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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(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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65
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Chain recurrent points of a gradient-like system
Let $X$ be a compact metric space and $f:X\to X
$ homeomorphism. Let $V:X\to \mathbb{R}$ be a Lyapunov function for $(X,f)$ (continuous function such that $(\forall x\notin Fix(f))\ \ V(f(x))<V(x))...
- 141
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73
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Nonintegrable classical dynamical systems and deterministic chaos
I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...
- 141
2
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191
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Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)
In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ...
- 356
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93
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Existence of infinite horizon values in dynamic programming
I am working through the book "Foundations of Stochastic Inventory Theory".
One of the results in the book is Theorem 11.2. The background to this theorem is as follows.
Given finite state ...
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86
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Gronwall-type bound for a mix-effect inequality?
This popped up in my research: we have the following mix-effect inequality that $\forall T \geq 1$
\begin{equation}\tag{*}
Y(T) - \frac{1}{100T^2}\int_1^T[\alpha^2 + e^{-(T - t)}]Y(t)dt
\lesssim \...
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91
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Examples of minimal almost 1-to-1 extension of torus having positive entropy?
It is well known that Toeplitz subshifts are minimal almost 1-to-1 extensions of an odometer, and that some of these subshifts have positive entropy. Thus, even if a system is an almost 1-to-1 ...
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110
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A quantity associated to a foliated manifold and its non-commutative interpretation
Let $M$ be a compact $n$-dimensional manifold. Assume that $F$ is a $k$-dimensional foliation of $M$.
The graph $G(M,F)$ of this foliation is a $(n+k)$-dimensional manifold. We recall its definition:
...
- 356
2
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0
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61
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Limit set for IFS has either empty interior or dense interior
Let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R^n$ be contracting affine maps. By the theory of iterated function systems, there is a unique minimal compact $K\subseteq\mathbb R^n$ such that $K=f_1(K)\cup\...
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Surreal numbers and the Collatz iteration as a game?
Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$.
Each number $n$ represents a game played by left $L$ and right $R$:
$$n = \{L_n | R_n \}$$
The rules ...
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91
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Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
This question, comes out of a question in MSE and I hope it is ok to ask it here:
Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
...
- 3,064
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131
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Rotation set vs existence of rotation number
Let $f\colon \mathbb{S}^{1}\to\mathbb{S}^{1}$ be a continuous function of degree 1 and $F\colon \mathbb{R}\to \mathbb{R}$ a lift of $f.$ One can define, for each $x\in \mathbb{R}$, the rotation number ...
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141
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Ergodicity of a dynamical system on the $n$-sphere
Let $v$ be continuous and nowhere-vanishing vector field tangent to the $n$-sphere $\mathbb{S}^n$ (hence $n$ is odd, w.r.t the Hairy-Ball Theorem). Let $x$ be a trajectory on $\mathbb{S}^n$, defined ...
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89
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Entropy of flow and time-1 map
Let $\Phi=(\phi_t)_{t\in \mathbb{R}}$ be a continuous flow on a compact metric space $X$. Let $\mu$ be a $\phi_1$-invariant measure. Then it is not hard to verify tht $\int_{0}^{1} \phi_t\mu dt$ is $\...
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Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?
Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...
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227
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Poincaré Recurrence Theorem for flows
Someone knows a book that have the proof of the Poincaré Recurrence Theorem for flows when the set have finite volume and when don't?
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83
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Center-stable manifold theorem on manifold with boundary
I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
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184
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Question on "semi-linear" dynamical systems
I am interested in understanding the convergence properties of a dynamical system at zero. We know that a dynamical system of the form $x_{k+1} = Ax_{k}$ where $A$ is a symmetric positive definite ...
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158
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Finding invariant closed subspace which are also subgroups for the action of $\text{SL}(2, \Bbb Z)$ on $\Bbb R^n\times \Bbb R^n$
I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as
$$
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\cdot \big(v,\,w\big)\...
- 203
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A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
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203
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Gurevich's entropy and topological entropy in a countable Markov shift
Good afternoon, I understand that Gurevich's entropy and topological entropy coincide when the countable Markov shift is topologically mixing (right?)
Does anyone know of an example or a reference ...
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125
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Does $\sum_{i\le k}\mathrm{frac}(n\alpha_i)<1$ hold infinitely often?
For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist ...
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100
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About geometric quantisation and application to real system
Quantisation is a important step to properly define a quantum system from a classical one. In a nutshell :
On a symplectic manifold $(M,\omega)$ and an algebra of function $f$ on $M$, one defines an ...
- 4,361
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165
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Run-away Volterra operator
For a continuous function $k:[0,1]^2\to \mathbb{R}$, let $A$ be the generalized Volterra integral operator on $C([0,1],\mathbb{R})$ defined by
$$
A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds, \quad t \in [...
- 5,405
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Understand the condition of transcritical bifurcation (Crandall-Robinowitz) geometrically
Consider the dynamical system $\dot{x}=F(x,\lambda), x\in\mathbb{R^n}$, and let $F(0,\lambda)=0$ for some neighborhood of $\lambda_{0}$, the transcritical bifurcation arises if we have $w\frac{\...
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On invariant cones of the Katok map
I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
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135
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Example shows that entropy is not upper semi continuous
Let $(X, \beta, \mu)$ be probabilty space of compact space $X$. Let $T:X \rightarrow X$ be continuous function, and expansive. It is well known that entropy $\mu \mapsto h_{\mu}$ is upper semi ...
- 1,043
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110
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Example of overtwisted contact manifold with finitely many periodic Reeb orbits
Are there examples of overtwisted manifolds with only a finite number of periodic Reeb orbits?
An example is given by the irrational ellipsoid in $(\mathbb{R}^4,\omega_\text{st})$, which is not ...
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Equivalence between Gibbs measures and conformal measures
I was reading an article about Gibbs measures, but the author defines Gibbs measures in a different way than the usual (which is done by using conditional expectations). The way that he defines I have ...
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184
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An analytic vector field on $S^3$ whose all orbits are dense (à la Seifert conjecture, 2)
Does there exist a real analytic vector field on $S^3$ all of whose orbits are dense? The second paragraph of page 285
Of this paper says that there is a vector field whose almost all orbits are dense:...
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100
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Homomorphism or derivation conserving irreducibility
Let $R$ be a integral domain and $\phi$ be an automorphism of $R$. For a given element $x \in R$, we consider a sequence $(\phi^n(x))_{n=0}^{\infty}$.
I wonder if there is any related theory to ...
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136
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Irrational closed orbits of vector fields on $S^2$ (Limit cycles and trace formula)
Motivations: We first introduce our motivations: We wish to find an operator-theoretical interpretation for the number of limit cycles of a polynomial vector field on the plane. We quote the ...
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258
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Why do unstable manifolds of two close point intersect each other in Baker map?
Let $M$ be $S^1 \times [-1,1]$, $f$ a baker map on $M$ and for $p, q \in M$ consider $W^s_p$ the stable manifold in $p$ (i.e. the set of points whose forward orbit tend to the forward orbit of $p$) ...
- 1,043
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150
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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 ...
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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 $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
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Baker map-like problem
Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...
- 2,069
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108
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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 ...
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151
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$C^1$-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 $W$),...
- 153
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105
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Bifurcations due to a nonlinearity parameter
Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...
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166
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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 ...
- 831
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139
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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 ...
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Stability of dynamical system via Lyapunov
I have a dynamical system which has the following form:
$\dot x=\mathcal F_1(m_1)x+\mathcal F_2(m_2)x$.
My objective is to find the parameters $m_1$ and $m_2$ via LMI (linear matrix inequality) using ...
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140
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Generalized right Perron-Frobenius eigenvector with rationally independent coordinates
Suppose you are given a directed graph $G=(V,E)$ which is strongly connected, i.e. for every two vertices $u,v \in V$ there exists a directed path between them. Consider the corresponding edge shift ...
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228
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coboundary in Dynamical system
a question about the definition:
given measurable dynamic system $ ( X, \mathcal{B}, T, \mu)$, $ \mu \circ T^{-1}=\mu$ ergodic.
$\phi \in L^{\infty}$ is coboundary with $\int \phi d\mu =0 $, means ...
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90
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Counting orbits of the standard map
Consider the standard map. Might it happen that for some nonzero parameter value $K$ and for some positive integer $q$ that there exist an infinite number of periodic orbits having period $q $ I ...
- 8,336
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92
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Does there exist a leaf of this holomorphic foliation with non trivial holonomy?
Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$.
Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, ...
- 356
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0
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293
views
Average of irrational flow on the torus
Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$
defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that
$$ \lim_{T\...
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Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems
I have a query regarding two equalities in the lemma in the book.
But first I'll provide two definitions that one needs for this lemma.
Definition 4.2.4: Consider the vector field $f(x,t)$ with $f:\...
- 1,594
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59
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Stability of ODEs with exponentials in the vector field
What is known about fine stability properties of ODEs of the following kind?
$$ \dot{x} = Ax + b + \phi(x),\quad x\in \mathbb{R}^d ,$$
where $d\geq 1$; $A$ is a constant matrix with all e.v. having ...
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Can this construction generate bounded aperiodic functions?
This question is based on this old MathOverflow question: How this set of functions is ordered?
In that question, Vladimir Reshetnikov asked about a class $S$ of functions $f:\mathbb{N}\to\mathbb{N}$ ...
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