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
1
vote
0
answers
179
views
Two mixing rates of random dynamical system
Given random dynamical system $(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$ where $(\Omega, \mathbb{P})$ is probability space with ergodic transformation $\sigma: \Omega \to \Omega$. Define ...
- 553
1
vote
0
answers
59
views
Regularity of the pdf of partial Birkhoff sums
Suppose that $T: X \to X$ is some measurable map on a Riemannian manifold $X$ (possibly with boundary). Let $\mu$ denote the Riemannian measure on $X$. For measurable, real-valued $g$ we may consider ...
- 403
1
vote
0
answers
89
views
Continuous self-maps of the plane are semiconjugate or conjugate?
Let $f : X → X$ and $g : Y → Y$ be continuous functions. We say that $f$ and $g$ are topologically conjugate if there exists a homeomorphism $α : X → Y$ such that $$f∘α=α∘g$$.
A related idea is the ...
- 1,197
1
vote
0
answers
86
views
Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
- 553
1
vote
0
answers
38
views
Rotation rates for a linear flow on a vector bundle
The following linear ODE on $\Bbb{C}$
$\dot{z} = (a + i b)z$
has solutions $z(t) = e^{(a+ib)t} z(0)$. Hence the real part $a$ captures expansion rate and the imaginary part $b$ captures rotation ...
- 501
1
vote
0
answers
71
views
Time-varying perturbations of continuous-time hyperbolic orbits
My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then ...
user85365
1
vote
0
answers
114
views
Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$
I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently:
Let $X$ be a compact $k$-...
- 121
1
vote
0
answers
108
views
stochastical stable
Given dynamic $f: S^1 \to S^1$ with Lebegue measure $dm$ on $S^1$. Assume it has unique SRB probability measure $\frac{d\mu_f}{dm} dm $. Given left shift space $([-\epsilon, \epsilon]^{\otimes \...
- 553
1
vote
0
answers
107
views
Relations between $\Omega$-groups, locally indicable groups, and right-orderable groups
We know that the class of right-orderable groups $\mathit{RO}$, is contained in the class of $\Omega$-groups (read it from "A note on group rings of certain torsion-free groups" by Burns-Hale).
A ...
- 51
1
vote
0
answers
80
views
Solutions of nonlinear equations with multiple parameters
In the course of analysing a particular three dimensional nonlinear dynamical system, I find the need to solve a nonlinear equation of the form:
$$ \mathcal{M}(x, \lambda) := x - f(x, \lambda_1, \...
- 213
1
vote
0
answers
43
views
definition of mixing component
definition of ergodic component: consider stationary dynamical system $(X, \mathcal{B}, \mu, T)$, each ergodic component is $m(\cdot)=\mathbb{E}_{\mu}^{\mathcal{I}}\mathbf{1}_{(\cdot)}$, $\mathcal{I} $...
- 553
1
vote
0
answers
100
views
Is $\partial M_d$ continuously determined by $d$?
This question is inspired by a question on math.stackexchange:
https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089
The animation ...
- 709
1
vote
0
answers
127
views
How many two-dimensional space filling Hilbert-like curves are there?
I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
- 355
1
vote
0
answers
38
views
Strong ergodicity of a countable subgroup of $PO(3,1)$
If we identify the boundary at infinity of the hyperbolic $3$-space $\mathbb{H}^3$ with the complex projective line $\mathbb{P}^1(\mathbb{C})=\mathbb{C} \cup \{ \infty\}$, we know that the ideal ...
- 121
1
vote
0
answers
53
views
Limit contration rates and expansion rate solenoid map
Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
- 1,043
1
vote
0
answers
92
views
Computing algebraic entropy
Could you recommend any reference for computing algebraic entropy?
Here algebraic entropy is defiened as $\lim_{n \to \infty}\log (deg (f^n))^{1/n}$ for a rational map $f $.
I saw that there are ...
- 663
1
vote
0
answers
218
views
Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow
However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
- 356
1
vote
0
answers
74
views
Has this type of pathwise (S)DE been studied before?
I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before.
Let $(G,\ast)$ be an abelian $C^1$ Lie group....
- 708
1
vote
0
answers
271
views
How to find the best convergence rate of a dynamical system $x_{n+1} = g(x_n),\ n\ge 0,\ x_0\in \mathbb{R}$?
Let $\{x_n\}_{n\ge 0}$ be a sequence of reals such that $x_{n+1}=g(x_n)$, where $g:\mathbb{R}\to \mathbb{R}$ is a continuous function such that $0$ is a fixed point of $g$. My question is the ...
1
vote
0
answers
79
views
Showing a modified system of quadratic equations is stable
I have and $n$ dimensional dynamical system, given by
$\dot{x} = M D(x) P x - \frac{c}{2}x$
$P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
- 111
1
vote
0
answers
126
views
Flows commuting with Anosov flows and further reference request
Hello respected members of Mathoverflow. I was reading the paper "Flots d’Anosov dont les feuilletages stables sont différentiables" by Etienne Ghys and there was a statement which he remarked was ...
- 229
1
vote
0
answers
95
views
A singular foliation analogy of the Riemann Hilbert problem
Note:
In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$.
...
- 356
1
vote
0
answers
125
views
Minimal period for a bounded Langton's ant moving on a tessellation
We consider Langton's ant on the 2D plane, but we replace the square lattice by a Voronoi tessellation obtained from a finite set of points (it could be another tessellation, however directions such ...
- 111
1
vote
0
answers
276
views
Stability when linearization fails
The dynamics of the $j$th system:
\begin{equation}
\begin{split}
\dot{\overline r}_j &= h (\overline r_j)
\,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...
- 33
1
vote
0
answers
83
views
Topological transitivity for a self-map of $\mathbb{R}$ with finitely many discontinuities
I started working with a map $f:\mathbb{R} \to \mathbb{R}$ such that it is continuous except on a finite set. I started looking for a definition of topological transitivity and topological mixing in ...
1
vote
0
answers
79
views
Dynamics for sets related to Brownian motion: zero set, fast points
For sets like the Cantor set, we have preserving maps (eg. the shift-maps and conjugates to it) that allows us to study dynamical quantities such as invariant measure and entropy. I am wondering if we ...
- 5,474
1
vote
0
answers
354
views
Is there a relationship between the Jacobian at a point and the curvature at that point?
The Jacobian $J$ For a dynamical system $\dot{\textrm{x}}=F(\textrm{x})$ determines the dynamics in the tangent plane at a given point. Intuitively speaking the Jacobian evaluated at a point should ...
- 11
1
vote
0
answers
55
views
Which planar smooth foliations are not smooth equivalent to a foliation arising from level sets of a harmonic function?
Is there an smooth foliation of the plane which is not smoothly equivalent to a foliation $dH=0$ where H is a harmonic function without critical values?
If the answer is negative then we conclude ...
- 356
1
vote
0
answers
308
views
A differential operator associated with a vector field on the torus
Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$.
We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows:
$T(f)=...
- 356
1
vote
0
answers
336
views
Existence of solution for Poisson equation in Markov chain
Consider $X_n\in \mathcal{X}$ a controlled Markov chain taking value in a compact set $\mathcal{X}$ with action $a\in \mathcal{A}$, where the action set $|\mathcal{A}|$ is finite.
(In particular, we ...
- 185
1
vote
0
answers
61
views
Stability of Fokker plank solutions with drift not coming from potential: Lyapunov analysis
Consider the FP equation on two dimensional space:
$\dfrac{\partial{\rho(x,y,t)}}{{\partial t}}+u(x,y)\dfrac{\partial\rho}{\partial x}+v(x,y)\dfrac{\partial\rho}{\partial y}=D\Delta\rho(x,y,t)$.
It ...
- 11
1
vote
0
answers
124
views
repeated addition and square root of fixed number
pick any real number $x$ and integer $k$ and do the following recursive :
1) $x_0 =x $
2) $x_{n+1} = x_n + \sqrt x_n$
using only $x$ and $k$ how to find the value of $x_k$ without going through ...
user95470
1
vote
0
answers
68
views
steady state distribution of a dynamical equation?
Given the following dynamical equation for $X(t)$ as follows:
$X(t+1) = X(t) - \min\{X(t), M\} + Y(t)$,
or can write it as follows:
$X(t+1) = \max\{X(t) - M, 0\} + Y(t)$,
Assume the PDF of $Y(t)$ ...
1
vote
0
answers
172
views
Generalizing approximate $\mathbb{Z}$-equivariance of a simple function
Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. https://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...
- 15.4k
1
vote
0
answers
134
views
Inverse limits of the interval with a single bonding map below the identity
My question is as follows.
QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...
- 6,548
1
vote
0
answers
197
views
A certain measure on Banach algebras
According to the comments of Nate Eldredge I did revise the question. In particular I change "$C^{*}$ algebras" to "Banach algebras".
Is there a reference who introduce the following measure on ...
- 356
1
vote
0
answers
40
views
Generalizing an expected increase in autocorrelation near a bifurcation point to a system of ODE
Near a bifurcation point, a stochastically forced dynamical system should show an increase in autocorrelation and variance. This is due to critical slowing (a loss in resilience to perturbations). ...
- 111
1
vote
0
answers
151
views
What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp
First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...
- 326
1
vote
0
answers
84
views
Hyperbolic PDE from total derivative?
Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...
1
vote
0
answers
172
views
Ergodic skew product on $\mathbb T^d\times U(2)$
Let $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ be the $d$-dimensional torus with normalized Haar measure $\mu_1$ and let $U(2)$ be the group of $2\times2$ unitary matrices with normalized Haar measure $\...
- 11
1
vote
0
answers
38
views
Boundedness of partial products for a divergent trig product
I am looking at a discrete dynamical system and I wish to show that it is bounded. I know that the displacement after $n$ iterations is given by the product
$$\Delta_n=\prod_{k=0}^n \left(1+\frac{2\...
- 11
1
vote
0
answers
62
views
Applications of systems with multiple time
A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space.
I am interested in informative examples and applications of such systems. I know ...
- 798
1
vote
0
answers
265
views
A Perron-Frobenius problem
Let $A$ be an irreducible nonnegative matrix with column sums equal to 1.
Let $b\in R^n$ have components summing to 0, and let $u$ be the solution of $u=Au+b$ with components summing to 1 (unique ...
- 3,873
1
vote
0
answers
64
views
Ordering periodic orbits
I want to prove the proposition:
Proposition- Let $f:I \to I$ be continuos, and let f have a (2n+1)- periodic orbit {$x_{k}=f^{k}(x_{0})$, $k=0,1,\dots,2n$}, but no (2m+1)-periodic orbit for $1\...
- 11
1
vote
0
answers
55
views
Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?
I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...
- 315
1
vote
0
answers
181
views
Discrete group action on the sphere
Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$
be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...
1
vote
0
answers
52
views
Small open sets around a point intersecting pieces of orbits
Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...
- 2,560
1
vote
0
answers
37
views
Bifurcations in flows on 2-dimensional torus
I am doing research on bifurcations which appear in flows on the
2-dimensional torus, in particular on such which do not appear in flows
on $\mathbb{R}^2$.
Can anyone provide some references on ...
- 615
1
vote
0
answers
72
views
Convergence to equilibrium for time in-homogeneous diffusions
Consider the long time behavior for a time in-homogeneous diffusion such as
$$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$
where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent ...
- 165
1
vote
0
answers
240
views
A Lie algebra associated with a one dimensional foliation
A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras $C(...
- 356