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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Why are critical points important for dynamical systems?
I have just started reading a little about (arithmetic) dynamics and it seems like critical points are very important - for instance, rational maps so that critical points have finite forward orbit (...
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Is this pair of coupled sequences known, and what are their properties?
I was examining the following pair of 'coupled' sequences (I don't know the correct terminology):
$a_{n+1}=a_n+b_n+\frac{a_n}{b_n}$
$b_{n+1}=b_n\left(1+\frac{b_n}{a_n}\right)$
Both sequences grow ...
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Size of the kernel (minimal ideal) of a finite semigroup
Let $A$ be an irreducible nonnegative $N\times N$ integer matrix with constant row sum $D$. Let $A_1, \dots, A_D$ be nonnegative integer matrices, each with constant row sum $1$, such that $\sum_k A_k ...
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Busy beaver sequence for a simple tag-like system
This question arose in the context of tag-like systems, specifically Bitwise Cyclic Tag (BCT). Consider the following discrete dynamical system:
Let $\mathbb{B} = \{\mathtt{0}, \mathtt{1}\}$. Let our ...
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Random walks on the Poincaré disk
Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \...
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Why are these sets divisible by n?
Suppose we have a polynomial $z \to f_c(z)$ defined over $\mathbb Z$ with a free parameter $c$, for instance $z \to z^2 + c$ and we consider the iterates $z \to f_c^{(n)}(z)$ and define the ...
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Real analytic recursion
Fix an analytic function $f:\mathbb{R}\to\mathbb{R}$. Assume $f(x)>x$ for all $x\in \mathbb{R}$.
Is there an analytic function $g:\mathbb{R}\to\mathbb{R}$ such that $g(x+1)=f(g(x))$?
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Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?
Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...
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When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
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Is the union of two proper flows proper?
Let $\varphi _t$ be a flow, aka. a one parameter group of homeomorphisms of the open subset $\Omega \subseteq {\mathbb R}^n$, which we assume to be continuous in the usual sense
that
$$
(t, x)\in {\...
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An angle between two vectors in Oseledets theorem
Let $f:\Sigma \to \Sigma$ be a two side shift map, where $\Sigma=\{1,2,3,4\}^{\mathbb{Z}}$ and let $A:\Sigma \to SL(2,\mathbb{R})$ be a function such that $A((x_{n}))=A_{x_{0}}$. Assume that there are ...
- 1,043
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Number of permitted words up to permutation in a subshift
Let $A$ be a finite set and let $X \subseteq A^{\mathbb{N}}$ be a subshift. Let $\mathcal{L}_n$ denote the set of words of length $n$ appearing in $X$. For a word $w \in \mathcal{L}_n$, one can ...
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Hamilton equations-Symplectic scheme [closed]
We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
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Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?
Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
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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 ...
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Dynamics of fiberwise starshaped hypersurface of Hamiltonian flows on $T^*M$
I have started reading the following paper arXiv link on Dynamical Systems and Symplectic Geometry and in page $3$ we have the following statement :
Let $\Sigma$ be a fiberwise starshaped ...
user174565
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Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?
Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics.
In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...
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Subset of the domain of attraction
Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
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Are all linear vector fields geodesible vector fields?
I had already asked this question in MSE then I ask here at MO.
Assume that $A\in M_n(\mathbb{R})$ is a non singular matrix.
Is the flow of the linear vector field $X'=AX$ a geodesible flow on $\...
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When is the unstable direction map $x\mapsto e^{u}(x)$ injective?
Let $f:M \to M$ be a $C^{2}$-Anosov diffeomorphism. Therefore, there exists an invariant splitting of the tangent bundle $T_{x}M = E^s(x) \oplus E^u(x)$ into a stable and an unstable directions, that ...
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Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map
Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...
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Prime generating arithmetical dynamical system
Is there a prime generating arithmetical dynamical system, by which I mean, is there a rational function $f$ and a prime $p$ such that the set of values of iterates of $f$ starting at $p$, $I(f) = \{f^...
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Question about the proof of Gromov's theorem in geodesic flows
I am trying to understand the following theorem from the book Geodesic flows :
Given a metric $g$ on a simply connected manifold $X$, there exists a constant $C_1>0$ such that given any pair of ...
user174565
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2
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Points attracting to 0 are dense in $\mathbb C$
I know that the following proposition is true, but at the moment I can't see how to prove it.
Define $f(z)=e^z-1$ for all $z\in \mathbb C$. Then $A:=\{z\in \mathbb C:f^n(z)\to 0\}$ is dense in $\...
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Continuity of Kneading invariants of generalised $\beta$-trasformations
For $\beta \in (1,2]$ and $\alpha \in [0,2-\beta]$ consider the generalised $\beta$-transformation $T_{\alpha,\beta}:[0,1] \to [0,1]$ to be $$T_{\alpha, \beta}(x) = \beta x + \alpha \mod 1.$$ It is a ...
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Are the orbits of this discrete dynamical system bounded?
Somehow I believe this should be true and easy to prove but cannot nail it down. A reference, proof, or counterexample will suffice. Didn't get any help over at MSE even with a bounty so I came here.
...
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Dynamical obstructions for a vector field $X$ whose adjoint operator $ad_X$ sends a global orthonormal frame to a set of mutually orthogonal vectors
Let $X$ be a vector field on a parallelizable manifold $M$.
Can we equipe $M$ with a Riemannian metric such that we have at least one global orthonormal frame $\{V_1,V_2,\ldots,V_n \} $ such that $[...
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Values appearing with density in an ergodic system
Values appearing with density in an ergodic system
Let $(X,\mu)$ be a probability space with invertible, measure preserving, totally-ergodic map $T:X \to X$. ($(X,\mu,T)$ is a $\mathbb{Z}$ dynamical ...
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About generalized continued fractions
Let us consider the sequences $(x_n), (a_n)$, starting with $n=0$ and $x_0\in ]0,1[$, defined by the following generalized Gaussian map:
$$x_{n+1}=\frac{\lambda_n}{x_n^{\alpha_n}}-\Big\lfloor \frac{\...
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Dynamical obstructions for a vector field whose derivation sends an orthonormal set to a mutually Sasakian orthogonal vectors
We ask two related questions which are inspired by this MO question Does $P_xP_y+Q_xQ_y=0 \implies$ "NONEXISTENCE OF LIMIT CYCLE for $P\partial_x+Q\partial_y$"? (Complex Dilatation and Limit ...
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Are topologically free and essentially free equivalent for minimal spaces with invariant measures?
Suppose $G$ is a discrete group acting by homeomorphisms on a compact Hausdorff space $X$, such that the action is minimal. Fix an invariant Radon measure $\nu$ on $X$. Is topologically free (the ...
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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 ...
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Orbits space of real-analytic planar foliations
Consider a foliation of $\mathbb{R}^2$, say coming from the trajectories of a vector field $X$. Its orbit space (the quotient of $\mathbb{R}^2$ by the relation "lying on the same trajectory")...
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Solve nonlinear, forced and damped Duffing oscillator
I am trying to solve a Duffing type equation by using Van Der Paul's method:
\begin{align}
\ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t)
\end{align}
with $$x(t) = Re[A(t) \...
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Existence of infinite horizon values in dynamic programming
I am working through the book "Foundations of Stochastic Inventory Theory".
One of the results in the book is Theorem 11.2. The background to this theorem is as follows.
Given finite state ...
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Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$
Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$.
You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does ...
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Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
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Is every gradient vector field a divergence free vector field?
What is an example of a gradient vector field $X$ on a Riemannian manifold $(M,g)$ which cannot be converted to a divergence free vector field via the following processes:
First we remove the ...
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1
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Non-minimal system in which every point is a full entropy point
Is there a discrete topological dynamical system $(X,f)$, where $X$ is a compact metric space (with distance $d$), which is transitive but not minimal, such that $h(f)>0$ and every point is a full ...
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Is it true that $(X,T^k)$ minimal for all $k\geq1$ implies $\mathrm{Aut}(X,T) = \mathrm{Aut}(X,T^k)$ for all $k\geq1$?
Let $(X,T)$ be a topological dynamical system ($X$ is compact metric space and $T\colon X\to X$ a homeomorphism). Recall that its automorphism group is
$$ \mathrm{Aut}(X,T) = \{g\colon X\to X : \text{$...
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Bounding the number of generic measures on an interval exchange transformation
In this paper by Jon Chaika and Howard Masur it is remarked at the end of page 1 that for an interval exchange transformation $T$ with $n$-intervals, one can bound the number of invariant measures ...
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Example of topologically transitive dynamical system with invariant non-ergodic Borel measure
Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which
$f : \Lambda \to \...
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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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2
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Construct a homeomorphism whose periodic points set is not closed
I'm looking for a simple example in discrete dynamical systems whose periodic points set is not necessary closed.
I've seen some example in websites but they are not that simple and discrete.
Note ...
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2
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488
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A functional equation involving the inverse function
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
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Reference for matrix Lyapunov function / matrix dynamic system / stability
We usually consider $\dot{x} = f(x)$, where $x$ is a vector.
Now, I want to consider $$\dot{X}=f(X,U),$$ where $X$ is a square matrix $\mathbb{R}^{n\times n}$ state, $U$ is a square matrix variable $\...
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1
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Mandelbrot set and logistic map connection
I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
3
votes
1
answer
237
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Invariant measure of a subgroup
Let $G$ be an abelian group with a $G$-invariant metric $d$. Let $H$ be a countable dense subgroup of $G$. Let $\mu$ be a non-atomic $\sigma$-finite Borel measure on $G$ that is $H$-invariant. Must it ...
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Lyapunov theory in coupled nonlinear dynamic system with input
Suppose I have the following nonlinear coupled dynamic system
\begin{align*}
&\dot{x}_1 = f_1(x_1,x_2)\\
&\dot{x}_2 = f_2(x_2) + u
\end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...
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Pocket billiards with balls in general position
There were at least two earlier MO questions about ideal pocket billiards.
(Ideal: frictionless, perfectly elastic collisions.)
Perfectly centered break of a perfectly aligned pool ball rack.
Does ...
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