw this problem some years ago and I would greatly appreciate any reference or solution. Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
3 votes
2 answers
384 views

Functional equations based on composition

I have asked this question here (*), but there are no answer. Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
14 votes
7 answers
3k views

Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure. I wanted to have a ...
3 votes
1 answer
171 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 ...
0 votes
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 ...
6 votes
1 answer
142 views

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 ...
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 ...
5 votes
2 answers
1k views

Textbooks or lecture notes about mean field games

I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal ...
0 votes
0 answers
188 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 $n$ ...
16 votes
6 answers
1k views

A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?

Let $f(x)$ be sufficiently regular (e.g. a smooth function or a formal power series in characteristic 0 etc.). In my research the following recursion made a surprising entrance $$ f_1(x) = f(x),\ f_{n+...
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 & \...
2 votes
0 answers
69 views

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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