Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,398
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Cryptography and iterations
Hi,
Here is a question in cryptography which is probably naive, and a reference request.
I was wondering about the following key-exchange scheme, which is a variant on Diffie-Hellman. Consider a ...
- 2,195
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votes
5
answers
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Recommended book for introduction to Chaotic dynamics? (application in probability distributions)
I'm just starting some research and I need a good introductory book in the topic of chaotic dynamics. Does anyone have a suggestion? Thanks.
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1
answer
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Is the geodesic flow on a Riemannian manifold conservative?
Let's consider a complete Riemannian manifold $\mathcal{M}$. The geodesic flow of $\mathcal{M}$ is a first-order flow on the tangent bundle $T\mathcal{M}$.
My question: Is it conservative? By ...
- 31
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3
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Computing the maximum modulus
For each $a\in \mathbb C$ define $f_a:\mathbb C\to \mathbb C$ by $f_a(z)=\exp(z)+a$. I am primarily interested in real values $a\in (-\infty,-1)$.
For each $r\in [0,\infty)$ define $M_a(r)=\max\{|f_a(...
- 3,055
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1
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Symmetries for Julia sets of perturbations of polynomial maps
This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...
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2
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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 ...
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2
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Systems similar to Erdős numbers?
As many mathematicians know, each person has an Erdős number (see: http://en.wikipedia.org/wiki/Erd%C5%91s_number). That is, Erdős himself has Erdős number zero, each person who published anything ...
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2
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$\mathbb{S}^2$ equivalent to frac$(n \alpha)$ equidistribution on $\mathbb{S}^1$
Let $\operatorname{frac}(x) = x - \lfloor x \rfloor$ be the fractional part of $x$.
Then, for $\alpha$ irrational, $\operatorname{frac}(n \alpha)$, $n=1,2,\ldots$, distributes
randomly in $[0,1)$, ...
- 149k
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2
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Points attracting to 0 are dense in $\mathbb C$
I know that the following proposition is true, but at the moment I can't see how to prove it.
Define $f(z)=e^z-1$ for all $z\in \mathbb C$. Then $A:=\{z\in \mathbb C:f^n(z)\to 0\}$ is dense in $\...
- 3,055
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special flows and Rudolph's theorem
The Rudolph's theorem confirm the existence of a special representation of an ergodic flow on the Lebesgue space.
(In the book of I.P.Cornfeld entitled Ergodic theory).
My question is: what is the ...
3
votes
1
answer
193
views
An explicit formula for a flat metric compatible to certain polynomial vector field with center
Let $X$ be the following vector field on the plane:
$$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$
The vector field $ (X)$ has a non isochronous center at the origin.The ...
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votes
1
answer
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A non vanishing vector field compatible to a Riemannian metric
Assume that $(M, g)$ is a connected Riemannian manifold which is either open or is compact with zero Euler characteristic.
Is there a non vanishing vector field $X$ on $M$ such ...
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1
answer
699
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On the Birkhoff ergodic theorem for geodesic flows
Let $S$ be a closed surface endowed with a Riemannian metric of negative curvature and let $US$ be the unit tangent bundle. Let $\mu$ be the Liouville measure on $US$.
Let $f: US\rightarrow\mathbb{R}$...
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Any relationship between Viswanath's constant and the Khinchine-Lévy constant?
It is well-known that if ${\{{F_n}\}}$ is a random Fibonacci sequence then we have almost certainly $\lim \limits_{n\to\infty}\sqrt[n]{|F_n|}=\tau$ where $\tau\approx 1.554682275$ is Viswanath's ...
- 13.2k
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The definition of simple eigenvalue
This question was posted a long time ago on the mathexchange, but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer.
I am ...
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2
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Elementary cellular automata in stochastic modes
There are several ways to run a given elementary cellular automaton in a stochastic way:
by giving for each of the eight local configurations 000,100,010 and so on a probability by which the rule is ...
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answer
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Solution to a Sylvester equation with positive definite coefficients
Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$
\begin{align*}
C = A^TXA + B^TXB.
...
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answer
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An example of deterministic sequence from Terence Tao's blog
The following is taken from a post by Terence Tao on the Chowla conjecture and the Sarnak conjecture
:
Given a bounded sequence ${f: {\bf N} \rightarrow {\bf C}}$, define the topological entropy of ...
- 1,553
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1
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Searching for the proof of a certain claim in Arnold's ODE book from 1992
I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition).
On pages 12-13 he writes the following:
Consider the following ordinary ...
- 1,524
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votes
2
answers
387
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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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331
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Convex combinations of Bernoulli Measures
How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?
I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\...
3
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2
answers
382
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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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240
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length comparison on negatively curved surfaces
Suppose $g_1$, and $g_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K_{g_1}$ $<$ $K_{g_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy ...
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2
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322
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Scale random variables in a way they have equal probabilities of being minimal
I have several positive random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these ...
3
votes
1
answer
528
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limit cycles of dynamical systems
Consider $2D$ dynamical systems $X' = F(X)$ where $X=(x,y)$ is a 2-vector, and there is an equilibrium point at the origin. Let $L$ be the set of numbers $x > 0$ such that a limit cycle of the ...
- 1,223
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When a sequence of coefficients converges to the coefficients of a rational function $R$, when does the sequence $R_n$ converge uniformly to $R$?
Let $R$ be a rational function of degree $d$ mapping the Riemann sphere to itself:$$R(z) = \frac{a_d z^d + a_{d-1} z^{d-1} + \cdots + a_0}{b_d z^d + b_{d-1} z^{d-1} + \cdots + b_0}$$
where $a_d$ and $...
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Analytic ODE with complex time
Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)...
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answer
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Topological amenability of actions - forgetting topology
Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there ...
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What are the right mathematical tools / language to analyse complex networks over time?
In this article about human physiology as a complex network the authors say that:
"Lacking adequate analytic tools and a theoretical framework to probe
interactions within and among diverse ...
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1
answer
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Boundedness of orbits and limit sets
Let $T: {\bf R}^n \rightarrow {\bf R}^n$ be an homeomorphism and $x$ a point in ${\bf R}^n$.
The positive orbit of $x$ is the set $\{T^n(x) \mid n \in {\bf N}\}$ and its
$\omega$-limit set is the set ...
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answer
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Invariant measure of a subgroup
Let $G$ be an abelian group with a $G$-invariant metric $d$. Let $H$ be a countable dense subgroup of $G$. Let $\mu$ be a non-atomic $\sigma$-finite Borel measure on $G$ that is $H$-invariant. Must it ...
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Using a poset or directed graph as input for a neural network
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 ...
3
votes
1
answer
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Run-away functions
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. We say that f has the run-away property if for every compact subset $K\subseteq \mathbb{R}$ there is some positive integer N such ...
- 4,969
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Entropy-minimal subshifts
Consider a subshift $X \subset \left\{0, \ldots, M \right\}^{\mathbb{N}}$. $X$ is said to be entropy-minimal if every subshift $Y \subsetneq X$ satisfies that $$h_{\mathrm{top}}(Y) < h_{\mathrm{top}...
3
votes
3
answers
255
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Example of a Chaotic discrete dynamical system in dimension 2
I am looking for examples of discrete dynamical systems in dimension 2 that are :
1) Chaotic dynamical system in Devaney's sense in dimension 2 ?
2) Chaotic dynamical system in Li-Yorke sense but ...
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votes
3
answers
530
views
Free ergodic probability measure-preserving actions of the free group
Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group.
An action of $\Gamma$ on $X$ is:
essentially free if for all $g \in \Gamma \setminus \{e \}$,...
- 25.9k
3
votes
2
answers
394
views
Subshifts of finite type of guaranteed positive entropy
Let $\Sigma$ be a subshift of finite type (SFT) with the alphabet $\{0,1\}$, which is given by the set of forbidden words $\mathcal F$, all of length $N$.
Question. Is there a $\delta>0$ such ...
- 2,085
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1
answer
233
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Translation surfaces & integer multiples of $\pi$
Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011),
defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...
- 149k
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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.
user11178
3
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2
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556
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Equivalence of two definitions of Lyapunov exponents
I saw in articles two different definitions for Lyapunov exponents of a discrete dynamical system.
Let's consider a discrete dynamical system
$$
x_{k+1}=f(x_{k}),\quad x_{k}\in\mathbb{R}^{n},\quad k=...
3
votes
1
answer
512
views
pseudo-Anosov surface in three manifolds
A surface $S$ in a three manifold $M$ is pseudo-Anosov means if there exists a homeomorphism
$f$ over $M$ for which $S$ is $f$ invariant and $f$ is a pseudo-Anosov on $S$. For example,
$M$---- any ...
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2
answers
369
views
How to detect frequency?
Let $J$ be an arc in $\mathbb{S}^{1}\subset\mathbb{C}$ (no matter open or
closed) and $\alpha\in(0,2\pi)$ be an angle such that $\alpha/\pi$ is
irrational. Consider in $\mathbb{S}^{1}$ the sequence $...
- 303
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votes
1
answer
695
views
Ergodic decomposition of quasi-invariant measure
I have a reference request concerning Proposition 1.6 in the following article Link
The setting: Let $G$ be a locally compact, second countable group. Let $S = (S, \mu)$ be a Polish space. Assume we ...
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1
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306
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Equivalent definitions of strongly proximal action
Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...
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answer
237
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Does an “almost weakly mixing” transformation admit a non-null ergodic component?
Problem set up:
Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space.
We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost weakly mixing if for ...
- 4,832
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votes
2
answers
210
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Showing that the inverse of a function is approximately equivalent to $\frac{1}{n^{1/\alpha}}$
I'm currently working with someone on my PhD, and last week they asked me to check that a certain approximation holds as an exercise. Unfortunately, I couldn't figure out how to do it, and we've since ...
3
votes
1
answer
190
views
Exact solution to a periodic linear ODE sought
We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these ...
- 33
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votes
2
answers
135
views
Does this strong form of being almost 1-to-1 imply injectivity?
Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that
$\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
$\tilde{Y}$ is a ...
3
votes
1
answer
254
views
A question on dynamics on complex algebraic curves
Let $X$ be a complex algebraic curve, assumed to be connected, smooth and complete. Let $f: X \rightarrow X$ be a surjective morphism. Define a backward complete set for $f$ as a subset $S$ of $X$ ...
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174
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Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?
Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...
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