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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A generalization of competitive systems
We consider the following standard partial order relation on $\mathbb{R}^n$:
We say $X=(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots,y_n)=Y$ iff $\sum_{i=1}^k x_i \leq \sum_{i=1}^k y_i,\quad \forall k: 1\...
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Continuous-time extension of a discrete dynamical system
It is clear that one can obtain a discrete dynamical system from a continuous one, but is the converse possible if the system is "nice"?
Define the discrete-time dynamical system on $\mathbb{R}^d$ by
...
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Proof that dynamical systems with bounded Kolmogorov complexity can't emulate all Turing machines
Motivation:
During a discussion with neuroscientists the question arose as to whether the human brain may emulate any Turing machine. If we assume that animal brains may be modelled as deterministic ...
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Limit of alternated row and column normalizations
Let $E_0$ be a matrix with non-negative entries.
Given $E_n$, we apply the following two operations in sequence to produce $E_{n+1}$.
A. Divide every entry by the sum of all entries in its column (...
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Supremum of $ a_n = a_{n-1}^3 - a_{n-2} $
Let $a_1=0$ and let $ - \ln(2) < a_2 < \ln(2) $
Define
$$ a_n = a_{n-1}^3 - a_{n-2} $$
Then
$$ \sup_{n>2} a_n = a_2 $$
And
$$ \inf_{n>2} a_n = - a_2 $$
How to prove that ?
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Examples of particle systems with higher-order collisions
In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), ...
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spectrum of multiplicative morphisms
Let $T:[0,1]\to[0,1]$ be a continuous map, which is neither surjective nor injective. Put
$$
C([0,1])\ni f\mapsto \Phi(f):=f\circ T\in C([0,1]).
$$
Notice that, under the above conditions, $0\in\sigma(...
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Finding monochromatic subset of triangular lattice
Let
$$ \Lambda = \{(x,y)\in\mathbb{N}^2:y\geq x\} $$
the upper triangular lattice and $d:\Lambda\to\{1,\dots,c\}$ a coloring (i.e. an arbitrary function) on $c$ colors. Let $k\geq2$. I am looking for $...
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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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How often can we renormalize unimodal maps?
A unimodal map is function $f:[0,1]\rightarrow [0,1]$ such that there exist $c\in (0,1)$ such that $f$ is strictly increasing on $[0,c)$ and strictly decreasing on $(c,1]$.
A unimodal map f is ...
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Fine structure of bifurcation diagram of logistic family
I'd like to learn about the period-doubling route to chaos of the logistic family $f_\lambda(x)= \lambda x (1-x)$ and got interested in the fine properties of the bifurcation diagram of this family as ...
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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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Example of zero Lyapunov exponentes
Assume that $(T, A)$ is a linear cocycle such that $T:X\rightarrow X$ is a homemorphism on compact metric space $X$ and $A:X\rightarrow SL(2, \mathbb{R})$ is a continuous function.
We say that an ...
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On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra
Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$.
Obviously the singularities of this systems are just the idempotents of the ...
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Closure of the periodic points in the logistic family
My question is about the closure of the periodic point for the logistic family $f_\lambda(x) = \lambda x(1-x)$ of maps of the interval $[0,1]$.
Is there an explicit parameter $\lambda$ for which the ...
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Are there existing approaches to estimate inherent dynamics of an unknown system?
Consider a set of measured discrete-time signals $y_t \in \mathbb{R}^L$, which are captured from a dynamic physical system with underlying states $x_t \in \mathbb{R}^N$. Let's assume we have more ...
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Showing that a certain level set of a continuous family of holomorphic maps is locally path connected
I'm working with a continuous function $P: [0,1] \times W \to \mathbb{C}^n$, where $W \subset \mathbb{C}^n$ is an open, relatively compact ball centred at the origin. The map $P$ satisfies the ...
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Invariant ergodic measure Volterra operator on Continuous Functions
This is a follow-up to this question.
Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int_0^{\sqrt{\cdot}} f(s)ds.
$$
Is there an example of an ...
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Invariant ergodic measure Volterra operator
Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int_0^{\cdot} f(s)ds.
$$
Is there an example of an ergodic and $V$-invariant Borel probability ...
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The currents homology of closed orientable surfaces and Birkhoff Ergodic theorem?
I just know very little about currents but I need vexedly. Thanks for your help.
Let $M$ be a closed orientable surface and $I=(f_t)_{t\in[0,1]}$ be an isotopies from identity to $f$. Suppose that $\...
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A linear map on $\chi^{\infty}(\mathbb{R}^2)$ arising from the Cauchy integral formula
The space of smooth vector fields on $\mathbb{R}^2$ and open unit disc $\mathbb{D}$ are denoted by $\chi^{\infty} (\mathbb{R}^2)$ and $\chi^{\infty}(\mathbb{D})$, respectively. A vector field on $\...
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Is there any foster-Lyapunov criterion for time varying Markov processes?
Suppose I have 2 Markov processes with transition kernels Q_1(y|x) and Q_2(y|x). Suppose i also have Lyapunov functions V_1, V_2 for these processes w.r.t. a common set, i.e. there exists a compact ...
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Invariant measure for composition on space of continuous functions
Let $C_g:C(\mathbb{R}^d;\mathbb{R}^d)\rightarrow C(\mathbb{R}^d;\mathbb{R}^d)$ be defined by $C_g(f)\triangleq f\circ g$ for some fixed $g \in C(\mathbb{R}^d;\mathbb{R}^d)$.
What are examples of ...
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Stationary distribution of Markov Chain with departure
I have a Markov Chain of $N$ states. Such states represent the energy levels in a molecule.
The states' connectivity is as follows:
States $j\in\{0,\ldots,N\}$ transition to $k\in\{\max(j-M,0),...,\...
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Reversal of open cover with topologically transitive dynamical system
Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel ...
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Topologically transitive dynamical system mapping space into ball
Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily ...
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Volume-preserving flows with cross section
Let $M$ be an orientable closed smooth manifold of dimension n. Let $\Omega$ be a volume form for $M$, i.e., a nowhere-zero smooth n-form. A smooth $\Phi_t$ flow defined on $M$ is volume-preserving ...
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Is there a research direction within dynamical systems theory / ergodic theory that concerns conjugability to a two-point motion?
Let $X$ be a set equipped with some structure (e.g. topological space, measurable space, probability space, etc.). We say that two endomorphisms $f,g \colon X \to X$ are conjugate to each other if ...
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Set operations over iterated function systems
An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically,
$$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$
is an IFS if each $...
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Non wandering sets and limit sets
Let $X$ be a compact metric space and $f:X\to X$ is a homeomorphism. A point $x$ is aid to be nonwandering if for any open set $U$ containing $x$, there is an $N>0$ such that $f^N(U)\cap U\neq \...
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Why is the billiard problem for obtuse triangles so hard?
This is an incredibly naive question so this may be closed. Nevertheless, I have been reading about the problem asking if every obtuse triangle admits a periodic billiard path, which has been open ...
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Do mixing homeomorphisms on continua have positive entropy?
I am trying to understand relations between various measures of topological complexity. I have read that expansive homeomorphisms on continua, for example, have positive entropy. But I do not know ...
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Are there multiple conjugacy classes of order 2 elements in the smooth automorphism group of $\mathbb{R}$?
Consider the group $\text{Aut}\mathbb{R}$ of smooth invertible maps from $\mathbb{R}$ to $\mathbb{R}$. If $f\in\text{Aut}\mathbb{R}$ has order 2 ($f$ is an involution), is $f$ conjugate to $g(x)=-x$?
...
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Examples of non-uniqueness of the equilibrium states
Let $f:X\rightarrow X$ be an Axiom $A$ diffeomorphism on a compact metric space $X$. Assume that $\phi:X\rightarrow \mathbb{R}$ is Hölder continuous. R. Bowen shows that there is a unique equilibrium ...
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How much more cyclic vectors are there than hypercylic vectors?
$\DeclareMathOperator\C{C}\DeclareMathOperator\HC{HC}$Definitions:
Let $T:X\rightarrow X$ be a bonded linear operator on a separable (infinite-dimensional) Banach space and define the sets:
$
\HC(T)\...
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Connection between entropy and the set of factors of a sequence
Let $a = (a_n)_{n=0}^\infty$ be a bounded real-valued sequence. By a factor of $a$ I mean a finite block $w \in \mathbb R^l$ that appears in $a$, that is, there exists $n \geq 0$ such that $a_n a_{n+1}...
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Hypercyclic vector for backshift operator
It is well-known that the weighted backshift operator $B_{\lambda}:\ell^p \rightarrow \ell^p$ is hypercyclic (with $\lambda>1$); that is, there exists a dense set of sequences $X\subseteq \ell^p$ ...
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A question about the proof of Hölder continuity for dominated splittings
There is a step in the proof of Theorem 4.11 of this set of notes that I don't quite see.
The set up is that $f$ is a $C^2$-diffeomorphism on some Riemannian manifold $M$, and that $E \oplus F = T M$...
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Find integer $k$ such that $k \alpha_i \bmod{1}$ are simultaneously small for all $i$
A classical result shows that if $\alpha$ is irrational, then $\{k \alpha \bmod{1}\}_{k \in \mathbb{Z}}$ is dense over $[0,1]$.
Can we extend this result as follows?
Suppose $\alpha_1,\dots,\...
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Cyclic vectors of translation operator
Let $H(\mathbb{C})$ be the space of holomorphic functions on the complex plane. Then it is well-known that for $a\neq 0$, the translation operator
$$
t_a(f)\triangleq f(x)\mapsto f(x+a),
$$
is ...
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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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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 ...
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139
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flow, stable manifold and tangent
Given vector field $f: \mathbb{R}^2 \to \mathbb{R}^2$, with $f(0)=0$
ODE: $\dot{x}=f(x)$ generates a flow $\Phi^{t}$. so $\Phi^{t}(0)=0$ for all $t \in \mathbb{R}$
So time-one map $\Phi^1$ is diffeo....
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3
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279
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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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Odometer actions of groups
If a group $G$ acts on a Cantor set $(X,\mu)$ by odometers, my question is: what is the explicit automorphism $\alpha_{g}$ for the extended Koopman action on $L^{\infty}(X,\mu)$, for $g$ $\in$ $G$? I ...
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Size of the orbit of a dense set
This question is a follow-up to: this post.
Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...
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261
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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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Attractivity of a system with state-dependent transitions
Let $A\in\mathbb{R}^{n\times n}$ and consider the following dynamical system:
$$
\frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n,
$$
where $\max\{\cdot\}$ acts ...
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Topology of the Yoccoz puzzles at depth-$n$
This paper "Local connectivity of Julia sets and bifuraction loci: three theorems of J.C.Yoccoz" (see http://pi.math.cornell.edu/~hubbard/Yoccoz.pdf) tells that it is quite easy to construct the ...
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