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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Solution of an ODE upon singular perturbation
The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.
The original system involves $N$ massless electric charges at position $\...
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2
votes
1
answer
121
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Examples of hyperbolic set and J-stable sets
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
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0
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85
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A 1 dimensional foliation which is Riemannian foliation with respect to no Riemannian metric
What is an example of a non vanishing smooth vector field on a manifold $M$ whose corresponding foliation is a Riemannian foliation with respect to no Riemannian metric on $M$
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161
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A closed leaf with two different index with respect to two different Riemannian metrics
Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question:
Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, ...
- 356
7
votes
1
answer
142
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Hamiltonian-ization of a dynamic system
On affine space, a sufficiently smooth continuous-time Hamiltonian dynamic system $\dot p = \nabla_q H, \dot q = -\nabla_p H$ conserves $H$, preserves the volume form (e.g. if we are looking for ...
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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 ...
- 1,409
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0
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210
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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 ...
- 356
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1
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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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1
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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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1
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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\}^...
- 633
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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 ...
- 633
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1
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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 neighborhood 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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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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103
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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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102
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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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2
votes
1
answer
129
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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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101
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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)\...
- 4,459
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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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2
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307
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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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130
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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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votes
1
answer
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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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votes
1
answer
177
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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 ...
- 481
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1
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331
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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 ...
- 4,459
3
votes
1
answer
193
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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,...,...
- 633
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355
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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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1
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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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273
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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. Even ...
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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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1
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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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Is this card shuffling process weakly mixing?
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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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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2
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329
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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 ...
- 633
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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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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 ...
- 633
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1
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435
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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 ...
- 9,720
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votes
0
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128
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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. ...
- 4,459
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101
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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
\...
4
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108
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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:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
- 1,759
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0
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100
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Generalizing an application of the Poincaré-Bendixson theorem
I was looking for applications of the Poincaré-Bendixson theorem and on this site I have found several examples almost all similar to this post. So I tried to make a quite natural generalization
$$
\...
- 143
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Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$
Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...
