Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations.

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7
votes
2answers
163 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 ...
3
votes
1answer
89 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 ...
-4
votes
0answers
61 views

Which branch should I choose for my master degree? PDE or Dynamical Systems [closed]

I'm finishing my undergraduate in Industrial Engineering and I've applied for a mathematical master in which I can choose all the subjects I want. https://mamme.masters.upc.edu/en/study-program ...
2
votes
1answer
103 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 ...
4
votes
2answers
140 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 reclects specularly at the boundary). ...
2
votes
1answer
248 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 ...
1
vote
2answers
182 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 ...
3
votes
2answers
88 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} ...
0
votes
0answers
73 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 ...
5
votes
0answers
76 views

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 ...
1
vote
1answer
206 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 ...
2
votes
0answers
45 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 ...
0
votes
0answers
51 views

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 ...
1
vote
0answers
66 views
+50

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 ...
9
votes
1answer
197 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 ...
23
votes
4answers
1k views

Stability of the Solar System

My question is simple: Is the Solar System stable? You can see this Wikipedia page. In May 2015 i was in the conference of Cedric Villani at Sharif university of technology (Iran) with this ...
0
votes
1answer
57 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 ...
6
votes
1answer
293 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 ...
0
votes
0answers
34 views

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 ...
0
votes
0answers
72 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 ...
3
votes
0answers
34 views

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 ...
5
votes
0answers
169 views

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 ...
1
vote
0answers
59 views

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 ...
5
votes
0answers
119 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 $\omega(x)$, ...
9
votes
1answer
451 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 ...
4
votes
1answer
290 views

for which values of $\theta$ does this equation $x_{n+1}=\cos(\theta)x^2_{n}-\sin(\theta)x^2_{n-1}$ have bounded solutions?

I would like to investigate the global behavior of the following equation : $$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$$ where $A(\theta)= \cos(\theta)$ and $B(\theta) =\sin(\theta)$ are nonnegative parameters ...
18
votes
1answer
951 views

Anti-Mandelbrot set

I clearly remember seeing a paper where the dynamic of the anti-conformal map $f(z)=\overline{z}^2+c$ was studied (the bar means complex conjugation). There was a picture of the analog of the ...
3
votes
2answers
146 views

Maximizing entropy under constraints

This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures. Consider the one-sided shift ...
7
votes
2answers
860 views

Boundedness of solutions of a difference equation

Is there someone who can show me how I can prove this conjecture? Or at least show me how to do the first implication ? Conjecture: Assume $\alpha,\beta, \lambda \in [0,\infty)$. Then every ...
4
votes
1answer
183 views

A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$

In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed ...
3
votes
0answers
67 views

Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space? More specifically given a smooth manifold of $M$ and a ...
3
votes
2answers
234 views

Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$

I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different. #1 Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$. Then ...
3
votes
1answer
114 views

Smooth conditional measures for strong stable foliations of Anosov flows

I am trying to prove an analytic result for gesodesic flows on negatively curved manifolds and I encountered the following dynamical-system porblem. Let $B^n$ be $n$-dimensional balls and ...
0
votes
0answers
47 views

Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.) Consider the linear time-varying system $$ \dot{x} = A(t) x, $$ where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...
2
votes
1answer
324 views

How to find representatives of $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$

While reading about the Teichmuller flow, I am reading about the space of lattices $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$. I could not a find a good way of computing the Teichmuller flow on this ...
8
votes
2answers
327 views

An algorithm for Poincare recurrence time

Define the function $[0,+\infty) \rightarrow R$: $$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$ I want a number $t $ bigger than $10^7$ such that $$ f(t) > 4 ...
0
votes
1answer
154 views

Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?

I am reading a blog which talks about a $C^1$ diffeomorphism $f: \mathbb{D}\{ x^2 + y^2 < 1\} \to \mathbb{R}^2$ and estimates the Hausdorff dimension of its image $\mathcal{H}_\sqrt{2}^d ...
4
votes
0answers
266 views

Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes: $f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is prime} \\ \lfloor n/2 \rfloor & \text{if} \;n ...
14
votes
1answer
466 views

A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows: $$ f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
1
vote
1answer
111 views

Stability of singularity in singular holomorphic foliation

For an open subset $U$ of $\mathbb{C}^{2}$ containing $0$ and a holomorphic map $f:U\to \mathbb{C}^{2}$ which has a unique zero at the origin we associate a natural singular holomorphic ...
6
votes
1answer
372 views

Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?
6
votes
1answer
126 views

Decay of cusps in geometrically finite groups

Let $X=\mathbb{H}^{n}/\Gamma$ be a quotient of hyperbolic space of a geometric finite subgroup. Let $\mu$ be the Bowen-Margulis measure on the unit tangent bundle, and $m$ its projection to $X$. Fix ...
8
votes
1answer
239 views

is there a diffeomorphism with only finite orbits but of infinite order?

I asked this in stackexchange, but got no answer, so I am trying here. Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties: All its orbits are ...
3
votes
2answers
141 views

Does the 2-shift map have a root automorphism?

By the 2-shift map I mean the map $T:\{0,1\}^\mathbb{Z}\to \{0,1\}^\mathbb{Z}$ that shifts the sequence leftwise. By a root I mean an homeomorphism $\psi:\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ that ...
8
votes
7answers
572 views

Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...
3
votes
0answers
83 views

Trapped Billiard trajectories on non-convex billiard tables

Let $\Omega$ be a domain in $\mathbb{R}^2$ with smooth boundary. A billiard trajectory is a continuous curve $c: \mathbb{R}\supseteq I \longrightarrow \overline{\Omega}$ such that $c(t) \in ...
2
votes
0answers
72 views

Non-ergodic Dye Theorem for orbit equivalent automorphisms

The Dye Theorem states that any two free ergodic p.m.p automorphisms of a standard probability space are orbit-equivalent. Question: Is there a version of the above theorem for non-ergodic ...
2
votes
0answers
44 views

Mappings between adaptive networks and Markov processes

Are there any known mappings between adaptive networks models (i.e. graph model representations of networks where the internal vertex dynamics and connectivity topology can change subject to specific ...
2
votes
0answers
88 views

Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form ...
1
vote
0answers
96 views

Size of the eigenfunction of Laplacian (reference request)

It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then ...