# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $... 1answer 103 views ### Ratner's orbit closure for a unipotent semigroup For Ratner's orbit closure theorem, one may refer to the following Wikipedia page. Let$\{u_t\mathrel: t\in \mathbb{R} \}\subset G$be a unipotent one-parameter subgroup of a connected Lie group. Let$...
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'...