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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Convergence of a recursively defined sequence (discrete selector mutator equation)
Let $\beta \in (0,1)$ and let $(u_n(k))_{n,k
\geq 0}$ be recursively defined by $u_0(k) = \mathbf 1_{k=0}$ and, for $n, k \geq 0$ :
$$u_{n+1}(k) = \beta u_n(k-1) \mathbf 1_{k \geq 1} + (1-\beta) \...
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Classification of real polynomial vector fields on R2, up to polynomial automorphisms?
A result of Brunella classifies complete complex polynomial vector fields on ${\mathbb C}^2$, up to polynomial automorphism, and relies heavily on an earlier work of Suzuki. I haven't fully digested ...
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The topological entropy of potential space filling curves on the unit interval
By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1$...
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Hilbert 16th problem and dynamical Lefschetz trace formula
I would like to apply the known version of the conjectural formula (11) page 10 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...
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"Saddle connection" on a translation surface
A saddle connection on a translation surface $\omega$ is a geodesic
in the flat metric determined by $\omega$ joining two zeros with no zeros in its interior.
Athreya, Jayadev S., and Howard Masur. ...
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Shub Conjecture and polynomial entropy
The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the ...
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Cocycles and the Collatz problem?
Let $T(n) = n+R(n)$, where $R(n) = -n/2 $ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$.
$R(n)$ is the Cantor ordering of the integers:
https://oeis.org/A001057
In the Collatz problem, one is ...
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Applicability of van Holten's algorithm for symmetries in classical mechanics
Background
van Holten's algorithm (see e.g. here and here) is a way of constructing or recognizing dynamical/hidden symmetries in classical mechanics by looking for Killing tensors on the ...
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Equivalence of Wind Forces: Intensity vs. Duration [closed]
The strongest tornado in the world happened recently in Greenfield Iowa with winds over 318 mph: https://www.facebook.com/watch/?v=2176728102678237&vanity=reedtimmer2.0
I am curious, are less ...
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“is topologically mixing” vs. “is topologically transitive” in the definition of chaos
This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours.
Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits"...
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Fixed points of maps defined on Teichmüller space
Let $\mathcal{T}_A$ be a Teichmüller space of the sphere $S^2$ with a finite set $A$ of marked points, and suppose that $f \colon \mathcal{T}_A \to \mathcal{T}_A$ is a holomorphic map that has a ...
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Why does an invariant measure define a Schwartzman cycle?
Let $M$ be a manifold with a flow $\phi_X:\mathbb{R}\times M\rightarrow M$ induced by a vector field $X\in \Gamma TM$.
Any Borel measure $\mu$ defines a $1$-current $c_\mu$, i.e. an element in the ...
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Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators
Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
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Morse Theory for Time-Periodic Constrained Path Spaces
Let $(M,g)$ be a smooth, compact Riemannian manifold of dimension $n \geq 2$. Define a time-periodic constraint field $\Phi: M \times \mathbb{R} \to \{0,1\}$ with period $T > 0$, where $\Phi(x,t) = ...
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Is it likely that gradient flow trajectories of a $G$-invariant function pass through degenerate points?
This question may be posed somewhat vaguely, but I'm interested to actually get an idea of what to expect, so I try to not target it at a specific result.
Assume that $G$ is a compact Lie group, ...
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Square root of an Anosov diffeomorphism
Let $T\colon \mathbb T^d\to\mathbb T^d$ be an Anosov diffeomorphism (that is, the tangent bundle splits into invariant stable and unstable bundles; the restriction of $DT$ to the unstable bundle is ...
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How to show the geodesic orbit of a badly approximable number are/are not homogeneously equidistributed on its orbit closure?
Let $x\in \mathbb R$ be a badly approximable number. By Dani's correspondence, $g_t u_x \mathbb Z^2$ is bounded away from the cusp in $X=\text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$, identified with ...
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More than one recurrence point (Birkhoff)
Birkhoff's recurrence theorem states that for a compact metric space $X$ and a continuous function $T: X\rightarrow X$, there is a recurrence point $x\in X$; the latter means that for any ...
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What means by "a hyperbolic measure has product structure"
I try to figure out what means by "a hyperbolic measure has product structure".In the Anosov system,Lebesgue measure has local product structure with respect to the Lebesgure measure on ...
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An example of a matrix whose eigenvalues fullfill 'No-resonance' condition
No-resonance for a matrix is defined in terms of its eigenvalue as (last para page-3 in ref.):
$$\lambda_i \neq \sum_{j=1}^N m_j\lambda_j;\ \forall m_j\in \mathbb{Z}\ \ and\ m_j\geq 0$$
$$such\ that\ \...
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Using a poset or directed graph as input for a neural network
I'm not sure if this is the right community to post this in but I would appreciate any help. As the title states, I'm trying to train a neural network using some unconventional input. I'm wondering if ...
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Neural networks over gadgets other than $\mathbb{R}$
Recently, I learned that neural networks (NN) can be defined over fields other than $\mathbb{R}$: for example, Khrennikov and Tirozzi wrote a paper in 1999 (!) on $p$-adic neural networks, or neural ...
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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 ...
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How quickly will billiard trajectories cluster?
Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic billiard trajectories
inside a convex polygon. Assume they all have the same speed.
Define an $\epsilon$-cluster as a ...
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Convergence of the sequence $s_{n+1}=s_n^2-s_{n-1}^2$
$s_{n+1}=s_n^2-s_{n-1}^2$, $s_0=\sqrt{x}$, $s_1=x$
This sequence seems simple, but is pretty confusing. If you try it with integers, you might think that it always diverges to infinity, but if you try ...
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What are the zero entropy invariant measures for an Anosov geodesic flow?
Let $M$ be the double-torus with a hyperbolic Riemannian metric. The geodesic flow on the unit tangent bundle $T^1M$ has many invariant Borel probability measures. In particular there are closed ...
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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}&= \...
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Derivatives of diffeomorphism whose iterates on an open set converge to a point
Consider a smooth manifold $M$, a diffeomorphism $\varphi\in\mathrm{Diff}^\infty(M)$, and an open subset $B\subseteq M$. Suppose that, when restricted to $B$, $\varphi^n$ converges uniformly to a ...
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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 ...
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Asymptotic growth rate for primitve S-adic systems
It is known that for a primitive substitution $S:\mathcal{A}\to \mathcal{A}^+$, there exists constants $c,C>0$ such that
$$ c\theta_S^n \leq \vert S^n(a)\vert \leq C \theta_S^n \quad \text{for all} ...
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Persistent homology of Markovian dynamical systems
Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information ...
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Exact approximation in $p$ adic
Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as
$W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\...
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Two questions on one-dimensional dynamical systems
(1) For $\beta>1$ let $T_{\beta}:[0,1)\to [0,1)$ be given by $T_{\beta}(x)=\{\beta x\}$, where $\{\cdot\}$ denotes the non-integer part of real number. It is classical result (due to Parry I think) ...
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Equidistribution on subvarieties of $\mathrm{SL}_n(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates ...
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Cohomology for extension problems in symbolic/topological dynamics?
Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is ...
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Question regarding characterization of linearly recurrent subshifts by Durand
I was looking at the following paper by Fabien Durand, Corrigendum and addendum to ‘Linearly recurrent subshifts have a finite number of non-periodic factors’.
I have somewhat of a basic question ...
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Deciding homomorphic images of De Bruijn graphs
The De Bruijn graph $B_n$ of
dimension $n$ (on the two-letter alphabet) is defined as the directed graph on
$2^n$ vertices and $2^{n+1}$ edges, where for every $w = w_0 \dots w_n \in
2^{n+1}$ we put ...
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Measuring how suboptimal control is
Suppose I have a linear dynamical system to control. I use PMP to find necessary conditions for the optimal control of the system wrt to some objective function. Now, suppose that the trajectory I ...
CommunityBot
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A technical lemma in the lecture notes of Yoccoz on interval exchange maps
I'm reading the elegantly written lecture notes "Continued Fraction Algorithms for Interval Exchange Maps" of Yoccoz, available through the link <www.college-de-france.fr/media/jean-...
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Confusion about Teichmüller curves and $\operatorname{SL}_2$-action
$\DeclareMathOperator\SL{SL}$Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega ...
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Topology of windings on the two-torus
In short my question is: what can we say about the quotient topology induced by the linear flow on a two-torus?
I know that an irrational slope leads to a dense winding and hence (if I'm not mistaken) ...
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Poincaré recurrence; Time Return
Hello everybody! Recently I start a reading of a survey by Benoit Saussol,
AN INTRODUCTION TO QUANTITATIVE POINCARE RECURRENCE IN DYNAMICAL SYSTEMS, I am interested in references (Papers) Basics ...
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Aligning frequencies
Let $\omega_1, \omega_2, \dots, \omega_n$ be frequencies between $1$ and $\log n$. I would like to find an upper bound for a point $t$ that align these frequencies up to a small error $\delta$, that ...
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measurability of a special set
I've been working on some questions in dynamical systems then I faced the following problem:
Consider the circle $\mathbb{T}^1:= \frac{\mathbb{R}}{\mathbb{Z}}$. We represent it as a union of disjoint ...
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Identifying Saddle-node bifurcation of a 3D system of ODEs
I am trying to understand and prove the results shown in the following article. However, I am stuck at a point where it is stated that saddle-node bifurcation of periodic orbits occurs at ...
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Chaotic behaviour of the secant method for $\sin(x)$
For not very serious reasons I was trying to understand the behaviour of the secant method for solving $\sin(x)=0$ starting with $x_0=2$ and $x_1=18$, so
$$ x_{n+2}=x_{n+1}-\sin(x_{n+1})\frac{x_{n+1}-...
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How much algebra is preferable for studying/doing research in dynamical/complex systems and networks
I asked this question earlier on Mathematics StackExchange (link), but I think it might be better to put it here.
This question seems quite broad to ask...
The situation here is that I'm a second-...
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The relay use of invariant set theory
For a dynamical system, set $A$ is an invariant set with a function $V_1$, whose derivative is semi negative definite on $A$, and the region where the derivative is $0$ is the set $B$, which is also ...
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Simple proof that exactness implies strong mixing
Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
Using the following definition of Dirichlet L-function
$$
L(1,\chi)=\begin{cases}
\dfrac{2\pi h}{w\sqrt{m}} & \textit{if}\ \chi(-1)=-1 \\\\
\dfrac{2 h \log{|\epsilon|}}{w\sqrt{m}} & \textit{...
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