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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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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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 ...
• 3,971
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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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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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1 vote
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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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1 vote
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Set of all real numbers $x$ satisfying $\lim_{t\to \infty} \lambda_1(g_t u_x \mathbb Z^2)$ exists

Let $g_t=\text{diag}(e^t,e^{-t})$ and $u_x=\begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix},x \in \mathbb R$. Dani's correspondence establishes the Diophantine approximation properties of $x$ and ...
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1 vote
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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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1 vote
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Continuous version of ergodic with integral

Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
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Proving Hopf bifurcations for 3D system

I am working with a 3D continuous system of ODEs. I have found Hopf bifurcation numerically for a certain value of parameter. However, I want prove it analytically. Is it enough to show that the ...
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1 vote
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Stability of rigid bodies spinning around $z$-axis under gravity

Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
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How should the "measure theoretic" Jacobians of a dynamical map be understood in Lai-Sang Young's "Recurrence Times and Rates of Mixing"

In Young's article: Recurrence Times and Rates of Mixing, she uses multiple times the notation $JF, JF^k, JF^R$ to mean the Jacobian of a dynamical map $F:\Delta\to\Delta$ w.r.t. a given reference ...
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Regularity and rigidity of stable/unstable distribution for geodesic flow on noncompact negatively curved manifolds

For a volume-preserving $C^\infty$ Anosov flow on a three-dimensional compact Riemannian manifold, it was shown by Hurder & Katok that the Anosov foliations are always of class $C^{1, \alpha}$. ...
63 views

Proving period doubling bifurcation

I am working with a 3D continuous dynamical system. I have plotted the bifurcation diagram and found that period-doubling bifurcation occurs at a certain parameter value. However, I also want to prove ...
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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 $\{ x_n \}$ ...
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Identifying bifurcation

[![enter image description here]] 1]1I am trying to analyze the bifurcation of a 3D continuous model. For a certain range of parameter values, the origin is always an unstable point, whereas the ...
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Criteria for extending vector field on sphere to ball

Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file. Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, ...
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Entire function of finite order with deficient value

There are some sufficient conditions for an entire function to have a finite deficient value e.g., if the order $\rho$ of an entire function $f$ is such that $2<\rho<+\infty$ with all but ...
1 vote
62 views

Multiply connected Fatou component of an entire function

This question may be trivial but still I want to know the answer. Question: Is there any necessary condition (except boundedness of the Fatou component) for the existence of a multiply connected Fatou ...
178 views

Is the geodesic flow on a Riemannian manifold conservative?

Let's consider a complete Riemannian manifold $\mathcal{M}$. The geodesic flow of $\mathcal{M}$ is a first-order flow on the tangent bundle $T\mathcal{M}$. My question: Is it conservative? By ...
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Same occupation measure $\Rightarrow$ same trajectory

Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ The occupation ...
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References on semigroup actions

I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994). I would like to ask for references on semigroup actions on ...
1 vote
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Is equal natural density on intervals with matching areas but opposite signs sufficient to use fixed-width part sizes for a simple Riemann sum?

Suppose we have a sequence $\theta_n$ which is dense on $\left(0,2\pi\right)$. Furthermore, if $A=(x,y)\subset(0,\pi)$ and $B=(x+\pi,y+\pi)$ for some $x,y$, and if we define the natural density of a ...
190 views

Variant of Collatz where one is allowed a series of choices

This question is prompted by thinking about this question and the answer I gave there. Consider the family of functions of the form f_a(n):=\begin{cases} n/2 & \text{if $n$ ...
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