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

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17
votes
4answers
648 views

Non-chaotic bouncing-ball curves

I was surprised to learn from two Mathematica Demos by Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel exhibits choatic behavior:     (The Poincaré map ...
1
vote
2answers
116 views

Estimation of number of ways to concatenate strings of the form $01^k2^k$ to create a string of length n

In symbolic dynamics, the context-free shift is the set of biinfinite concatenations of strings of the form $01^k2^k$ for $k\in\mathbb{N}\cup\lbrace 0\rbrace$. I've reduced a certain problem to ...
4
votes
0answers
56 views

Is a closed set with orbit capacity zero automatically thin?

Let $G$ be a countably infinite amenable group. Let $\alpha: G\curvearrowright X$ be a continuous group action. (Mostly free and minimal, though!) Definition 1: Let $A\subset X$ be closed and ...
3
votes
2answers
211 views

Unstable Foliations

Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$. Definition. $\mathcal{F}$ is said to be a unstable ...
5
votes
1answer
790 views

Beyond Collatz: A $5n+1$ conjecture? [closed]

Let $$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
5
votes
1answer
155 views

Uniform hyperbolicity decay estimate

I have been trying to understand the proof of the following result, which is considered well-known. Theorem: Fix a compact metric space $X$, a homeomorphism $T:X \to X$, and a continuous map $ A : ...
7
votes
2answers
218 views

Transitive geodesics on closed surfaces of genus greater than one

A well-known result of Hedlund and Morse states that if a Riemannian metric on a closed surface of genus $g > 1$ has no conjugate points, then it carries transitive geodesics (i.e., geodesics whose ...
2
votes
1answer
210 views

Non-linear 1st order difference equation

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$ I have tried various substitutions, simplifications ...
11
votes
0answers
141 views

Fundamental groups of reduced subgroup lattices

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
1
vote
1answer
190 views

Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)

I have difficulties finding an appropriate reference for the following question (which I hope that it to be true). Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexification and $\tau: ...
4
votes
1answer
159 views

How many different states of Nash equilibrium?

So there is this quite well known Prisoner's dilemma where two parties can both defect and cooperate (and get points based on their decisions). In my presently used example I take it that cooperating ...
5
votes
0answers
90 views

Equivariant zero dimensional extension recovering a given measure

Let $X$ be a compact metrizable space and $\alpha: \mathbb{Z}^d\curvearrowright X$ a continuous group action. Then it is well known that there exists a zero dimensional compact space $Y$, an action ...
19
votes
2answers
1k views

How does it End?

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context. Let $\mathcal{C}$ be a category. ...
2
votes
0answers
65 views

Rotation number of perturbated equation

I have a differential equation on torus $(t,x)$ and well studied it's Arnold tongues for Poincare map of the circle $x(t=0) \to x(t=2\pi)$. The question is how changes rotation number when I add small ...
11
votes
2answers
410 views

Book on the Three body Problem

Hi all, I am looking for a good book about the famous (infamous perhaps?) three body problem - both theoretical and numerical hardless and accomplishments. can you help? Thanks
10
votes
1answer
651 views

Analysis of the boundary of the Mandelbrot set

Motivation: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite ...
0
votes
0answers
164 views

Rope simulation with Position Based Dynamics

First off I apologize if this is the wrong stack exchange for this question, it seems to be like halfway between programming and math. But it leans more on the math side so hopefully I'm not out of ...
3
votes
2answers
173 views

Mapping nonlinear systems into linear ones

This is a request for references. I'm not asking for the answer... Just give me a starting point! I'm interested in global linearization of linear equations. I guess the problem is well understood in ...
3
votes
2answers
147 views

Hyperbolic sets that are not locally maximal

I would like, if possible, a simple example of a hyperbolic set that is not locally maximal. What kind of dynamic phenomenon should occur for the appearance of hyperbolic set that is not maximal.
3
votes
0answers
84 views

“Spectral decomposition” action on the unitary group

Consider a matrix $U$ from the unitary group $U_N(\mathbb{C})$ and consider the map $f:U_N(\mathbb{C})\rightarrow U_N(\mathbb{C})$ where $f(U)$ is the matrix of the eigenvectors of $U$. What is ...
2
votes
1answer
351 views

C* Algebras, Foliations and Dynamical Systems

I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles so one ...
0
votes
0answers
162 views

Spectral radius of products of matrices of a certain form

Background Let $\mathcal{D}$ be the set of $S\times S$ diagonal matrices with elements in $[0,1)$. Let $\mathcal{G} = \{\Gamma \in \mathcal{D} : tr(\Gamma) = 1\}$, and let $B$ be an $S\times S$ ...
5
votes
0answers
169 views

Quasicompactness of transfer operators associated to IID matrix products

Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
2
votes
1answer
275 views

Homeomorphism of the circle with rational rotation number

I am sorry because it is probably a trivial question. I consider a homeomorphism of the circle that preserves orientation and that has a rational rotation number p/q (it is an irreducible fraction). I ...
11
votes
3answers
328 views

Dynamics in one matrix variable

Are dynamical systems $$X \mapsto F(X)$$ studied where $X \in \mathrm{M}_n$, $\mathrm{M}_n:=\mathrm{Mat}(n,\mathbb{C})$ or $\mathrm{Mat}(n,\mathbb{R})$, and $F$ is a (properly defined ...
2
votes
2answers
134 views

A notion of a 'coarse', parametrized dimension of an object, where the parameter determines how finely we can distinguish (say) a very thin rod from a line

I apologize for the clumsy wording of the title-- what I'm looking for is a notion of an integer-valued dimension $d_{\epsilon}$, which we parametrize by a real positive number $\epsilon$, of, say, a ...
0
votes
0answers
45 views

Why this synchronization error dynamic for Krasovskii-Lyapunov?

I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", but I'm missing something basic up front. The problem is to take a ...
3
votes
1answer
309 views

Phase transition in dynamical systems

There are several occasions in the study of dynamical systems that are called phase transitions. For example consider a homeomorphism $f:X\to X$ and a potential function $\phi\in C(X,\mathbb{R})$. ...
3
votes
1answer
168 views

Hyperbolic sets

I recently started reading about hyperbolic dynamics in the notes of L. Wen, http://www6.cityu.edu.hk/rcms/publications/ln5.pdf and in this (page 8) there is the following statement: If the ...
1
vote
1answer
148 views

Relation between volume entropy and Hausdorff dim of limit set?

I have a very stupid question: I often see that the volume entropy of a compact Riemmannian manifold with negative curvature coincide with the Hausdorff dim of the limit set or Patterson sullivan ...
6
votes
2answers
372 views

Variational Principle for the Entropy

Theorem: Let be $f$ a homeomorphism of a compact metric space $X$, then $$ h_{top}(f)=\sup_{\mu\in \mathcal{M}_{f}}~ h _\mu (f) $$ Question: The above theorem is the famous variational principle ...
0
votes
0answers
91 views

Application of Morse theory to second order systems

Hello I'm looking for some applications of Morse theory to second order differential system,( or boundary value problems ) Someone can help me with a pdf or a book which has these applications ? ...
3
votes
0answers
176 views

Is it difficult to prove that nature is chaotic?

If we have a Markov coding or another symbolic description of a dynamical system it is usually easy to prove that the system is chaotic (in the sense of of Li-York, Devaney, positive entropy of what ...
1
vote
0answers
87 views

null controllability of linear wave equation

Consider the linear wave equation : $$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$ Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...
4
votes
2answers
269 views

Sz.-Nagy dilation for uniformly convex Banach spaces

The Sz.-Nagy dilation theorem says that for a Hilbert space $H$ with nonexpansive operator $T$, there is a larger space $H'$ containing $H$ and a unitary operator $U$ on $H'$ such that for all $x \in ...
1
vote
1answer
107 views

Computing saddle connections in flat structures

Background: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described here. Curves of constant argument on $S(P)$ which start and end at a ...
5
votes
1answer
333 views

Conley Theorem (or fundamental theorem of dynamical systems)

Notations: $\mathcal{R}(f)$ denotes the chain recurrent set of $f$ $NW(f)$ denotes the non wandering set of $f$ $R(f)$ denotes the recurrent set of $f$ ($x: x\in \omega(x)$) Given compact ...
1
vote
1answer
117 views

Extension of power bounded operators over a finite subspace

Suppose $Y$ is a Banach space and $X$ is a finite-dimensional subspace of $Y$. Further assume $T:X \rightarrow X$ is a linear operator which is power bounded from above and below, in other words ...
3
votes
0answers
97 views

Are irrational multiples of central sets again central?

Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact metric space, $T$ a ...
3
votes
2answers
212 views

Uniqueness of fixed points for rational transformations

Background Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by $$ T(x,y) := \left( \frac{1}{1 + ax + by}, \frac{1}{1 + cx + ...
4
votes
2answers
184 views

How do we recognize a Markov partition?

I'm looking for theorems that can be used to show that a topological partition for a given expanding map is Markov. Here are the relevant definitions: Let $\phi\colon\mathbb{R}^m\to\mathbb{R}^m$ be ...
1
vote
1answer
328 views

Ricci flow as a gradient flow and its Lyapunov function

In study of Ricci flow, for making Ricci flow as a gradient flow I faced $\mathcal{F}(g,f)=\int (R+|\nabla f|^2)e^{-f}$. I know that if we suppose $\frac{df}{dt}=-R$, then ...
1
vote
0answers
53 views

strong stability for the wave equation

Consider the $n-$dimensional wave equation $$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$ where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
5
votes
3answers
427 views

Integer dynamics hitting infinitely many primes

I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). To make it specific, ...
1
vote
0answers
103 views

Langevin equation with position-dependant damping: existence of an invariant measure?

The usual Langevin equation for a particle in a 1D harmonic potential $dq(t) = p(t)~dt$ $dp(t) = -q(t)~dt + a ~dW(t) - b~p(t)~dt$ admits as an invariant measure the Gibbs measure ${1\over ...
0
votes
1answer
290 views

Sierpinski Triangle and the Chaos Game

The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
1
vote
1answer
86 views

On solution of a recursion with rectangular matrices

Greetings to members here. The question is how to calculate the solution $S(k)$ of the following recursive equation $$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$ where $J$ and $A$ are rectangular not ...
27
votes
4answers
723 views

Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$, except leave the origin $(0,0)$ unoccupied by a disk. Q. Is it the case that every disk can be ...
3
votes
0answers
59 views

Do identical orbit tiles imply identical combinatorial types?

Given a periodic trajectory on a triangle, we can associate to this trajectory a sequence of integers $1,2$ and $3$ by labeling the edges of the triangle and taking the sequence of edges the ...
0
votes
1answer
202 views

Recurrence and transience of cocycle over a dynamical system

Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and ...