Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
854 questions with no upvoted or accepted answers
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Uniform convergence for pointwise ergodic theorem
Let $K$ be a compact set of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(K,\mathbb{R}^n)$ be a speed field on $K$ such that for any initial condition $x_0\in K$, the following dynamical system
\...
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Discrepancy estimate for $3$-interval exchange or $n$-interval exchange map, $n\geq 3$
We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called ...
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Proof of property for Fiedland entropy
I am working with Friedland entropy and there is a proof I cannot figure out how to do.
Friedland entropy is defined for $\mathbb{Z}^k$ continuos actions $\mathcal{T}$ on a topological metric space $X$...
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124
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Can non wandering sets be connected?
I know that the alpha and omega limit sets of a flow on a compact connected invariant subset of a manifold must be connected and these limit sets are contained in the non wandering set.
My question is ...
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Distribution of the values of the product $\prod_{k=1}^n |1-e(k\alpha)|$ for an irrational number $\alpha$
For an irrational number $\alpha$, let $e(k\alpha):=\exp(2k\pi i\alpha)$. It was indicated in this thread that
$$\limsup_{n \to \infty} \prod_{k=1}^n |1-e(k\alpha)|=\infty$$
(actually a weaker result ...
- 1,565
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92
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A polynomial formed from the roots of another polynomial ad infinitum
Let $P(x)$ be a monic polynomial of degree $d$ with complex coefficients. Let $r_1(P),r_2(P),\dots, r_d(P)$ denote the set of roots, ordered so that $|r_1(P)| \leq |r_2(P)|\leq\dots\leq |r_d(P)|$. ...
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369
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Definition of generic point
I am trying to read a paper named D.S. Ornstein, B. Weiss, Subsequence ergodic theorems for amenable groups, Israel J. Math. 79 (1) (1992) 113–127, doi:10.1007/BF02764805. In this paper the authors ...
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55
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Dense stratification of a separable Hilbert space
Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
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What kind of differential equation problem is this?
I have a function $f(x,t;k)$, a starting point $x_0$, a gradient $\operatorname{Grad}(f)$, and an equilibrium point $x^*$. I can adjust the parameter $k$ freely, and I know that for any $k$ the ...
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Systems with trivial cohomology
If $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ is an irrational number, then the rotation $X = (S^1, +\alpha)$ has "trivial" cohomology i.e.
$$H^1(X) := C(X,\mathbb{C})/\beta C(X,\mathbb{C})$$
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Probabilistic Approximation of non-linear Dynamical System by Diffusion Process
Setting
Suppose I have a discrete dynamical system given by:
$$
X^{n+1} = f(X^{n})
\qquad X^0 =x
,
$$
where $f$ is some diffeomorphism from $\mathbb{R}^{d}$ to itself, and some $x \in \mathbb{R}^d$. ...
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For which parameters is the logistic map chaotic?
The logistic map is $f_\lambda(x)=\lambda x (1-x)$. It is known that the map is chaotic for $\lambda=4$ (on $[0;1]$) and also for $\lambda>0$ (on some hyperbolic subset of $[0,1]$). My question is:
...
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Derivatives in unusual support domains
Originally posted on Math.StackExchange, here, but I was advised to post it on MathOverflow as it is a research question. Now two final, great answers have been posted, see on Math.StackExchange.
I ...
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$||g_n||_{\infty} < \delta_{n-1}(g)$
It may be a simple question to post it here, but I posted this question in the Math Stack Exchange forum and no one answered me.
Let $E$ be a (possibly infinite) alphabet and consider $X = E^{\...
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Unique poine in holonomies
Let $\Lambda$ be Axiom A for $C^{1+\gamma}$ $f$. I am reading this paper. I have a problem to undestand holonomies. The holonomy mapping
$$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...
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Existence of the eigenvalue of the dual operator of the transfer operator
In the passage that I marked in green apparently the author uses a relationship between fixed point and eigenvalues. The result that I know of to ensure the existence of this eigenvalue requires that ...
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87
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Semi-conjugacies between interval and circle maps
There are examples of self-maps of the circle which are semi-conjugate to self-maps of a compact interval. A famous one is the covering map $z\mapsto z^2$ of the unit circle which is semi-conjugate to ...
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393
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Topological entropy of logistic map $f(x) = \mu x (1-x)$, $f:[0,1] \to [0,1]$ for $\mu \in (1,3)$
As stated in the question, I want to find the topological entropy of the logistic map on the interval $[0,1]$ for a "nice" range of the parameter $\mu$, namely $\mu \in (1,3)$. I think the fact that $...
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117
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Global stability question for system with a unique locally-asymptotically-stable steady state
I have an ordinary differential system of dimension 3 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is strictly-positively invariant and bounded.
Now, ...
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Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
- 5,405
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On the measure of regular and chaotic regions in a phase space
Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...
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324
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Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
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184
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Oja's rule gives unit eigenvectors
Does Oja's rule for normalized Hebbian learning always result in a unit eigenvector which corresponds to the largest eigenvalue? Or are there any specific conditions or assumptions under which this is ...
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What is a self-consistent equation in percolation theory
I was reading papers about percolation theory in which I was confused by the expression "self-consistent equation", for example in Temporal percolation in activity-driven networks. I read some ...
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Convergence of a positive sequence controlled by a difference inequality involving quadratic map
I have a sequence $\{x_n\}_{n\ge 0}$ with $x_0>0$, controlled by the difference inequality: $$x_{n+1}\le ax_n^2+b$$ where, $a,b>0$. Had $b$ been $0$ and $a<1$, I would find $x_n\to 0$ as $n\...
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164
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What is the attracted locus in this recursion?
Consider $R: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $R(a,b) = (b,2.5b-a).$ Let $p_0 = (x_0, y_0)$ be arbitrary and $p_{i+1} = R(p_i).$ Most starting points $p_0$ give a divergent path. One ...
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Geodesic flows on affine two-dimensional tori
I am looking for a reference here. Consider a two-dimensional torus $\mathrm{T}^2 =S^1 \times S^1$ together with an affine structure, that is a $(Aff(\mathbb{R}^2), \mathbb{R}^2)$-structure. Such a ...
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Non-linear mixed-degree integro-differential equation
In my research i came across this certain ODE and I've reduced it to this form:
\begin{equation}
\sum_{i=1}^N c_i \frac{\partial e_i(t)}{\partial t} + \sum_{i,j=1}^N \sigma^i(\sigma^j)^te_i(t)\int_0^t ...
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Equivalence classes on an ordered Bratteli diagram
Let $S$ be the adic transformation preserving a probability measure $\mu$ on the set $\Gamma$ of infinite paths of a $\mathbb{N}$-graded ordered Bratteli graph.
For every $n \geq 0$ define the ...
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114
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Transformations whose product with the odometer are ergodic
Let $T$ be an invertible ergodic transformation on a Lebesgue space $X$ and $O$ be the dyadic odometer on $(0,1)$. Is it true that $T\times O$ is ergodic if and only if $T^{2^n}$ is ergodic for every $...
- 2,560
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An almost periodic point must be a unifomly recurrent point?
$(X,G)$ is a topological semi-group action, $G$ is a topological (abelian) semigroup, and $X$ is a Hausdorff space.
$x\in X$ is called almost periodic of $(X,G)$, if for any neibourhood $U$ of $x$, ...
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A question from One Dimensional Dynamics book by De-Melo and van-Strien
In One Dimensional Dynamics, on page 27 I don't understand how does $(1.7)$ follow; anyone care to explain this to me?
Thanks in advance.
I am adding some information from the text below:
We are ...
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112
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Integrability of the orthogonal complement of a holomorphic vector field on $\mathbb{C}^{2}$
Assume that $$\begin{cases}\dot x=P(x,y)\\\dot y=Q(x,y)\end{cases}$$ is a non vanishing holomorphic vector field on an open subset $U$ of $\mathbb{C}^{2}\simeq \mathbb{R}^{4}$. It defines a two ...
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On a certain set of probability measures on a shift
Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2.
Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where $...
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128
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A heat equation approach to the perturbation of vector field with center
Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.
We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=...
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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 ...
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149
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Invariant mesures for expanding maps of the circle
Is there any characterization for the support of T-invariant measures? where T is a C¹ expanding map of the circle i.e. T'(x)>Lambda>1 for all x in the circle.
I know there are periodic and total ...
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173
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Product of two foliations
1.What is an example of a manifold $M$ with two foliations $F$ and $F'$ which are not topological equivalent but the product foliations $F\times F$ and $F'\times F'$, as foliations on $M\times M$, ...
- 356
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107
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Asymptotic pseudo orbit of an action
Let $G$ be finitely generated group (i.e $G= <S>$ $S=\{ s_1, ...,
s_n\}$) and $\varphi:G\times M\longrightarrow M$ is an action then
$f:G\longrightarrow M$ is called $\delta$- pseudo orbit if $...
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126
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Rational dynamical system with nonnegative paramaters
let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
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Does the number of zero eigenvalues correspond to the dimension of equilibrium manifold for nonlinear system?
Consider nonlinear dynamical system with $m \times m$ Jacobian matrix $J(x)$ that has $k$ zero eigenvalues for all $x$. The rest $m-k$ eigenvalues have negative real part for all $x$. Is that true ...
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Is an odometer action on a product space always conjugate to its inverse by an involution?
This is a follow on question from
Is an non-singualr invertable ergodic transformation on a measure space isomorphic to its inverse?
Given a measure $\mu$ on the product space $X = \prod_{i=1}^\infty ...
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88
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Question on center-stable manifold
Assume that you have a gradient system smooth enough and a fixed point $x_{0}$. Is it true that if $x_{0} \in \omega(x)$ then $\gamma^{+}(x)$ must intersect the local center-stable manifold of $x_{0}$?...
user31317
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117
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excplicit formula of iterates of an interval exchange
Let $f$ be an interval exchange transformation of $[0,1]$. Is there an explicit formula giving $f^k(0)$ in function of $k$?
If not, are there particular cases where this formula is simple? (except ...
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255
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Express measurable entropy in terms of Fourier coefficients of the measure
Let $S^1$ be the unit circle and $T:S^1\to S^1$ be a continuous map. Suppose $\mu$ is a $T$-invariant Borel probability measure on $S^1$, that is, $\mu(T^{-1}A)=\mu(A)$ for every Borel subset $A$ of $...
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320
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Prove that origin is globally exponentially stable with Lyapunov Indirect Method
I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially stable?...
- 11
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61
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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 ...
- 101
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129
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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 ?
...
- 27
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182
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Degree of freedom restricted by inequalities
Motivational example
Consider a polyhedral graph $G$. A realization of $G$ is given by a convex polyhedron which is - essentially - characterized by the angles between the edges emanating from each ...
- 9,720
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183
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Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
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