# Questions tagged [discrete-dynamical-systems]

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90
questions

4
votes

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### Asymptotic decay rate of an oscillator integral

Question:
I want to evaluate the decay estimate of the integral
$I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $
for ...

0
votes

0
answers

44
views

### Is there a correlation between the bifurcation points of dynamical system and the integral points of the elliptic curve $E_d$?

Motivated by an interest in the interplay between dynamical systems and elliptic curves also On a question of Mordell, I derived a dynamical system corresponding to the elliptic curve:
$
E_d: Y^2 = X^...

0
votes

0
answers

53
views

### Recognizability of a substitution implies aperiodicity

Is there a good reference, aside from the book of "Tilings and Patterns" by Grunbaum and Shephard, on the fact that recognizability\unique-composition of a tiling implies aperiodicity? I ...

0
votes

0
answers

118
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### coupled discrete dynamical system -- bifurcation analysis

Suppose you have the following coupled discrete dynamical system:
\begin{align*}
e_{k+1}&=e_k - 2~\alpha~e_k~\lambda^2~\alpha_k^2 + \alpha^2~e_k^2~\lambda^3 \alpha_k^3\\
\alpha_{k+1}&= \...

0
votes

0
answers

16
views

### Nature of unbounded initials in polynomial symplectic maps

Is the following statement true? How it can be proved/rejected?
Initial conditions that correspond to unbounded orbits in polynomial symplectic mappings, which exhibit chaotic behavior (exponential ...

2
votes

0
answers

208
views

### separation time of points have 'positive upper density' under positive topological entropy?

Let $(X,d)$ be a compact metric space and $g:X\to X$ be a continuous map on $X$. A subset $K$ of $X$ is called $(n,\epsilon)$ separated if, for any two distinct points $x$, $y$ of $K$ there exists $i\...

0
votes

0
answers

69
views

### Extension of automorphism of shift of finite type

$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be two subshifts of finite type and $X\subset Y$, and $\phi:X\rightarrow X$ be a homeomorphism commuting with the shift map. Is there any homeomorphism $\...

1
vote

0
answers

34
views

### When does an optimal input sequence for a discrete-time system exist?

Suppose an LTI discrete-time system is given by the equations
$$
x_{k+1} = Ax_k + Bu_k,\\
y_{k} = Cx_k + Du_k
$$
with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\...

0
votes

0
answers

90
views

### Renormalization group in condensed physics and field theory

Renormalization group in field theory is differenct from the one in condensed physics in that the former satisfies a differential equation, but the latter does not.
Does Renormalization group in ...

0
votes

0
answers

47
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 $\{ x_n \}$ ...

3
votes

0
answers

117
views

### Finite approximability of graphs with finitely many automorphisms

In this question, all graphs are understood to be simple and undirected, and have countably many vertices and edges, but not necessarily finite.
Let $G = (V, E)$ be a graph. It is clear that any ...

12
votes

6
answers

1k
views

### Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map $$\begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned},$$
starting ...

1
vote

1
answer

122
views

### Does every proximal dynamical system have zero topological entropy?

A dynamical system is proximal if $$\:\forall (x,y) \in X \times X, \: \liminf_{n \rightarrow \infty} d(f^{n}(x),f^{n}(y)) = 0 $$ (where $X$ is a compact metric space with metric $d$). Is it true that ...

0
votes

0
answers

92
views

### How to show that the map $ R $ here is measure-preserving

Assume that $ (X,\mathcal{B},m,T) $ is a measure-preserving dynamical system, where $ (X,\mathcal{B},m) $ is a probability space, $ \mathcal{B} $ denotes all the measurable sets in $ X $, $ m $ is the ...

1
vote

0
answers

40
views

### The boundedness of dynamical systems discretized from Hamiltonian systems

Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e.,
\begin{align}
&\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\
&\frac{dq}{dt}...

4
votes

1
answer

233
views

### Equivalence of bounded gradient flow solutions and uniformly bounded gradient descent trajectories for definable functions

I am reading paper [1] by C. Josz regarding the global convergence of the gradient method. The main result is the following:
$\textbf{Theorem}$: For a definable differentiable function $f : \mathbb{R}^...

2
votes

1
answer

167
views

### A sensitive 2-dimensional cellular automaton with a blocking word

I'am a Ph.D student in the domain of discrete dynamical systems. My thesis is about spectral properties of cellular automata in higher dimension.
Kurka gives a classification for one dimensional ...

3
votes

1
answer

143
views

### Condition for 3×3 block matrix to be stable

Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times ...

0
votes

0
answers

60
views

### How to prove that pseudo entropy and topological entropy are the same with only Markov inequality and continuity?

Let $(X,\rho)$ be a compact metric space and $f:X\to X$ a homeomorphism. We say $(x_1,\ldots,x_{n})\in X^n$ is a partial $n$ orbit if $f(x_i)=x_{i+1}$. Let $Sep_{\epsilon}(X,\rho_n)$ be the maximal ...

2
votes

1
answer

186
views

### How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?

I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...

1
vote

1
answer

144
views

### Existence of center-stable manifold when the Jacobian is singular?

The following is a result from Shub's monograph "Global Stability of Dynamical Systems".
I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}_{\rm loc}$ does not ...

1
vote

1
answer

194
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 $...

1
vote

0
answers

60
views

### A family of dynamical systems similar to Conway's Game Of Life

Given a dimension $d\geq 1$ and two subsets $\mathcal B$ (the 'birth' set)
and $\mathcal D$ (the 'death' set) of $\lbrace 0,\ldots,2d\rbrace$, we consider the
dynamical system $\lbrace \mu_0,\mu_1\...

1
vote

0
answers

121
views

### Frog game on tree graphs is in NP but not in P (NP-complete)?

Problem
We can restrict ourselves to tree graphs. What is the complexity of the following problem?
Let $G$ be simple connected graph with vertices in $V$, edges in $E$, and a vertex weighted function $...

1
vote

0
answers

146
views

### Time-scale calculus (an similar approaches - measure chains) on more general "time" sets

Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...

8
votes

1
answer

341
views

### State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...

5
votes

1
answer

271
views

### Unbounded solution but bounded Euler discretization

Is there an ordinary differential equation in $\mathbb{R}^d$ induced by a gradient vector field with unbounded solutions, for which the difference equations obtained by using the forward Euler method ...

3
votes

1
answer

147
views

### Is there a classical textbook/reference on numerical discretization schemes?

I found that it is relatively easy to find a book that discusses Euler discretization or Runge-Kutta discretization, but I am not aware of one that is well-known and/or common knowledge (i.e., field-...

6
votes

2
answers

339
views

### Topological dynamical systems with only zero-entropy factors

Suppose the dynamical system $(X,T)$ has only proper factors (i.e. not $(X,T)$ itself) of zero topological entropy. Does the system $(X,T)$ also have zero entropy?

46
votes

3
answers

7k
views

### Does Conway's game of life admit a notion of energy?

(I am not sure if this is a mathematics or physics question so I am not sure where to post it. I am posting it here because the chief subject is an unreal universe that is purely a subject of ...

1
vote

0
answers

24
views

### Reduced $H_{\infty}$ problem for nD systems

Let $G(z)$ denote the (rational not necessarily square and unstable) transfer function of an nD system, where $z=(z_{2},...,z_{n})$, of a discrete spatial-temporal recurrence Givone-Roesser type ...

6
votes

0
answers

157
views

### Construction of minimal zero entropy measure-theoretically strong mixing subshift?

Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is
(1) minimal
(2) zero (topological) entropy
(3) measure-theoretically strong mixing (for some measure)?
I am in particular ...

0
votes

0
answers

82
views

### Distortion estimates to control Hausdorff measure of a curve

I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent.
I have a problem to understand how the distortion estimates are used. The ...

4
votes

2
answers

329
views

### Is there half an iteration of the QR algorithm?

Every real square matrix $M$ has a QR decomposition $M = QR$ where $Q^{-1}=Q^T$ and $R$ is an upper triangular matrix with non-negative reals on the diagonal. Call the function $f(QR)=RQ$ the Francis ...

6
votes

0
answers

478
views

### Existence of an explosive prime

The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below).
Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \...

0
votes

1
answer

144
views

### Why all the coefficients of the center manifold of this system are zeros?

I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...

1
vote

1
answer

174
views

### A completely positive equivariant map $\varphi: A \to B$ induces a map on the full crossed products

Let $G$ be a discrete group. Let $(A,\alpha)$ and $(B,\beta)$ be $G$-$C^*$-algebras and $\varphi: A \to B$ be $G$-equivariant and completely positive. All crossed products in this post are full (= ...

3
votes

0
answers

121
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### Description of Anderson-Putnam CW-complex construction

I have been trying to read the paper, Topological invariants fo substitution tiling and their associated $C^*$-algebras, to learn more about a construction of Anderson-Putnam complexes. However, it ...

1
vote

0
answers

34
views

### $L^p$-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{...

1
vote

1
answer

174
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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 ...

0
votes

1
answer

130
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(...

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}...

0
votes

0
answers

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 $M_j=[t_j,t_{j+1})$, $j=0 , \cdots ,n , t_0=t_n$....

1
vote

0
answers

195
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### 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?

2
votes

0
answers

58
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))...

1
vote

1
answer

164
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### 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 ...

2
votes

1
answer

163
views

### Busy beaver sequence for a simple tag-like system

This question arose in the context of tag-like systems, specifically Bitwise Cyclic Tag (BCT). Consider the following discrete dynamical system:
Let $\mathbb{B} = \{\mathtt{0}, \mathtt{1}\}$. Let our ...

2
votes

0
answers

128
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### Is this variant of bitwise cyclic tag Turing-complete? [closed]

Cross-posted from Theoretical Computer Science.
CT is an extremely minimalist programming language that can simulate arbitrary tag systems, and is therefore Turing-complete. A program consists simply ...

12
votes

1
answer

932
views

### Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$.
You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does ...

1
vote

0
answers

171
views

### Example of topologically transitive dynamical system with invariant non-ergodic Borel measure

Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which
$f : \Lambda \to \...