Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
1,549 questions
1
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0answers
41 views
Global solution of second order ODE defined on riemannian manifold
Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
1
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1answer
135 views
On the 2002 paper “Dynamics of polynomial automorphisms of $\mathbb{C}^k$” by Guedj and Sibony
I desperately need to read the paper [1] before meeting a would-be supervisor, but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, not even one ...
1
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0answers
93 views
Quasi homomorphism from integers to reals
Reference : Lemma 5.3 from "Groups acting on circle" by Etiyenne Ghys.
Let $f:\mathbb{Z}\rightarrow \mathbb{R}$ be a quasi-homomorphism, i.e $|f(a+b)-f(a)-f(b)|\leq D$ $\forall$ $a$ and $b$ in $\...
4
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0answers
103 views
Absolute oscillator in Langton's Ant
We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...
2
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0answers
85 views
A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory
The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
1
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1answer
69 views
Does the following percolation model have a name?
Consider the following model for percolation in an infinite graph: each vertex has a certain region (set of vertices) associated with it, which at the beginning contains only the vertex itself, and ...
5
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0answers
200 views
Geometric bang-bang theorem for nonlinear optimal control
The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems ...
1
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1answer
75 views
Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive
Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.
Let $S : X → X$ and ...
4
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0answers
109 views
+50
mixing time of Young tower
Given Young tower $(\Delta, m, F)$ with base $\Delta_0$, return map $R$, distortion of $F^R$, g.c.d($R$)$=1$.
This system is exact, so mixing, and implies:
given fixed $\epsilon$, exists $N$, s.t. $...
2
votes
2answers
192 views
Ergodic theorem and products
If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that
$$ \lim_{n \rightarrow \infty} \frac{f_n}{...
0
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1answer
101 views
Probabilistic approach for cellular automata
Few months ago my scientific adviser asked me to use probabilistic ideas in such problem :
Consider a matrix NxN. Each element of matrix is a number 1 or 0. We may change all elements of this matrix ...
4
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0answers
56 views
Weighted distribution of irrational rotation
Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
2
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0answers
65 views
Random Young Tower
the following setting is the same as Baladi's paper "ALMOST SURE RATES OF MIXING FOR I.I.D. UNIMODAL MAPS" (2002), define random Young tower ($\Delta_{\omega})_{\omega\in \Omega}$ sharing the same ...
1
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0answers
88 views
Can a local extremum of a function be an asymptotically stable equilibrium of corresponding gradient dynamics?
Let's first describe the setup: we consider a(say smooth enough) function $f: \mathbb{R}^d \to \mathbb{R}$ and write it as $(x,y) \to f(x,y)$, where $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
2
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0answers
61 views
How to compute expansion factors for hyperbolic rational maps?
It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...
7
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2answers
305 views
Is it possible to prove unboundedness of 3rd order ODE?
Consider the 3rd order ODE
$$\dddot{x}+A\ddot{x}-\dot{x}^{2}+x=0$$ where $\dot{x}\equiv \frac{dx}{dt},\ddot{x}\equiv \frac{d^{2}x}{dt^{2}}, etc$. $A$ is a constant.
If we multiply this equation by $\...
2
votes
1answer
94 views
Irreducible subcontinuum of Lorenz attractor?
In my first question Lorenz attractor path-connected?, some are saying the Lorenz attractor $\mathscr L$ is not path-connected.
But suppose $x$ and $y$ are two points in different path components of ...
9
votes
1answer
510 views
Lorenz attractor path-connected?
Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.
EDIT: The answer below is unsatisfactory, and possibly ...
7
votes
1answer
97 views
Under which conditions do ellipsoids have a focal property?
Given an ellipsoid $E$, we consider the trajectories of light inside $E$ assuming that $\partial E$ would be a mirror. In other words, let a light trajectory be piecewise linear path $\gamma:[0,\infty)...
2
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1answer
93 views
Increasing union of embedded submanifold is immersed manifold
While working on the proof of the stable manifold theorem, I came across a problem that I'm not able to really grasp. Given some Anosov map $f: M \to M$ on a compact Riemann manifold $M$, one can ...
5
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1answer
152 views
Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$
If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...
1
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2answers
85 views
Quantitative bound on irrational rotation recurrence time
Given an irrational $a$, the sequence $b_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified.
Given a point $p$ in $\mathbb ...
1
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0answers
67 views
$C^1$-foliation are absolutely continuous
Brin & Stuck defined in Introduction to dynamical system two notions:
That of a absolutely continuous foliation : given any foliated chart $U$ on some Riemannian manifold $M$ (with foliation $W$),...
10
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0answers
205 views
Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?
Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
0
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0answers
20 views
Splittings for generic flows on a vector bundle
Question: Consider a smooth vector bundle $\pi:V\to B$ and the space $\mathcal{F}$ of $C^k$ linear flows $\Bbb{R}\times V \to V$ endowed with the strong $C^k$ Whitney topology. Is it true that for all ...
1
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0answers
27 views
Rotation rates for a linear flow on a vector bundle
The following linear ODE on $\Bbb{C}$
$\dot{z} = (a + i b)z$
has solutions $z(t) = e^{(a+ib)t} z(0)$. Hence the real part $a$ captures expansion rate and the imaginary part $b$ captures rotation ...
6
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0answers
70 views
Gradient flows on Hilbert manifolds
I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded.
To be more precise, a ...
2
votes
2answers
53 views
Lower bound of positive entropies of automorphisms on tori
Let $A$ be an automorphism on tori $\mathbb{T}^d$. It is well known that the topological entropy
$$
h(A)=\sum_{\lambda} \max\{0, \log|\lambda| \}
$$
where $\lambda$ goes through all eigenvalue of $A$ ...
3
votes
1answer
83 views
On Krieger's Embedding Theorem
This is Theorem 10.1.1 of Lind & Marcus's book, An Introduction to Symbolic Dynamics and Coding. They say that is "straightfordward" to go from
Let $X$ a shift of finite type and $Y$ a mixing ...
0
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0answers
56 views
What's equal the below power nested radical?
The same copy of this question is montioned here in SE with no convinced Answer , I want to know what MO will say about the below nested radical as a power form
it is well known that
$$\frac{2}{\...
3
votes
0answers
58 views
Flow of zeros in the shifted exponential generating function?
Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...
1
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0answers
56 views
Time-varying perturbations of continuous-time hyperbolic orbits
My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then ...
1
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0answers
84 views
Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$
I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently:
Let $X$ be a compact $k$-...
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0answers
85 views
Bifurcations due to a nonlinearity parameter
Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...
9
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1answer
171 views
Topological amenability vs amenability of an action
Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...
1
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0answers
84 views
stochastical stable
Given dynamic $f: S^1 \to S^1$ with Lebegue measure $dm$ on $S^1$. Assume it has unique SRB probability measure $\frac{d\mu_f}{dm} dm $. Given left shift space $([-\epsilon, \epsilon]^{\otimes \...
7
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0answers
167 views
Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?
Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
2
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0answers
102 views
A generalized Furstenberg's $\times p,\times q$-conjecture
Let $p,q$ be two positive integers such that $\frac{\log p}{\log q}\notin\mathbb{Q}$. Furstenberg's $\times p,\times q$ conjecture says that the only ergodic nonatomic $\times p,\times q$-invariant ...
1
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0answers
51 views
Relations between Omega, Local Indicable and Right Orderable groups
We know that the set of Right-Orderable groups $RO$, is contained in the set of $\Omega$- groups (Read it from "A Note on Group Rings of Certain Torsion-Free Groups" Burns-Hale).
A Group $G$ is a ...
1
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0answers
40 views
Solutions of nonlinear equations with multiple parameters
In the course of analysing a particular three dimensional nonlinear dynamical system, I find the need to solve a nonlinear equation of the form:
$$ \mathcal{M}(x, \lambda) := x - f(x, \lambda_1, \...
7
votes
1answer
176 views
Distribution of $\{cn^a\}$
Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) ...
11
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1answer
317 views
Do solenoids embed into Möbius strips?
I found a strange attractor which looks a lot like a solenoid. The attractor continuum is the closure of a continuous line which limits onto itself, and it is locally homeomorphic to Cantor set times ...
3
votes
2answers
179 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 \}$,...
1
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0answers
82 views
size of local strong stable manifold is measurable
Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
7
votes
1answer
466 views
Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?
Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
2
votes
0answers
74 views
Is the Mandelbrot set weakly self-similar?
A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...
1
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1answer
59 views
Lotka Volterra existence of Caratheodory solution
I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$
has a unique Caratheodory solution ...
0
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0answers
141 views
About the limit of transverse intersection
Let $n$ be a fixed positive integer, and let $W^{s}(R_{q})$ and $W^{u}(R_{q})$ be the stable and unstable manifolds of a fixed point $R_{q}$ of a discrete 2-D mapping $f_{q}$. Notice that the sequence ...
4
votes
2answers
226 views
Newton method and Siegel disks
I am looking for a degree 3 polynomial $P$ whose associated Newton's method $z \mapsto z - P(z)/P'(z)$ has a Siegel disk. Is there an explicit example of such polynomial $P$?
8
votes
3answers
202 views
Random reflections unexpectedly produce banded distributions
Start with $p_1$ a random point on the origin-centered unit circle $C$.
At step $i$, select a random point $q_i$ on $C$, and a random mirror line
$M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
