# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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### better correlations and a special flow

A Gauss map $T: [0,1]\to [0,1]: Tx=x^{-1}-[x^{-1}]$, and a suspension $S=\{(x,t) \in [0,1]\times [0, \infty): 0\le t < -\ln x\}/{\sim}$, where the equivalent relation $\sim$ is defined by $ (x, -\...

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### Analyticity of central stable manifolds

Let $X$ be a real analytic vector field defined on $\mathbb{R}^n$. Assume the origin $0 \in \mathbb{R}^n$ is a zero of $X$. Assume, furthermore, that we know that the center-stable manifold (in the ...

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### Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...

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### Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity

Let's say I have a nonlinear system of ODEs, where one of equations looks like:
$$
\frac{dX_i}{dt} = a_1X_0+\dotsb+a_j\frac{X_j^{0.5}}{X_j^{0.5}+b_j^{0.5}}+\dotsb.
$$
And equilibrium point is 0. I ...

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### Shift-ergodic stochastic processes in continuous time

Let $\mathscr{C}:=\{\gamma : \mathbb{R}_+\rightarrow\mathbb{R}^n \mid \gamma \ \text{ continuous}\}$ be the set of all $\mathbb{R}^n$-valued paths over $[0,\infty)$. Endow $\mathscr{C}$ with the $\...

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### Lower bounds for pattern complexity of linearly repetitive aperiodic subshifts

I recently asked in this thread about lower bounds on the complexity in the case where we have an aperiodic subshift. If I denote $c_n(\Omega)$ as the number of possible patterns on $Q_n=\{0,...,n−1\}^...

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2
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### A variation of domino tiling problem with fusions

I know several specific variations of the domino tiling problem has been determined to be decidable or undecidable, such as the seed domino problem. I have a variation which I have not been able to ...

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### naturality of the Godbillon-Vey class

This is a problem from Lawrence Conlon's differential manifolds a first course. I do not know how to prove in the following problem
If $f: N \rightarrow M$ is transverse to $\mathcal{F}$, prove that
$$...

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### How well does Sharkovskii's theorem hold up for isometries over p-adic integers?

Question
Let $T$ be a bijective isometry $\Bbb Z_p\to\Bbb Z_p$ then does Sharkovskii's theorem hold over the periodic points of $T$? i.e. for all points of period $p$ do there exist
points of period $...

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### 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 neighhorhood of $...

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### Strongly constant divergence vector fields

Inspired by this question on homothety vector field we ask the following question
Let $M$ be a manifold equiped by a volum form $\Omega$. A strongly constant divergence vector field is a vector ...

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### Bibliography about vector fields defined by an oriented double covering

I am trying to study the $\omega-$limit set of a trajectory on a connected no oriented manifold so my idea is to use an oriented double covering to lifting the trajectory and analyze its $\omega-$...

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### Growing gliders under rule 110

I found a glider in the evolution space of rule 110 that grows constantly in size. Normal gliders live in the so-called ether, e.g. the so-called E-glider:
Other – often complex – gliders exist in an ...

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### Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity

Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{...

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### Homothety vector fields generating a foliation of $S^3$

Inspired by this question on homothety vector fields we realize that non homotheticity is some how an intrinsic property of the foliation associated to the vector field. See the comment by Prof. ...

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### Reference on relation between SFTs and Wang-tiles

I've been looking at several papers which allude to a relation between SFTs. Namely, given an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ with allowed patches $\mathcal{F}$, we can associate a ...

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1
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### Dynamical systems with disjoint $\omega$-limits of single points

For $X$ compact metric spaces and $f:X\to X$ continuous, is there a nice characterization of the systems $(X,f)$ for which, for every pair of points $x,y\in X$ with disjoint orbits, we have $\omega(x)\...

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### Getting analytic center manifolds

The center manifold of a degenerate zero of an analytic vector field need not be unique nor analytic. But say I want it analytic. Does anyone know of additional conditions to be imposed on the ...

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### Easiest self-contained proof of the Jewett–Krieger theorem?

Does anyone have a go-to reference for a proof of the Jewett–Krieger theorem in dynamical systems/ergodic theory? It's quite technical and I'd like to have something to show students. The best I ...

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### Relation between symbolic substitution and cellular automata

I recently asked this on Math Stackexchange recently in this thread. I was told that there is a relation between symbolic substitutions and cellular automata. I'm vaguely familiar with Cobham's ...

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### Approximate range of Radon-Nikodym derivative in a dynamical system

Suppose $(X, G, \Omega, \mu)$ is a dynamical system where $(X, \Omega, \mu)$ is a Borel measure space and $G$ is acting on $X$ such that each group action $x\mapsto g\cdot x$ defines a measurable ...

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### Non-twist maps of the annulus and their lack of fixed points

I have been wondering about the existence of a kind of `counterexample' to a modification of the Poincare-Birkhoff theorem which badly breaks the twist condition.
Let me state a variant of the ...

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1
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### Density of points in the torus whose iterates under a matrix converge to zero

In Yves Benoist and Jean-François Quint's notes Introduction to random walks on homogeneous spaces (top of page 11),
the following is listed as a step in the non-Fourier analytic proof of ergodicity ...

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### First visit of intervals for an irrational rotation

I suppose that what I look for is known, but I can't find it.
Let $\left\lbrace I_n=[a_n,b_n)\right\rbrace$ and $\left\lbrace J_n=[b_n,c]\right\rbrace$ ($n\in\mathbb{N}$) be two countable families of ...

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### Stability of a special singular perturbation problem

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a lower bounded smooth function, i.e., $\inf_{x\in\mathbb{R}^n} f(x)>-\infty$. Consider the following singular perturbation problem:
$$\begin{cases}\dot{...

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1
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### 'Trivial' lower bounds for pattern complexity of aperiodic subshifts

I recently asked in this thread about lower bounds on the complexity in the case where we have an aperiodic subshift. If I denote $c_n(\Omega)$ as the number of possible patterns on $Q_n= \big\{ 0,...,...

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### Arithmetic Teichmüller curves, first eigenvalue of the Laplacian, McMullen's expander conjecture

$\DeclareMathOperator\SL{SL}$By a result due to Ellenberg and McReynolds, any finite index subgroup $\Gamma$ of $\Gamma(2) \subset \SL\left(2,\mathbb{Z}\right)$ is the Veech group of an arithmetic ...

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### The most general (but useful) definition of "attractor" for dynamical systems

Consider J. Milnor's paper: On the concept of attractor.
There he writes: "A less restrictive definition" [than some of the previous ones he had considered] of the concept of an attract is &...

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171
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### Anosov flow on the 2-sphere

Is there a simple proof that there is no Anosov flow on $S^2$? Where can I find it?

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### Lower bounds for pattern complexity of aperiodic subshifts

In the setting of symbolic dynamics over $\mathbb{Z}^d$, one can define for the $n$-th pattern complexity of a given a subshift $\Omega\subseteq \mathcal{A}^{\mathbb{Z}^d}$ as
$$ c_n(\Omega):= \Big\...

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### Topological full groups of subshifts: differences between one-dimensional and multi-dimensional subshifts

For a multidimensional subshift $X$ over $\mathbb Z^d$, the topological full group $[X]$ is the set of homeomorphisms $f$ of $X$ that can be written as $f : x \mapsto \sigma_{c(x)}(x)$ with $c : X \to ...

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### General references on dynamics of continuous, piecewise linear interval maps

I want to find a general reference on topological and measurable dynamics of continuous piecewise linear interval maps. I am particularly interested in cases with only three pieces. I know there are ...

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### Reason behind the names of sub and supercritical bifurcations

What is the reasoning behind the names sub- and super-critical bifurcations that occur in the context of pitchfork and Hopf bifurcations? Textbooks seem to introduce this terminology without any ...

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### On the correct definition of attractors

It is well-known in dynamical systems that the concept of "attractor" differs in the literature.
My question is whether attractors need to be defined as subsets of $\omega$-limit sets of ...

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### Ergodic properties of a random shuffling process

Consider the following continuous analogue of a card shuffling process:
Let $Y_i, Z_i$ ($i \in \mathbb Z^+$) be sequences of jointly independent uniformly distributed random variables on $[0, 1]$. ...

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### Are arbitrary collections of ergodic measures "strongly mututally singular"?

I'm quite embarrassed not to know the answer to this question, but I think someone else will.
Suppose that $(X, T)$ is a topological dynamical system, and $\mathcal{E}$ is the collection of ergodic $T$...

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### Elementary cellular automata in stochastic modes

There are several ways to run a given elementary cellular automaton in a stochastic way:
by giving for each of the eight local configurations 000,100,010 and so on a probability by which the rule is ...

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### Sufficient conditions for periodic tiling by Wang tiles

I'm recently interested in whether a sub-shift of finite type contains a doubly-periodic problem, when the set of configurations is of the sort $\mathcal{A}^{\mathbb{Z}^2}$. When $Q_2=\{0,1\}^2$, and ...

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### On bounded solutions of a given fourth-order linear ODE

Consider the fourth-order linear ODE
$$
\label{eq1}
v^{(4)} + \frac{-C_2 - 2\alpha \phi}{C_4}v'' + \frac{4\alpha \phi'}{C_4}v' + \frac{k_1 + 3k_3\phi^2 -2\alpha \phi''}{C_4}v = 0.
$$
Without getting ...

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1
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### Entropy of $f^{m(x)+n}$ of full shift

Let $(X,\mu,f)$ be a two-sided full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $...

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### Asymptotic behaviors of equilibrium points of a switching SDE with Levy jumps?

Consider the following paper titled: Stochastic regime switching SIR model driven by Lévy noise, authored by Yingjia Guo.
Link: https://www.sciencedirect.com/science/article/pii/S0378437117302145
The ...

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### Possible weaker version of the Domino/Wang tiling problem

This may be a dumb question, but I was wondering whether the question of 'periodically tiling the plane from a finite set of tiles' is the same as the domino tiling problem or a weaker version. I ...

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### Showing convergence of an infinite ODE system

Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(0) \leq 1$ for all $n\in \mathbb N$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ ...

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### Asymptotic behavior of a dynamical system of density functions

On September 24, 2022, I asked the question below on Mathematics Stack Exchange, linked here:
Link to question on Mathematics Stack Exchange.
I received two up-votes, but no comments or answer. I ...

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1
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### A special kind of pseudo-garden eden states in cellular automata

I'm currently investigating Wolfram's elementary cellular automata on finite grids with periodic boundary conditions, i.e. on $\mathbb{Z}/k$ for different $k$.
It is clear that for each rule $R$ and ...

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### When is a composition of homeomorphisms topologically transitive provided one of the two is?

Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...

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### Turing reaction diffusion equations and neural networks

Suppose you have a Turing-type reaction-diffusion system
$$
\begin{cases}
\partial_t \phi = & f(\phi, \psi) + D_\phi \nabla^2\phi \\
\partial_t \psi = & g(\phi, \psi) + D_\psi \nabla^2\psi
\...

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### Build up of an external Kalman filter with closed form solution of an ODEs

When using an external Kalman filter, what are the benefits and advantages if, instead of a physical model, e.g. PP-p2D model - described by a system of nonlinear coupled differential equations - a ...

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### The logistic elliptic equation

Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...

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### Generalizing an application of the Poincaré-Bendixson theorem

I was looking for applications of the Poincaré-Bendixson theorem and on this site I have found several examples almost all similar to this post. So I tried to make a quite natural generalization
$$
\...