Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
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Does this sequence of Blaschke Product have rescaling limit $z-1$?
Background: The conformal conjugacy class of parabolic isometry of upper half plane $\mathbb{H}$ consists of $f(z) = z+1$ and $g(z)=z-1$.
Consider surjective proper holomorphic $F_n: \mathbb{H} \...
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Cohomology of foliations and closed forms along the leaves
Let $M$ be a manifold equipped with a codimension one, transversely orientable, regular foliation $F \subset M$. Let $\alpha\in \Omega^k(M)$ be a differential form on $M$ that is not closed on $M$ ...
- 2,792
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The Hausdorff codimension of singular matrices vs. the Hausdorff codimension of points with divergent trajectories
Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$.
(1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular}...
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Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term
The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
- 12.9k
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Recognize this metric? Do you have a name for this metric on the product of spheres?
Take the product $S^2 \times S^2$ of two two-spheres,
but perturb the product metric as follows.
Think of each $S^2$ as the unit two-sphere in Euclidean 3-space
in the standard way
so that for $p ...
- 8,336
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Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 63.9k
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Why "no wandering domain" fails in parabolic basin?
Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$
I am familiar with the proof: spread around ...
- 91.8k
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Is there an equivalent to the logistic map for a nonlinear path through some of the other nodules of the Mandelbrot set?
The logistic map can be related to the real axis of the Mandelbrot set, looking at the different cycle lengths as you pass through all the various nodules along the real axis. But there are other ...
- 63.9k
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Alternative proof of parabolic implosion
I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation.
Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
- 63.9k
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Conditions required for the orbit of a set of positive measure to cover state space?
Suppose $(X, \mathcal{M}, \mu, T)$ is a measure-preserving dynamical system with $T$ invertible.
I am wondering what properties the dynamical system would need to have in order for the following to be ...
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
- 12.9k
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Multidimensional intersection property
Consider the multidimensional annulus $\{(p,\theta)\} = \mathbb R^n\times\mathbb T^n$ endowed by the $1$-form $\omega=p\,d\theta$. A diffeomorphism $A$ of this annulus onlo itself is said to be exact ...
CommunityBot
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Chaotic dynamics of maps on unit square that are NOT Triangular
We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $...
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Existence of asymptotic sequence in ergodic measure-preserving transformations
Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \...
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Transitive map on a profinite group
Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
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(Exponential) mixing property for Gauss map — going from cylinders to intervals
I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step.
The Gauss map $T$, ...
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Local dimension of stationary measures for iterated function systems with an expanding map
Consider the iterated function system (IFS) $X_n$ on $I = [0,1] $generated by the functions $\Phi = \{f_1,f_2,f_3\}$ and the probability vector $P = (p/2,p/2,1-p),$ where:
$f_1,f_2: I\to I$, where $...
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For which values of $\mu$ is the Standard Map $f_{\mu}(x,y)=(x+y+\frac{\mu}{2\pi} \sin (2\pi x),y+\frac{\mu}{2\pi}\sin (2\pi x))$ non wandering?
Let $f_{\mu}(x,y)=(x+y+\frac{\mu}{2\pi} \sin (2\pi x),y+\frac{\mu}{2\pi}\sin (2\pi x))$, with $\mu > 0$ and considered in the cylinder $\mathbb{R}/\mathbb{Z} \times \mathbb{R} $. For which values ...
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Precise location of the Mandelbrot Bulb Attachment to the main Cardioid
Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...
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Restrict sigma algebra in measure-preserving system
Consider a measure space $(X,\mathcal{A},\mu)$ and a measure-preserving transformation $\phi \colon X\rightarrow X$, that is, $\phi$ is measurable and $\phi_*\mu = \mu$.
My intuition tells me that we ...
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Repelling invariant manifold of a discrete dynamical system
Given a $C^\infty$ map $Q: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following properties
$Q$ fixes the $x_1$-axis, i.e. $Q(x_1,0,\dotsc,0) = (x_1,0,\dotsc,0)$.
For $x_1$ in a neighborhood of $...
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What is the name for a point that is periodic to within $\varepsilon$?
Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...
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Asymptotic behavior of positive solution to nonlinear scalar field equation
It is well-known that the radial positive solution
$u=u(r)$ to nonlinear scalar field equation $$-\Delta u+u=u^p\text{ in } ~\mathbb{R}^d, 1<p<\frac{d+2}{d-2}$$ has the following asymptotic ...
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Singular continuous ergodic measures for the map $z \to z^2$
Where can I find the details of constructing singular continuous ergodic measures for the map $z \to z^2$ on the unit circle? I know that it was done by Furstenberg, but I could not find it explicitly ...
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Examples of discrete-space continuous-time dynamical systems
Something that I see occur repeatedly in my work is the need for formal notions of discrete-space continuous-time dynamics — these are generally realized as digital oscillators that are interact using ...
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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:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...
- 4,713
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Positiveness of the largest Lyapunov exponent
Let $\alpha\in \mathbb{R} / \mathbb{Q}$, let $A(x)$ be the $2$-by-$2$ matrix
$$
A(x)=\begin{pmatrix}
\dfrac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\dfrac{1}{{\lambda}} \\
\dfrac{...
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Dynamical properties of cellular automata of small diameter
Let $f\colon\{0,1\}^k\to\{0,1\}$ be a function, $j$ an integer, and define the cellular automaton $F\colon\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ by $F(x)_i=f(x_{i+j},\dotsc,x_{i+j+k-1})$.
I wonder ...
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Stability Problem in a Nonlinear Dynamical System
Consider the nonlinear dynamical system given by the following differential equations
\begin{cases}
\dot{x} = y, \\
\dot{y} = x - x^3 - \gamma y + \delta x^2 y.
\end{cases}
I want to demonstrate that, ...
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Convergence of the Cesàro mean of iterated continuous functions
Does anyone have a counter-example of the following statement :
Let $f : [0;1] \to [0;1]$ a continuous function w.r.t. the usual topology. Let $A_n(x) = \frac{1}{n} \sum_{k=0}^{n-1} f^k(x)$ for $n \ge ...
- 12.9k
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Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
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Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
CommunityBot
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Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
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Number of periodic points of subshift of finite type
Let $(X, \sigma)\subset (\{0, 1, 2, 3\}^\mathbb{N},\sigma)$ be a subshift of finite type. Let $P_n$ be the set of $n$-periodic points. If $|P_n|=2^n$ for all $n\ge 1$, then it is true that $(X, \sigma)...
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Borel complexity of the set of generic points for an invariant measure in a minimal system
I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is ...
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Is this card shuffling process weakly mixing?
Consider the following continuous analogue of a card shuffling process:
Let $Y_i, Z_i$ ($i \in \mathbb Z^+$) be sequences of jointly independent uniformly distributed random variables on $[0, 1]$. ...
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Is a random circle rotation weak mixing almost surely?
Consider the random circle rotation $x \to x + Z \text{ mod 1}$ on $([0, 1], \text{Lebesgue})$ where at each rotation, $Z$ is uniformly distributed on $[0, 1]$ and independent of previous rotations.
...
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Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
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Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
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Do sets of big returns contain sets of returns?
We say a subset $E$ of $\mathbb{N}$ is a set of returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B}$ with $\mu(A)>0$ such that $E=\{n\in\mathbb{N};\...
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Great literature on discrete dynamical systems and/or qualitative theory of difference equations
I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some ...
- 63.9k
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Stability of a nonlinear dynamical system with non-elementary dynamics
I am trying to prove stability and get a non-asymptotic upper bound on the convergence rate of a nonlinear discrete-time dynamical system, whose dynamics are stated in terms of the (non-elementary) ...
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Proving positive invariance
I need to prove that set $D$(A picture for Set $D$) given by
$$D=\{(x,y):0\leq x\leq L_0,~0\leq y\leq X_0,~0\leq x+y \leq R_0\}\subseteq \mathbb{R}_+^2$$ of the system:
$$\dot{x}=k_1(R_0-x-y)(L_0-x)-...
- 6,367
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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 ...
- 6,367
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Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
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On reproducing the Poincare section figure in a paper by Sato, Akiyama and Doyne Farmer [closed]
I am trying to reproduce Figure 1 in the paper "Chaos in learning a simple two-person game" (English) Proc. Natl. Acad. Sci. USA 99, No. 7, 4748-4751 (2002) (MR1895748, Zbl 1015.91014), by ...
- 6,367
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On the maximal difference between points in orbit
Let $(X, T, \mu)$ be an ergodic measure preserving system with finite measure, and $f \in L^{\infty} (X)$.
Define the maximal orbit deviation function $D_f: X \to \mathbb R$ by
$$D_f := \sup_{n, m \...
- 6,155
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Analysis of sensitivity to initial conditions in dynamic systems
Consider the iterative function defined by:
$$
x_{n+1} = f(x_n)
$$
where $x_0\in [0, 1]$ and
$$
f(x) = \sin\left(\pi \left(b^{rx(x-1)}\mod 1 \right)\right)
$$
with $b, r > 0$. We aim to demonstrate ...
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For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,155
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What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
- 6,155