Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,388
questions
4
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2
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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 $$\...
1
vote
0
answers
1k
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
Using the following definition of Dirichlet L-function
$$
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
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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
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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
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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
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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 ...
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
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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 ...
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
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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 ...
2
votes
1
answer
59
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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 ...
2
votes
0
answers
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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}$. ...
0
votes
0
answers
39
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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 \}$ ...
-1
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0
answers
39
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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 ...
10
votes
1
answer
306
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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
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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}$$
The inverse ...
5
votes
5
answers
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Fractal questions: Weierstraß-Mandelbrot
Coming from a specific field in algebraic geometry, I am a total noob in Fractal Theory and I'd like to learn it a bit. I hope I am tolerated for my maybe-trivial questions. I just read about the ...
5
votes
1
answer
376
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Positiveness of the largest Lyapunov exponent
Let $\alpha\in \mathbb{R} / \mathbb{Q}$, let $A(x)$ be the $2$-by-$2$ matrix
$$
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
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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,...
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:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
6
votes
1
answer
407
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How can I catalog these generalized Collatz problems?
The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$:
$x + x + 1 \rightarrow x+x+x+1+1;$
$x + x \rightarrow x;$
Whenever a number matches the LHS ...
0
votes
0
answers
22
views
How to control the angles of Kuramoto model by controlling its order parameter?
Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $A$ such that all second-order critical points of $E(\theta)$ are in two opposite quadrants, by saying ...
2
votes
2
answers
196
views
Devaney chaos and topological entropy
I am searching for dynamical systems on compact spaces which are Devaney chaotic but have topological entropy zero. On the interval such systems do not exist. I think on the Cantor space and on the ...
6
votes
2
answers
559
views
If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?
Let $\{a_k\}_{k\in \mathbb{Z}} \subset \mathbb{R}$ a real sequence and $a\in \mathbb{R}$ such that $$ \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
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(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
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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
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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
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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 ...
20
votes
2
answers
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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
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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
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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
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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 ...
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
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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
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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 $n$ ...
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
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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 & \...
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{\...