# 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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### Distribution of the values of the product $\prod_{k=1}^n |1-e(k\alpha)|$ for an irrational number $\alpha$

For an irrational number $\alpha$, let $e(k\alpha):=\exp(2k\pi i\alpha)$. It was indicated in this thread that
$$\limsup_{n \to \infty} \prod_{k=1}^n |1-e(k\alpha)|=\infty$$
(actually a weaker result ...

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43 views

### Vandermonde shift

I'm looking for any known results on a Vandermonde matrix that acts as a kind-of shift, Specifically, I am interested in solving
$\begin{bmatrix}1 & c_0 & c_0^2 & \cdots \\
1 & c_1 &...

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37 views

### Uniform stability of linear operators - reference request

Let $T$ be a bounded linear operator on a complex Banach space $X$. I am looking for a reference for the following result:
Theorem 1. Let $p \in [1,\infty]$. The following assertions are equivalent:
(...

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367 views

### Can an "almost injective'' function exist between compact connected metric spaces?

Let $\pi: X\to Y$ be a surjective continuous function between the compact, metric and connected spaces $X$, $Y$, and $Y_0 = \{y\in Y: \#\pi^{-1}(y)>1\}$. Suppose that:
$Y_0$ is dense in $Y$,
$Y\...

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61 views

### Rotation set vs existence of rotation number

Let $f\colon \mathbb{S}^{1}\to\mathbb{S}^{1}$ be a continuous function of degree 1 and $F\colon \mathbb{R}\to \mathbb{R}$ a lift of $f.$ One can define, for each $x\in \mathbb{R}$, the rotation number ...

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**1**answer

73 views

### Using a poset or directed graph as input for a neural network [closed]

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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66 views

### Ergodicity of a dynamical system on the $n$-sphere

Let $v$ be continuous and nowhere-vanishing vector field tangent to the $n$-sphere $\mathbb{S}^n$ (hence $n$ is odd, w.r.t the Hairy-Ball Theorem). Let $x$ be a trajectory on $\mathbb{S}^n$, defined ...

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40 views

### Central manifold and fixed point theorems

Let us consider a real dynamical system $s′=g(s)$. In order to study the stability of the central manifold, we reformulate the problem as follows: Let $d=d(y,z)$ be a function verifying $u<d<w$ ...

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615 views

### Why the least action principle is always (?) used in this particular form?

The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...

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257 views

### Is the series $\sum_{n=1}^{\infty} \sin(n^4)\sin(4^n)$ convergent or divergent?

Is the series $$ \sum_{n=1}^{\infty} \sin(n^4)\sin(4^n) $$ convergent or divergent?
I tried expanding the sine functions and got no clue, and any test that I know of isn't helping me with this series. ...

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73 views

### Area preserving diffeomorphisms of surfaces with only hyperbolic periodic points

This is a (probably very naive question) about area-preserving maps of surfaces.
Does there exist a Hamiltonian diffeomorphism
$$ f: \Sigma \to \Sigma $$
of a symplectic surface (real dimension $2$), ...

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115 views

### Classification of Lagrangians with given Euler-Lagrange equations

In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of ...

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40 views

### Entropy of flow and time-1 map

Let $\Phi=(\phi_t)_{t\in \mathbb{R}}$ be a continuous flow on a compact metric space $X$. Let $\mu$ be a $\phi_1$-invariant measure. Then it is not hard to verify tht $\int_{0}^{1} \phi_t\mu dt$ is $\...

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39 views

### What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?

When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...

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81 views

### Foliations and locally free action of $\mathbb{R}^{n-1}$

Let $M$ be a $n$-dimensional closed manifold endowed with a foliation ${\cal F}$ suppose that the leaves of ${\cal F}$ are diffeomorphic to $\mathbb{R}^{n-1}$ are the leaves of ${\cal F}$ defined by a ...

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61 views

### When are all average trajectories of $w_{k+1}=Aw_k+b$ bounded?

Below is an open-problem in my field, and I'm wondering if someone has insights I'm missing. (cross-posted on math.se)
Suppose observation $x$ is drawn from some distribution $\mathcal{D}$, $w_0\in \...

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23 views

### Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; \boldsymbol{x}(t) \in \mathbb{R}^n, \;\; t \geq 0, \; \...

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96 views

### A unique equilibrium state which does not have Gibbs property

Let $T:\Sigma \rightarrow \Sigma$ be a topologically mixing subshift of finite type and let $f:\Sigma \rightarrow \mathbb{R}$ be a continuous functions over $(T, \Sigma)$. Assume that there is a ...

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98 views

### Difficulty of homeomorphism of effective Cantor dynamics

Let $X = \{0,1\}^{\mathbb{N}}$ with the product topology. Given a Turing machine $M$ and $x \in X$, define $M(x) \in \{0,1\}^* \cup X$ as the sequence of bits output by $M$ when given an oracle for $x$...

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499 views

### What are some foundational authors/papers in dynamical systems?

I have just begun my first dynamical systems class, and I would like to try out the advice in the top answer here. To summarize, the answer suggests that when studying a new field, one should look at ...

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32 views

### Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?

Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...

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53 views

### Composite canonical transformation using Lie operators

Context: According to a Lie theorem in canonical transformation, if $f$ and $S$ are arbitrary functions of the canonical variable set $(\xi,\eta)$ (i.e. meaning that $\xi_{i},\eta_{i}$ are the 2n ...

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80 views

### Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?

My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider,
$\dot{x}=Ax$, where $x$ is the infinite dimensional ...

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223 views

### Does there exist a vector field on the disk whose flow has constant singular values?

$\newcommand{\tr}{\operatorname{tr}}$
$\renewcommand{\div}{\operatorname{div}}$
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi_t$ be its flow.
...

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66 views

### Proving positive invariance

I need to prove that set $D$(A picture for Set $D$) given by
$$D=\{(x,y):0\leq x\leq L_0,~0\leq y\leq X_0,~0\leq x+y \leq R_0\}\subseteq \mathbb{R}_+^2$$ of the system:
$$\dot{x}=k_1(R_0-x-y)(L_0-x)-...

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151 views

### Rigorous results on chaos in a driven damped pendulum

The harmonically driven damped pendulum is often used as a simple example of a chaotic system, the equation is just $\ddot{\phi}+\frac1q\dot{\phi}+\sin\phi=A\cos(\omega t)$. As long as $A$ and $\omega$...

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128 views

### Local cross-sections for free actions of finite groups

Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...

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311 views

### Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?

It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form
$$
x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)}
$$
for ...

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223 views

### Topological objects associated to Steinerberger's 4-regular graphs

Very recently, in arXiv:2008.01153, Steinerberger has associated to any sequence $(x_n)_{n\in\mathbb{N}}$ of distinct real numbers a 4-regular graph.
In the case irrational multiples, like $x_n=n\sqrt{...

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147 views

### Modified energy method for transformed Fokker-Planck equation (tricky integration by parts…)

I came across Villani's paper titled "Hypocoercive diffusion operators" and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed ...

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60 views

### Balls in minimal systems II

If $(X,T)$ is a minimal system uniquely ergodic with $\mu$. Is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$ for some metric $d$ (with the same topology)?
This question is ...

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97 views

### Balls in minimal systems

If $(X,T)$ is a minimal system uniquely ergodic with $\mu$, is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$?

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31 views

### Approximation of vector field by vector fields with all trajectories closed

Let $\vec j$ be a smooth compactly supported vector field in $\mathbb{R}^3$ such that $div(\vec j)=0$. Are there known either necessary or sufficient conditions such that there exists a sequence of ...

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112 views

### Lagrange’s interpolation formula: Theoreme and Example [closed]

I would like to know where they come up with the formula of Lagrange interpolation (Lagrange’s interpolation formula),Lagrange_polynomial because I did some research, but I find a different definition ...

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433 views

### Why is a dynamical system not a dynamic system? [closed]

This is a research question in the history of math, I suppose.
As a non-native english speaker I became used to mathematical expressions like 'dynamical' and 'tangential'. When using them in daily ...

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136 views

### What are the hypotheses we should add for the generalizations of Furstenberg recurrence theorem?

In my question here I suggest a possibility for generalization of Furstenberg recurrence theorem needing some hypothesis for that generalization to be hold in the side of convergence of the below ...

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42 views

### Convergence of iteration of a convex program

Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}, \ \ \mathbf{E} \in \mathbb{R}_{+}^{n \times m}$, with $\mathbf{V} \mathbf{1}_{m} = \mathbf{1}_{n}$ and $\mathbf{E}^{T} \mathbf{1}_{n} = \mathbf{1}_{m}$...

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97 views

### Algorithm/iterative procedure for constructing hypercyclic vectors?

Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...

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80 views

### Poincaré Recurrence Theorem for flows

Someone knows a book that have the proof of the Poincaré Recurrence Theorem for flows when the set have finite volume and when don't?

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29 views

### Hitting Time-Analogue for Chaotic Systems

Let a topologically mixing dynamical map $f$ on $\mathbb{R}^n$, and define the dynamical system with initial value $x \in \mathbb{R}^n$ by
$$
x_{t+1}^x = f(x_t^x),\, x_0^x=x
.
$$
Fix $y\in\mathbb{R}^n$...

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80 views

### Explicit transitive flow on disc

$D_n\triangleq \left\{x \in \mathbb{R}^n:\, \|x\|\leq 1\right\}$ with its subspace topology. By a transitive flow on $D_n$ I mean a continuous function
$$
\phi: [0,1]\times D_n\rightarrow D_n,
$$
...

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**1**answer

93 views

### On proving the absence of limit cycles in a dynamical system

I'm studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I'm stuck at this two-variable problem which I've struggled with for too long now.
$$ \dot M ...

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80 views

### Regular singular point of non-linear ODE: $\dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$

Consider a system of ordinary differential equations of the form
$$
\dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t))
$$
where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}_{n\times n}(\mathbb{C})$ is a constant ...

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142 views

### Random products of $SL(2,R)$ matrices and Furstenberg's theorem

I was recently learning Furstenberg's theorem on random products of $SL(2,R)$ matrices, and came across with a simple example that confused me:
Considering random products of two matrices $A=\begin{...

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59 views

### Entropy of Markov measure using marginal distribution

Let $S$ be a finite set and let $\mathcal{X}$ be the set of all bi-infinite sequences over $S$. Let $\eta_1,\eta_2$ be two shift invariant 1-step Markov measures over $\mathcal{X}$. For a finite word $...

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103 views

### Ratner's orbit closure for a unipotent semigroup

For Ratner's orbit closure theorem, one may refer to the following Wikipedia page.
Let $\{u_t\mathrel: t\in \mathbb{R} \}\subset G$ be a unipotent one-parameter subgroup of a connected Lie group. Let $...

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154 views

### Infinite composition of continuous functions

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F_n:= f_n\circ \dots\circ f_1$. Then $F_n$ is continuous. However, the pointwise limit need not be (consider Mateusz'...

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138 views

### Search for a general formula from known iterative relation

$F$ is a mapping among $\{\theta_{n_1n_2}\}$, with $\eta_{1/2}$ being arbitrary constants involved.
$F: \theta_{n_1n_2} \rightarrow \theta_{n_1+1n_2}+\theta_{n_1n_2+1}+\eta_{1}n_1\theta_{n_{1}-1n_{2}} ...

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93 views

### Characterization of topological entropy?

Let $V$ be a smooth Anosov vector field on a compact $n$ dimensional manifold $X$. Let $D(X)$ denote the set of distance functions $d$ on $X$ that are equivalent to fixed Riemannian distance. For each ...

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158 views

### On the density map of the abundancy index

Let $σ$ be the sum-of-divisors function. Let $σ(n)/n$ be the abundancy index of $n$. Consider the density map $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...