Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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A Perron-Frobenius problem

Let $A$ be an irreducible nonnegative matrix with column sums equal to 1. Let $b\in R^n$ have components summing to 0, and let $u$ be the solution of $u=Au+b$ with components summing to 1 (unique ...
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1answer
111 views

Difference between Hopf-Turing bifurcation and traveling-waves bifurcation in reaction-diffusion systems

I am perturbing a dynamical system with a perturbation that looks like $$\vec u e^{-i k x} e^{\sigma(p;k)t}$$ where $\sigma$ is a function of the parameters of the dynamical system and the wavenumber, ...
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109 views

cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...
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1answer
86 views

Definition o branched 1-manifold [closed]

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia: "A finite graph whose edges are smoothly embedded arcs in a surface, ...
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1answer
308 views

electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...
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63 views

Problem on differential inclusion

For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the ...
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154 views

Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence $$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3} ...
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Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...
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54 views

Ordering periodic orbits

I want to prove the proposition: Proposition- Let $f:I \to I$ be continuos, and let f have a (2n+1)- periodic orbit {$x_{k}=f^{k}(x_{0})$, $k=0,1,\dots,2n$}, but no (2m+1)-periodic orbit for $1\...
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125 views

Multi dimensional symbolic dynamics

I want to learn Multi dimensional symbolic dynamics. can you point to any recent thesis containing a good exposition or lecture notes?
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46 views

Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?

I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...
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213 views

The converse of von Neumann's mean ergodic theorem

Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following. Let $\{F_n\}_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ ...
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Which polynomial's roots are its coefficients?

Start with any polynomial of degree $n$ with complex coefficients, e.g., $$z^3+z^2+2 z+3 \;.$$ Find its $n$ roots, and list them in order of their modulus: $$-1.28, (0.14\pm 1.53 i)$$ Now form a new ...
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90 views

Perturbations to a vector field

I ran into some problems while working through a proof of the Poincare-Hopf theorem that essentially boiled down to the following question: given a smooth vector field $V$ on a (compact Riemannian) ...
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50 views

When do positively invariant subset contain a given set?

Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't ...
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122 views

Discrete group action on the sphere

Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$ be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...
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2answers
122 views

Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
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1answer
166 views

Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense: A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...
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227 views

The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$. We consider the following polynomial vector field on ...
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1answer
87 views

Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...
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138 views

When is the time one map of a suspension flow ergodic?

I'm sure the answer to the following question is well known but I couldn't find the answer I needed. Let $(\Sigma,\sigma)$ denote the full shift on $k$ symbols and let $\mu$ be an invariant measure ...
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2answers
153 views

Do ergodicity, minimality and equicontinuity on a compact space imply total ergodicity?

Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ? According to some results I found in some books, a rotation on a ...
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1answer
138 views

Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof): In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...
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47 views

Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...
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244 views

Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
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1answer
176 views

Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. For any $x \in E$, there is a ...
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1answer
111 views

Determining whether $k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$

My REU partner and I are working on a problem involving iterations of quadratic rational maps over an algebraically closed field $K$ that is complete with respect to a non-trivial non-archimedean ...
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2answers
212 views

Are periodic billiard trajectories stable on a manifold with strictly convex boundary?

Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary. Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reflects specularly at the boundary). ...
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2answers
304 views

Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
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322 views

Hilbert 16th problem, distribution of Limit cycles

Edit: Can one help for translation of the link in Russian(comment by Dimitry Todorov)(Or at least a summary of it)? It seems that the second part of the Hilbert 16th problem is solved or is going to ...
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2answers
223 views

Growth of the size of iterated polynomials

I have been working independently on a project but now I am stuck and need to seek an expert's wisdom for a part of it. I am basically looking for theorems related to growth of the size of polynomials....
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3answers
115 views

Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by \begin{eqnarray*} s_0 &=& a, \\ s_1 &=& b, \text{and} \\ s_{n+2} &=& s_{n+1} ...
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115 views

The complex leaves containing real limit cycles of Lienard equation

According to the answer and comments to this question we realize that a useful approach to such type of questions is to consider algebraic vector field which possess algebraic solutions. On the ...
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Quantitative approximation of invariant measures by periodic ones

It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Is there some knowledge ...
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1answer
235 views

Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post I apologize in advance, if this question is obvious: 1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...
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59 views

Any minimal WAP dynamical system is distal

I'm trying to show that any minimal WAP dynamical system $(X, G)$ is almost periodic. By Ellis's joint continuity theorem, it suffices to show that any minimal WAP system is distal. There are many ...
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lattice basis reduction of the orbit of a rational vector on the torus

LEt $v=(p_1/q,...,p_n/q)$ be a vector of the torus $\mathbb{T}^n$, such that for any $i$, $p_i$ and $q$ are relatively prime. Let $L= \{ kv \mod \mathbb{T}^n , k=0,...,q-1 \}$. What is the lattice ...
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An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
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1answer
211 views

Is an explicit $c$ known to lead to a noncomputable Julia set?

Braverman & Yampolsky have shown that there exist noncomputable Julia sets, i.e., there exist $c \in \mathbb{C}$ such that the Julia set of $f(z) = c + z^2$ is not computable. "A set is computable,...
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4answers
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Stability of the Solar System

Is the Solar System stable? You can see this Wikipedia page. In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and ...
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1answer
140 views

Convergence to equilibrium via gradient descent

J. B. Rosen proved that in concave games of n players (which assumes that Cartesian product of strategy profiles is convex) if the game satisfies the condition of diagonally strictly concave then ...
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1answer
703 views

Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...
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Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times \...
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Bifurcations in flows on 2-dimensional torus

I am doing research on bifurcations which appear in flows on the 2-dimensional torus, in particular on such which do not appear in flows on $\mathbb{R}^2$. Can anyone provide some references on ...
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76 views

Two vector fields are cojugate but not take orbits

Let $X$ and $Y$ be $C^1$ vector feilds on $R^m$. Suppose that $0$ is an attracting hyperbolic singularity for $X$ and $Y$. Show that there exists a homemorphism $h$ of a neighborhood of origin which ...
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Limit Behavior of Iterated Curvature-Function

What can happen, if one defines an infinite sequence of functions as follows $f_0\in C^\infty: x\in\mathbb{R}\mapsto y\in\mathbb{R}$ $f_{n+1}: \int_0^x \sqrt{1+(f_n'(t))^2}dt\mapsto\frac{f_n''(x)}{\...
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Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...
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Convergence to equilibrium for time in-homogeneous diffusions

Consider the long time behavior for a time in-homogeneous diffusion such as $$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$ where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent ...
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1answer
184 views

Is there a universal $\omega$-limit set?

For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$. For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...
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508 views

A weakening of the Littlewood conjecture

For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by $$\ell(\alpha,\beta)=\liminf_{n\rightarrow\infty}n\|...