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
6
votes
2
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Continuity of the period for a periodic dynamical system
Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 1)$ a velocity field such that every solution $(x_t)_{t\geq 0}$ of $(d/dt)x_t=v(x_t)$ is periodic. Denote, for a non-stationary point $x\in\...
- 449
4
votes
1
answer
272
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A sufficient condition for an ergodic system to be weakly mixing
Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n_k\}$ of natural numbers with positive lower density, ...
- 6,213
3
votes
0
answers
271
views
Approximating rotations on a torus with irrational rotations
Consider a rotation of the form $x\mapsto e^{i\theta}x$, for $x$ on the unit circle. By iterating this rotation, one can approximate any other rotation $x\mapsto e^{i\phi}x$ arbitrarily well, as long ...
- 181
0
votes
1
answer
262
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Non-asymptotic convergence rates for gradient descent
I'd like to know how the number of steps needed for gradient descent depend on properties of the Hessian in non-asymptotic regime.
More specifically, number of gradient descent steps needed to obtain ...
- 2,252
1
vote
1
answer
208
views
Robustness of ergodic dynamical systems
Let $\mathbf X := (X, \mathcal F, \mu)$ be a standard probability space.
For an ergodic measure preserving transformation $T$, we define the ergodic robustness $\mathcal R(T)$ of $T$ as follows:
For $...
- 6,213
0
votes
0
answers
58
views
The solutions of a system of differential equations
Let $P(x,y) = \frac{x}{y}^{\frac{x^2}{y-x}}$ for $x \neq y$ and using the proper limits $P(x,y)=e^{-x}$ for $x=y $, $P(x,y)=0$ for $x\neq0, y=0,$ and $P(x,y)=1$ for $x=0, y\neq0.$
Consider this system ...
- 31
2
votes
0
answers
116
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Birth of chaos due to nonautonomous perturbation
Let $\sigma, b>0$. I want to study the dynamics of the map
$$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that
$$T_{\sigma,b}(n,\theta,y) = (\...
2
votes
1
answer
104
views
Operators "building" linear independant sets
Let $E$ be a separable Banach space and let $T\in L(E,E)$.
Is there a condition on $T$ ensuring that:
$$
\mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow
\{T(x_n)\}_{n=1}^N\cup \...
- 93
6
votes
2
answers
437
views
Matrix-valued ordinary differential equation with symmetry
I am considering the following equation
$$\begin{pmatrix} -\frac{d}{dx} + \lambda \sin(2\pi x) & \lambda - \lambda \cos(2\pi x) \\ -\lambda-\lambda \cos(2\pi x) & -\frac{d}{dx} - \lambda \sin(...
- 192
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votes
1
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138
views
Do measure-valued dynamical systems correspond to marginals of Markov processes?
Let $(\mu_n)_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}_1(X)\rightarrow \mathcal{P}_1(...
- 5,405
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votes
1
answer
282
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Limiting distribution in $M_t/M_t/1$ queue
Consider a $M/M/1$ queue with a constant arrival rate $\lambda$ and service rate $\mu$ with $\lambda < \mu$. We know that in this case the limiting distribution exists and it is a geometric ...
- 31
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134
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Is there a non geodesible vector field $P\partial_x+Q\partial_y$ which satisfies $P_xP_y+Q_xQ_y=0$
Inspired by the following two posts
Finding a 1-form adapted to a smooth flow
Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex ...
- 356
0
votes
0
answers
72
views
Li-Yorke sensitivity Vs Li-Yorke dense chaos
Let $X$ be a compact metric space, $X*X$ its cartesian product, and $A$ a subset of $X*X$.
Are the following two properties the same, or e.g. one is stronger than the other?
$A$ is dense and residual ...
- 141
1
vote
0
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117
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Is a "global period" similar to a "local period"?
Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the ...
- 449
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votes
3
answers
690
views
Escaping from infinitely many pursuers
The fugitive is at the origin. They move at a speed of $1$. There's a guard at $(i,j)$ for all $i,j\in \mathbb{Z}$ except the origin. A guard's speed is $\frac{1}{100}$. The fugitive and the guards ...
- 2,619
1
vote
0
answers
47
views
Hypercylic operators have very typical cyclic vectors
Let $W$ be the Wiener measure on $C_0(\mathbb{R})$ and let $T\in L(C_0(\mathbb{R}),C_0(\mathbb{R}))$ be a hypercylic operator; i.e. there exists some $f \in C_0(\mathbb{R})$ such that $\{T^n(f)\}_{n=1}...
- 5,405
11
votes
0
answers
344
views
Tauberian Theorem for 1-parameter groups of operators
The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...
- 5,405
1
vote
1
answer
187
views
Is a “uniformly minimal” dynamical system ergodic?
Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly transitive in the sense ...
- 6,213
4
votes
0
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149
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Cyclic vectors for the translation operator
Let $b\in \mathbb{R}\neq 0$, and consider the translation operators:
$$
\begin{align}
T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\
f &\mapsto f(\cdot + b).
\end{align}
$$
*Are there known ...
- 5,405
2
votes
0
answers
95
views
Persistence of homoclinic points in the non-compact case
It is well known that a transverse homoclinic point of a hyperbolic periodic point of a $C^1$-diffeomorphism of a compact manifold $M$ persists under small $C^1$ perturbations. This follows easily ...
- 111
4
votes
1
answer
2k
views
Summary of “Almost All Orbits of the Collatz Map Attain Almost Bounded Values”
Terence Tao's 2019 paper ``Almost all Orbits of the Collatz map attain almost bounded values" is pretty famous. However, it's also long and complicated. I think there are useful techniques to ...
- 188
3
votes
0
answers
73
views
What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?
Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of ...
- 111
44
votes
3
answers
3k
views
Is there an elementary proof that distal maps are invertible?
Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.
Then it is true that $T$ is bijective.
Question: Is there an ...
- 6,213
1
vote
0
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80
views
Dynamics of composition of reflections
Let $C$ be a curve defined by $y = f(x)$, and define the vertical reflection over $C$ to be the map $(x,y) \mapsto (x,y')$, where $y' = 2 f(x) - y$. In other words, the vertical distance from $(x,y)$ ...
- 213
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0
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63
views
a lemma on interval translation map
Consider the map $S:T^1 \to T^1$ where $x \mapsto x+c_j$ , mod 1 where $c_j$'s are real numbers. We represent $T^1$ as a union of disjoint subsegments $M_j=[t_j,t_{j+1})$, $j=0 , \cdots ,n , t_0=t_n$....
- 145
5
votes
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144
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Examples of transitive geodesic flows that are not ergodic
What would be an easy example of a transitive geodesic flow (defined as: there is a geodesic whose velocity vectors are dense on the unit tangent bundle) that is not ergodic?
Motivation. A well-known ...
- 13.5k
6
votes
1
answer
492
views
Uses for Sharkovskii's theorem
Sharkovskii's theorem is a fascinating result and ready to stand on its own. But is it also used somewhere? Are there other theorems that rely on it or somehow use the Sharkovskii ordering?
- 887
3
votes
1
answer
238
views
Does an “almost 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 mixing if for every $...
- 6,213
4
votes
1
answer
259
views
Reaction-diffusion systems treated as dynamical systems
I wonder if there is a good reference on reaction-diffusion systems on $\mathbb{R}^N$, that treats them as dynamical systems.
I have the books of Alain Haraux – Systèmes dynamiques dissipatifs et ...
- 1,759
2
votes
2
answers
209
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Does ergodic theorem apply to trajectories outside of attractor?
Ergodic theorem says that $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{t=1}^nf(T^tx) = \displaystyle\int f\,\mathrm{d}\mu$ for $\mu$-almost every $x$. In many cases, the support of $\mu$ ...
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7
votes
1
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274
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Uniqueness of stationary measures for $(G,\mu)$ boundaries
Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup.
Let $\nu$ is a $\...
- 481
2
votes
1
answer
382
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$, ...
- 21
4
votes
1
answer
507
views
Are these topological sequence entropy definition equivalent?
I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s $(X,T)$ that it is defined ...
- 139
6
votes
2
answers
530
views
What are the current research directions in the geometric theory of dynamical systems?
By geometric theory of dynamical systems, I mean the kind found in the book by Palis, or papers like this one. In other words, dynamics on manifolds, but not specifically hyperbolic dynamical systems.
...
- 6,213
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votes
2
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431
views
Necessary and sufficient curvature condition for a regular planar curve to be simple and closed
Given a smooth, $2\pi$-periodic function $\kappa(s)$, the associated planar curve $\gamma(s)$ for which $\kappa(s)$ is the (signed) curvature, is uniquely determined up to Euclidean invariance: a ...
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203
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Terminology question in group actions
Given a continuous group action $G \times X \rightarrow X$ on a topological space $X$, is there a standard term for the subsets $K \subset X$ for which
Every open neighborhood of $K$ intersects every ...
- 13.5k
0
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0
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64
views
Implications for a simple deterministic chaos definition
Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties:
sensitive dependence to initial ...
- 141
5
votes
1
answer
425
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{...
- 131
5
votes
1
answer
177
views
Can you use Oseledet's theorem to numerically approximate the Lyapunov spectra?
Let's say you have a set of first order differential equations with known Jacobian $J$. Let $x_0, x_1, ..., x_n$ be sampled points on the trajectory near the attractor.
Let $T_n = J(x_{n-1})J(x_{n-2})....
- 188
1
vote
0
answers
210
views
Is there a condition for a subshift of finite type to be uniquely ergodic?
Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic?
- 111
5
votes
0
answers
155
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Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?
It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...
- 481
5
votes
2
answers
433
views
On which closed Riemannian manifolds are geodesics always recurrent?
Let $M$ be a closed Riemannian manifold. What are the necessary and sufficient conditions on $M$ to ensure that for every point $p \in M$, and every geodesic $\gamma: [0, \infty) \to M$ with $\gamma(0)...
- 6,213
1
vote
1
answer
982
views
Turing's fixed-point theorem
Motivation:
It recently occurred to me that Turing's analysis of the halting problem may be formulated as a fixed-point theorem. Might this intuition from theoretical computer science have informed ...
- 3,871
0
votes
1
answer
213
views
Uniformity of convergence in the pointwise ergodic theorem
Definitions and some motivation:
Let $X$ be a compact metric space, and $T$ a uniquely ergodic measure preserving transformation on $X$, with associated invariant ergodic probability measure $\mu$. ...
- 6,213
3
votes
1
answer
336
views
Can every ergodic map be approximated by ergodic maps close to the identity?
Let $\mathbf X := (X, \mathcal S, \mu)$ be a probability space without atoms. We say two measure preserving transformations $T$ and $F$ on $\mathbf X$ are $\delta$-close, for $\delta > 0$, if $ \...
- 6,213
7
votes
2
answers
318
views
Planar flow with bounded orbits and a single equilibrium point
Is there a $C^1$ flow $\varphi_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x,
$$\lim_{t\rightarrow +\infty}\varphi_t(x) = 0,$$
$$\lim_{t\rightarrow -\infty}\varphi_t(...
- 18.7k
2
votes
0
answers
65
views
Chain recurrent points of a gradient-like system
Let $X$ be a compact metric space and $f:X\to X
$ homeomorphism. Let $V:X\to \mathbb{R}$ be a Lyapunov function for $(X,f)$ (continuous function such that $(\forall x\notin Fix(f))\ \ V(f(x))<V(x))...
- 141
3
votes
0
answers
69
views
Non-closed trajectories in convex billiards
This is a weak version of this problem, written down in Lviv Scottish Book.
I start with necessary definitions.
Let $K=-K$ be a centrally symmetric compact convex body in the Euclidean space $\mathbb ...
- 41.9k
1
vote
1
answer
169
views
Gradient-like dynamical systems
I've tried asking this question on Mathematics site, but I only got an upvote and no answer. I've searched online, tried to find something about this topic, but I haven't found much (and the things I ...
- 141
18
votes
1
answer
495
views
Iterated antiderivatives of polynomials having many real roots
Question For which polynomials $p_n:\mathbb{R} \rightarrow \mathbb{R}$ having $n$ distinct real roots can we find an infinite sequence of polynomials
$$ p_n, p_{n+1}, p_{n+2} , p_{n+3}, \dots, $$
such ...
