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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Seeking a Lyapunov function for a SIR model with immunity loss
We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\...
- 2,239
4
votes
2
answers
109
views
Minimal subshift with some $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?
There exists a minimal subshift $X$ with a point $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?
6
votes
0
answers
267
views
Elliptic foliations of the plane
A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties:
The differential operator ...
- 356
2
votes
0
answers
480
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A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
- 356
2
votes
1
answer
231
views
A reference to the fact that a topologically transitive action of a group on a compact metrizable space has a dense orbit
I need a proper reference to the following obvious fact:
An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $...
- 41.9k
1
vote
0
answers
106
views
Existence of large first return times
Let $(X,T,\mu)$ be a measure preserving system, with $\mu$ a probability measure. Let $E \subset X$ of positive measure and $\tau_E$ be the first return time to $E$. Then the Kac Lemma asserts that $$\...
- 2,283
1
vote
1
answer
136
views
A permutation group inducing a topologically transitive action without dense orbits on $\omega^*$
Let $G$ be a subgroup of the permutation group $S_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech ...
- 41.9k
5
votes
1
answer
524
views
A topologically transitive dynamical system without dense orbits
By a dynamical system I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$.
We say that a dynamical system $(K,G)$
$\bullet$ is ...
- 41.9k
3
votes
1
answer
331
views
Applying the Abramov-Rokhlin skew product entropy formula to a bounded-to-one factor
Let $(X, \mathcal{B}, \mu, S)$ and $(Y, \mathcal{C}, \nu, T)$ be invertible probability-measure-preserving systems, with a measurable factor map $\pi: X \to Y$, i.e. $\pi \circ S = T \circ \pi$. ...
- 695
6
votes
1
answer
222
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Sliding block code on irreducible sofic shift
I was looking at the following exercise from Lind/Marcus book An Introduction to Symbolic Dynamics and Coding that I cannot solve. Can someone give me a hint?
Find an example of a pair of irreducible ...
- 61
3
votes
3
answers
257
views
Computing the maximum modulus
For each $a\in \mathbb C$ define $f_a:\mathbb C\to \mathbb C$ by $f_a(z)=\exp(z)+a$. I am primarily interested in real values $a\in (-\infty,-1)$.
For each $r\in [0,\infty)$ define $M_a(r)=\max\{|f_a(...
- 3,317
2
votes
0
answers
203
views
Gurevich's entropy and topological entropy in a countable Markov shift
Good afternoon, I understand that Gurevich's entropy and topological entropy coincide when the countable Markov shift is topologically mixing (right?)
Does anyone know of an example or a reference ...
- 193
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0
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299
views
Possible Birkhoff spectra for irrational rotations
Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
- 1,658
6
votes
2
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237
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Movement of repelled particles in a ball
EDIT:
Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that ...
- 21.8k
5
votes
2
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311
views
Ergodicity of induced system
Suppose $(X,\mathcal{F},\mu,T)$ is an ergodic measure preserving dynamical system.
Let $Y\subset X$ be such that $\mu(Y)>0$ and suppose there is an integrable function $R:Y\to \mathbb{N}$ such that ...
- 53
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votes
3
answers
363
views
Fully invariant measures for rational functions
Let $f(z)$ be a rational function of degree $d \geq 2$, with complex coefficients. I am interested in fully invariant measures for the dynamical system $(\mathbb C_\infty,f)$, where $\mathbb C_\infty$ ...
- 26k
1
vote
1
answer
263
views
Ergodic automorphism is mixing of all orders
I'm trying to solve an exercise in "Ergodic Theory with a view towards Number Theory".
The exercise is suppose $T$ is an ergodic automorphism on an compact abelian group, show that it is mixing of ...
- 515
1
vote
1
answer
262
views
Smooth dynamics with zero Lyapunov exponents
Apologies if this is a vague question.
It seems that a lot of the literature in smooth dynamics is focused on understanding systems that exhibit hyperbolic/non-uniformly hyperbolic behavior. In other ...
- 1,601
7
votes
1
answer
632
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On complex dynamics in high dimensions
I am a fresh Ph.D student and I'm interested in complex dynamics in high dimensions. I have the following questions.
What research directions are there in several complex dynamics and what problems ...
user153571
1
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0
answers
45
views
Nonlinear fixed-point equation with linear solutions?
Let $S$ be an $N\times N$ row-stochastic matrix and let $w'$ be the left Perron eigenvector of $S$ (i.e., $w$ is the stationary distribution of the Markov chain represented by $S$). Let $T$ be the ...
- 31
4
votes
1
answer
568
views
Topological weak mixing vs measure-theoretic weak mixing
Let $X$ be a compact metric space and $T$ a continuous map from $X$ to $X$. The system $(X,T)$ is called topologically weakly mixing if the product system $(X\times X,T\times T)$ is topologically ...
- 395
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votes
1
answer
343
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Entropy-minimal subshifts
Consider a subshift $X \subset \left\{0, \ldots, M \right\}^{\mathbb{N}}$. $X$ is said to be entropy-minimal if every subshift $Y \subsetneq X$ satisfies that $$h_{\mathrm{top}}(Y) < h_{\mathrm{top}...
0
votes
1
answer
155
views
Conditions to determine sign of real roots
From a delay system, I obtain the following as part of a characteristic equation:
$$f(\lambda) = \lambda - a + be^{-c\lambda},$$
where $a, b,$ and $c$ are positive number and $a<b, ac<1$. My ...
- 513
2
votes
1
answer
258
views
Absolutely continuous invariant measures for a minimal flow
Fix $(X, g)$ to be some compact Riemannian manifold and $V \in \Gamma(TX)$ a smooth, non-vanishing vector field. Suppose the flow is minimal, i.e. every orbit is dense in $X$, and volume-preserving.
...
- 1,601
2
votes
1
answer
132
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Examples of minimal topological systems which are not intrinsically ergodic
Consider dynamical systems $(X,T)$ where $X$ is a compact metric space, $T:X\rightarrow X$ is continuous, the system is minimal and finally, $0<h_{\rm{top}}(X)<\infty$. I am looking for examples ...
- 3,599
26
votes
3
answers
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Unexpected behavior involving √2 and parity
This post makes a focus on a very specific part of that long post. Consider the following map:
$$f: n \mapsto \left\{
\begin{array}{ll}
\left \lfloor{n/\sqrt{2}} \right \rfloor & \...
0
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1
answer
206
views
Can the immediate basin of attraction of super-attracting fixed point at 0 of a polynomial contain non-zero roots?
Let $f$ be a polynomial with a super attracting fixed point at $x=0$. Can the immediate basin of attraction of the fixed point contain other roots? If so, please provide a specific example with the ...
4
votes
1
answer
114
views
This almost periodic condition implies equicontinuity?
Let $X$ be a metrizable compact space and $T\colon X\to X$ a minimal homeomorphism, i.e.
$$ \mathrm{orb}(x) := \{T^kx:k\in\mathbb{Z}\}$$
is dense in $X$ for every $x \in X$. Assume that the following ...
11
votes
0
answers
810
views
Borderline Collatz-like problems
The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$.
We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...
7
votes
2
answers
901
views
Is this a new strange attractor?
I recently made some experiments in programming strange attractors, and I found this (very simple) equations, which create a nice strange attractor:
...
- 101
6
votes
1
answer
611
views
The "Chaos Game" as a particular series of i.i.d. random variables
Fix a parameter $\alpha\in(0,1)$ and take an i.i.d. sequence $X_0,X_1,\ldots$ of $\mathbb{R}^n$ valued random variables. Construct the limiting random variable
$X_\infty = (1-\alpha)\sum_{k=0}^\infty ...
- 646
1
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0
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76
views
Decomposition into distal and proximal
For a topological group $G$ and a bounded real- or complex-valued function $f$ on $G$, the orbit closure of $f$ is the pointwise closure in the space of all bounded functions on $G$ of the orbit of $f$...
- 1,074
1
vote
1
answer
86
views
Entropy spectrum is not concave
Let $T:[0, 1]\rightarrow [0, 1]$ be map such that $T(x)=4x(1-x)$. For any $\alpha \in \mathbb{R}$, we define the level set as follows
$$F(\alpha)=\{x\in [0,1]: \lim_{n\rightarrow \infty}\frac{1}{n}\...
- 1,043
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votes
0
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87
views
What kind of differential equation problem is this?
I have a function $f(x,t;k)$, a starting point $x_0$, a gradient $\operatorname{Grad}(f)$, and an equilibrium point $x^*$. I can adjust the parameter $k$ freely, and I know that for any $k$ the ...
16
votes
1
answer
502
views
Group actions and "transfinite dynamics"
$\DeclareMathOperator\Sym{Sym}$I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ ...
- 4,265
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1
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165
views
Is the set of non-escaping points in a Julia set always totally disconnected?
I am looking for examples of transcendental entire functions $f:\mathbb C\to \mathbb C$ such that the set of non-escaping points in the Julia set of $f$ is not totally disconnected. I denote this set $...
- 3,317
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votes
0
answers
44
views
Systems with trivial cohomology
If $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ is an irrational number, then the rotation $X = (S^1, +\alpha)$ has "trivial" cohomology i.e.
$$H^1(X) := C(X,\mathbb{C})/\beta C(X,\mathbb{C})$$
...
3
votes
1
answer
160
views
Example of minimal connected weakly mixing dynamical system
I am looking for an example of a dynamical system $(X,T)$ such that:
$X$ is a connected topological space,
$(X,T)$ is minimal and weakly mixing.
Does there exists one?
2
votes
1
answer
129
views
Extension of semi-flows on a locally compact metric space
Theorem 2.3, p. 26 from W Shen, Y Yi "Almost automorphic and almost periodic dynamics in skew-product semiflows" states that if there is a semi-flow $\varphi^{t}$, $t \geq 0$, on a locally compact ...
- 798
22
votes
3
answers
1k
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Cyclic action on Kreweras walks
A Kreweras walk of length $3n$ is a word consisting of $n$ $A$'s, $n$ $B$'s, and $n$ $C$'s such that in any prefix there are at least as many $A$'s as $B$'s, and at least as many $A$'s as $C$'s. For ...
- 24.2k
1
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0
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45
views
On the center-stable manifold theorem for sets
Suppose I have a dynamical system $f:S \to S$ where $S \subset \mathbb{R}^n$ and $S$ is compact and $f$ is twice differentiable. Assume there exists a function $V$ such that $V(f(x)) < V(x)$ unless ...
- 105
14
votes
1
answer
957
views
On the iterated automorphism groups of the cyclic groups
Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
1
vote
1
answer
88
views
Example of connected factor of symbolic system that is not a rotation
I am looking for an example of a factor $f\colon (X,T) \to (Y,T)$ between topological dynamical systems, where $(X,T)$ is a minimal subshift and $Y$ a connected topological space such that $(Y,T)$ is ...
12
votes
2
answers
750
views
Algorithm for computing external angles for the Mandelbrot set
Let $M$ be the Mandelbrot set: there exists a unique series
$$
\psi(z) := z + \sum_{m=0}^{+\infty} b_m z^{-m} = z - \frac{1}{2} + \frac{1}{8} z^{-1} - \frac{1}{4} z^{-2} + \cdots
$$
which defines a ...
- 32.5k
2
votes
0
answers
125
views
Does $\sum_{i\le k}\mathrm{frac}(n\alpha_i)<1$ hold infinitely often?
For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist ...
- 1,511
16
votes
2
answers
2k
views
Physical interpretation of the Manifold Hypothesis
Motivation:
Most dimensionality reduction algorithms assume that the input data are sampled from a manifold $\mathcal{M}$ whose intrinsic dimension $d$ is much smaller than the ambient dimension $D$. ...
- 3,871
1
vote
0
answers
72
views
Distance between value function of deterministic and stochastic control problems
Suppose that one wants to control a diffusion process
$$
dX_t^u = \mu(X_t^u,u)dt + \sigma dW_t; \qquad X_0^u=x
$$
in order to optimize a stochastic control problem with value function
$$
V_T(u)=\...
- 5,405
0
votes
0
answers
52
views
Probabilistic Approximation of non-linear Dynamical System by Diffusion Process
Setting
Suppose I have a discrete dynamical system given by:
$$
X^{n+1} = f(X^{n})
\qquad X^0 =x
,
$$
where $f$ is some diffeomorphism from $\mathbb{R}^{d}$ to itself, and some $x \in \mathbb{R}^d$. ...
- 5,405
4
votes
1
answer
342
views
Questions about a return map
Consider the following map in the interval $u\in[-1,1]$ ($U\in[-1,1]$ also)
$$U=u-\frac{64}{3}u^{3}+64u^{5}-64u^{7}+\frac{64}{3}u^{9}$$
It has 3 fixed points at $u=0,\pm 1$. If we compute the ...
- 213
2
votes
1
answer
158
views
A vector field $X$ on $\mathrm{GL}(n,\mathbb{R})$ with $\begin{cases} X.\mathrm{trace}=\mathrm{Det} \\X.\mathrm{Det}=-\mathrm{trace} \end{cases}$
Is there a vector field $X$ on $\operatorname{M}_n(\mathbb{R})$ or $\operatorname{GL}(n,\mathbb{R})$ with the following condition:
$$\begin{cases} X\cdot \operatorname{trace}=\operatorname{Det} \\X\...
- 356