Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
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Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
- 708
3
votes
1
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293
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Filling cups and buckets continuously
There are $n$ cups labeled $1,\dots,n$, each with a water tap that adds water into it at the same rate. There are also $k$ buckets, and $k$ sets $S_1,\dots,S_k\subseteq\{1,\dots,n\}$. At any point, if ...
- 1,209
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Change of Reeb orbits after scaling the contact form
Let $M$ be a contact manifold with a contact form $\theta$ with Reeb vector field $X$ and $f$ be a positive function on $M$. If $\mathcal{L}_X f\neq 0$ the Reeb vector field $X'$ of $\theta'=f \theta$ ...
- 271
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34
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$L^p$-continuity for discrete linear causal systems
Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by:
\begin{...
- 41
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Equivalence between smoothly regular and analytically regular
I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...
- 803
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0
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59
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Reference of the fact that Hoelder cocycles are associated to Hoelder potentials in Ledrappier's correspondence
Let $\tilde{M}$ be the universal cover of a compact pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
...
- 481
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54
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Statistical characteristics of low complexity subshifts
I am looking for calculations of statistical characteristics (variance, entropy, etc.) of the $n$-dimensional distributions of the invariant measures of low complexity subshifts (e.g., the Sturmian or ...
- 17k
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1
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The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system
I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum.
For a compact ...
- 41.9k
6
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2
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Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?
Consider a continuous ODE,
$$\dot x = f(x), f \in C^1$$
$\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
- 205
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Behavior of $n^\alpha \sin^{\circ\, n}(n^{-\alpha}x)$
I'll write it formally: Let $\sin^{\circ\, 1}(x) = \sin(x)$ and $\sin^{\circ n+1}(x) = \sin\bigl(\sin^{\circ n}(x)\bigr)$ for $n\in \Bbb N$ with $n>1$.
What is the limit as $n \to \infty$?
It's ...
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955
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Open problems in symbolic dynamics
I would like to know which are some noticeable open problems in symbolic dynamics, including substitution dynamics. I'm especially interested in connections with topological chaos of various forms. ...
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3
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Neural networks over gadgets other than $\mathbb{R}$
Recently, I learned that neural networks (NN) can be defined over fields other than $\mathbb{R}$: for example, Khrennikov and Tirozzi wrote a paper in 1999 (!) on $p$-adic neural networks, or neural ...
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What is a holomorphic foliation?
For a smooth foliation $F$, there are three equivalent definitions:
the leaves of $F$ are tangent to a smooth vector field;
the foliation chart $\phi:U\to \mathbb R^k\times \mathbb R^{n-k}$ is ...
- 307
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Oscillator in Langton's ant
First of all, see Langton's ant Wikipedia page.
If we place a pair of ants looking north (using Golly or any another prog) on the coordinates $(x_1,y_1)$ and $(x_2,y_2)$ under the conditions:
$p=|x_1-...
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What are desirable properties that data should satisfy to reasonably use the dynamic mode decomposition?
In the dynamic mode decomposition, we consider a sequence of data vectors $\{z_0, \dots, z_m\}$ where $z_k \in \mathbb{R}^n$ for all $n$. We assume that the data satisfies the linear relationship $z_{...
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Existence of a continuous ergodic dynamical system for a given distribution?
It seems to me that given a distribution (which is well-behaved), there should be at least an ergodic dynamical system that its time average would create this distribution. Is this question already ...
- 163
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1
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The mean ergodic theorem for weakly mixing extension
I asked this question in https://math.stackexchange.com/q/4236870/528430, but did not get any help.
I got stuck with the following while going through the proof of Lemma 3.21 from the book 'Ergodic ...
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Has the von Neumann entropy ever been used in classical mechanics?
After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience [2], it occurred to me that this entropy might have ...
- 3,871
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Almost every $m\times n$ real matrix is Dirichlet approximable
Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces.
Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities
$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^...
- 1,565
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1
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Intuition for almost periodic solution and Poincaré recurrence theorem
I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer.
Suppose that we have a PDE that admit a solution $u$ that can be ...
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153
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Ergodic action on product spaces
Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
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A property of rapid sequences of natural numbers
$\newcommand{\IR}{\mathbb R}$
$\newcommand{\IT}{\mathbb T}$
$\newcommand{\w}{\omega}$
$\newcommand{\e}{\varepsilon}$
Taras Banakh and me proceed a long quest answering a question of ougao at ...
- 5,409
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126
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Dynamical phenomena in $\mathbb{R}^n$ first arising for n > 3?
For differentiable dynamical systems defined on, say, an open ball in $\mathbb{R}^n$, when $n=2$ Poincaré-Bendixson tells us a lot about what can happen. In particular, P-B precludes chaos and strange ...
- 3,230
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0
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35
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Under reasonable assumptions, is a closed invariant graph with only negative Lyapunov exponents necessarily stable?
Let $\Omega$ and $M$ be compact $C^\infty$ manifolds, let $\theta \colon \Omega \to \Omega$ be a $C^\infty$ diffeomorphism, and let $\Theta \colon \Omega \times M \to \Omega \times M$ be a $C^\infty$ ...
- 708
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0
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190
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Uniform distribution modulo 1 and probability [closed]
Define counting function $A(E; N; \omega)$ as the number of terms $x_n, 1\leq n\leq N$, for which $\{x_n\}\in E$.
Then the sequence $\omega=(x_n), n=1,2,...,$ of real numbers is
said to be uniformly ...
- 111
1
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Invariant distributions for iterated random variables (stochastic dynamical systems)
This is related to discrete dynamical systems, with the initial condition $X_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X_{n+1} = g(X_n)$ for ...
- 3,098
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The busy Star Guardian
On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of $1$ in their ...
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About Lie group $G$ has this escape property？
Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.
...
- 71
3
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0
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213
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Dynamical degree and spectral radius
Let $X$ be a smooth, projective surface over an algebraically closed field $k$ of characteristic zero, and let $f \in \mathrm{Bir}(X)$ a birational map.
Let's denote $f_{\ast} : \mathrm{NS}(X) \...
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92
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Homoclinically related hyperbolic periodic points gives the same pesin homoclinic class up to null sets
In MINIMALITY AND STABLE BERNOULLINESS IN DIMENSION 3 by Nunez and Hertz, the first paragraph in the proof of Corollary 2.4 says the above statement follows by using a "$\lambda$-lemma". ...
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Selecting a suitable Lyapunov function for the following systems?
i) SI MODEL
Consider
\begin{align}
\frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex]
\frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I
\end{align}
Where $N=S+I$ is the total population.
If ...
- 185
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1
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613
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Searching for the proof of a certain claim in Arnold's ODE book from 1992
I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition).
On pages 12-13 he writes the following:
Consider the following ordinary ...
- 1,594
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0
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88
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Sequences generated from commuted quaternions and general commuted linear transformations
Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...
- 1,547
7
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3
answers
521
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Count of non-trivial ergodic measures of a topological dynamical system
Given a compact Hausdorff space $X$ and a continuous mapping $\varphi: X \to X$. We denote by $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. A probability measure $\mu$ on the Borel-$\...
- 467
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0
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61
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Holomorphic dynamical systems defined on a contractible bounded open subset of $\Bbb{C}^n$
Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$:
Attracting Case: There is an ...
- 3,599
3
votes
1
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361
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Equivalent definitions of strongly proximal action
Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...
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1
answer
243
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Does an “almost weakly mixing” transformation admit a non-null ergodic component?
Problem set up:
Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space.
We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost weakly mixing if for ...
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1
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Irrational rotations are rank 2 by intervals without spacers
Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$.
S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
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Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds
Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
When $M$...
- 481
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0
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53
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Solve $(A(x).\nabla)u+cu=0$
ِDoes the equation
$y\partial_x u(x,y)-x\partial_y u(x,y)+cu=0$
have complex-valued compact-supported or vanishing-at-infinity $C^1$ solution defined on the whole plane without any singularity? Here $...
- 107
8
votes
1
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647
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How to analytically prove chaos
Consider the following map
\begin{align*}
T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\
(x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\left(\theta+\...
18
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2
answers
2k
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Renormalization in physics vs. dynamical systems
I am studying complex dynamics, so to me renormalization of a dynamical system means something like a rescaled first-return map on (a subset of) the underlying space. I understand that in quantum ...
- 181
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Dynamics of a curious bijection of $\mathbb N$
The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows:
Given an ...
- 17.9k
5
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Model theory and dynamical system (open problems)
I am curious about the open problems which are between model theory and dynamical system. I mean the open problems that are interesting for both groups and there are some evidences showing there might ...
- 51
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0
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43
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Self-maps (dynamical systems) in several variables induced by functions $X^{n+1}\to X$
Self-maps $F:X\to X$ can be viewed as dynamic systems.
A function $f:X^{n+1}\to X$ induces a self-map $F:X^{n+1}\to X^{n+1}$,
$$ F(x_0\dots x_n):=
(x_1, \ldots, x_n, f(x)) $$
for every $x:=(x_0, \...
- 4,786
6
votes
2
answers
2k
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Textbooks or lecture notes about mean field games
I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal ...
- 61
2
votes
0
answers
143
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inverse of moment-generating function in terms of moments
Let $\{h_i\}$ be decreasing sequence of $n$ positive reals. Define distribution $p(X=h_i)\propto h_i$ and let $g(s)=E_X[e^{sX}]$ be the moment generating function. For instance, for $h=\{1,\frac{1}{4},...
- 2,252
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0
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221
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Measure concentrated on the $\omega$-limit set
Let $(X,F)$ be a dynamical system with $X$ a compact metric space and $F: X\to X$ continuous. By $\omega$-limit set of a subset $A\subset X$ I mean:
$$\omega(A):= \bigcap_{n=0}^\infty \left(\overline{\...
- 4,459
2
votes
1
answer
102
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On the existence of regular orbit cylinders
Let $(M,\omega,H)$ be a Hamiltonian system and assume that $\gamma$ is a periodic orbit on a regular energy hypersurface. Then the regular orbit cylinder theorem (see for example Abraham/Marsden: ...
- 548
3
votes
1
answer
171
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Does full shift have the local product structure?
We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is ...
- 1,043