Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
6
votes
0answers
58 views
Pursuit-evasion with many slow pursuers
Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...
0
votes
0answers
92 views
Book recommendations beyond an introduction [on hold]
So I’ve scraped the surface of many topics, but I would like to go further. Can anyone recommend some continuations to the following introductory books? It’s okay if necessarily it needs to be a ...
5
votes
0answers
123 views
Looking for a counterexample for Ruelle's inequality on compact manifold
Let $M$ be a compact differentiable manifold, and $f:M\to M$ be a $C^1$- smooth diffeomorphism.
If Assume that $\mu$ be a $f$-invariant probability measure on $M$.
Then D.Ruelle proved that
$$
h_\...
2
votes
0answers
68 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 ...
1
vote
1answer
126 views
A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf
Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.
4
votes
0answers
48 views
Obstructions for finding a volume form for a given flow
Let $X$ be a non-singular $C^\infty$ vector field on a three manifold $M$.
There are some obvious obstructions for finding a volume form that is preserved under the flow given by $X$:
If $X$ is ...
1
vote
0answers
36 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 ...
0
votes
0answers
87 views
A complex limit cycle not intersecting the real plane(2)
Inspired by this question and the counter example provided in its answer we ask:
Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the ...
1
vote
1answer
36 views
Measuring how suboptimal control is
Suppose I have a linear dynamical system to control. I use PMP to find necessary conditions for the optimal control of the system wrt to some objective function. Now, suppose that the trajectory I ...
3
votes
0answers
46 views
Types of triangles admitting periodic billiard orbits
It is an open problem in dynamical systems if every triangle has a periodic billiard orbit. So far it has been proven that equilateral triangles, isosceles triangles, right triangles, and obtuse ...
2
votes
0answers
70 views
From quenched to annealed decay of correlation
If we have quenched decay of correlation, can we transfer it to annealed decay of correlation? To be precise, let us consider following setting:
Given transformations $T_{\omega}: (S^1, dm) \to (S^1, ...
3
votes
0answers
48 views
Flow lines of a real analytic vector field convergent to a point
Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...
0
votes
0answers
60 views
A problem in trying to show that a system is ergodic
If $f:[0,1]\to [0,1]$ is given by
$$
f(x)=
\begin{cases}
2x & \mbox{ if } x\in [0,1/3)\\
& \\
2x-\frac{2}{3} &...
0
votes
1answer
34 views
Nonsmooth dynamical system (DAE) - systematic way to calculate period numerically?
What I have in mind is a mechanical system that is described by an implicit system of ODEs or a system of DAEs (differential algebraic equations). The system is asymptotically stable, meaning that ...
5
votes
0answers
106 views
Extension of Vector Field in the $\mathcal{C}^r$ topology
This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...
0
votes
0answers
37 views
Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
8
votes
0answers
188 views
Structural Stability on Compact $2$-Manifolds with Boundary
I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...
3
votes
1answer
117 views
Symmetries for Julia sets of perturbations of polynomial maps
This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...
3
votes
0answers
61 views
Piecewise linear expanding maps
Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose ...
1
vote
1answer
112 views
Trajectory leaving a set
Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
1
vote
0answers
62 views
Nearly eventually almost periodic functions
Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges.
Suppose $f: ...
2
votes
0answers
66 views
Does singularity confinement imply a fixed pattern of irreducible factors?
Consider a rational map
$f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...
2
votes
1answer
81 views
Discrete dynamical system and bound on norm
Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...
2
votes
1answer
344 views
Formal group law and Koenigs function conjecture?
Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). \tag{1}$$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$
This equation has many ...
1
vote
0answers
44 views
About strange invariant set of the Lozi mappings
Consider the Tent map: $f_{μ}(x)=μx$, if $x<0.5$ and $f_{μ}(x)=μ(1-x)$ if $x≥0.5$. In this page (https://en.wikipedia.org/wiki/Tent_map) it was stated that:
If $μ$ is greater than $2$ the map's ...
8
votes
2answers
247 views
Dimension of orbit versus invariant functions
$\def\CC{\mathbb{C}}$Let $K = \CC(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$; in the cases I care about, $G \cong \mathbb{Z}$. Then the field of $G$-invariants, $K^G$, ...
1
vote
1answer
105 views
A special oscillatory orbit in space
Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
2
votes
1answer
95 views
6-periodic billiards trajectory in acute triangle
We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards?
5
votes
0answers
88 views
$C^{1+\epsilon}$ conjugacy of expanding map on circle
A continuously differentiable map $f:S^{1}\rightarrow S^{1}$ is called expanding if $|f^{'}(x)|>1$ for all $x\in S^{1}$.
We can define the degree of f, def(f) to be number of preimage $f^{-1}(x)$, ...
2
votes
1answer
149 views
Corollary of the Malgrange Preparation Theorem
(This question was previously posted on MSE and I decided to post it here too.)
Let $f:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ be a smooth function, such that
$$f(0,0)=0,\ \frac{\partial f}{\...
4
votes
1answer
255 views
How are the fields of dynamical systems, stochastic processes and additive combinatorics, inter-related?
Currently I’m interested in a couple of fields, namely dynamical systems, stochastic processes, and additive combinatorics. I was wondering if it’s feasible to keep pursuing all 3, and whether I can ...
12
votes
1answer
225 views
Poincaré on analytic dependence on parameters of solutions of linear differential equations
There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...
0
votes
0answers
40 views
Periodicity of oscillators in Langton's Ant and powers of $2$
This question based on previous one by me. As Christopher Purcell noticed in his comment, there exist conjecture (which has a lot of counterexamples) that if you take a pair of ants $(n,n+1)$ apart (...
4
votes
0answers
87 views
Existence of (generalized) Crossed Products
Compare the following construction with the crossed product construction for $C^*$-dynamical systems:
Let $W\curvearrowright X$ be a group action of a discrete group on a compact Hausdorff space $X$ ...
1
vote
0answers
55 views
Continuous self-maps of the plane are semiconjugate or conjugate?
Let $f : X → X$ and $g : Y → Y$ be continuous functions. We say that $f$ and $g$ are topologically conjugate if there exists a homeomorphism $α : X → Y$ such that $$f∘α=α∘g$$.
A related idea is the ...
1
vote
0answers
63 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$ ...
4
votes
3answers
369 views
Reference request: Dynamical systems
I’m currently reading Brin and Stuck’s Introduction to Dynamical Systems, and I think I like the field a lot so far. I haven’t finished it quite yet, but what are some other good textbooks I can read ...
2
votes
0answers
77 views
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 ...
1
vote
1answer
191 views
On the 2002 paper “Dynamics of polynomial automorphisms of $\mathbb{C}^k$” by Guedj and Sibony
I desperately need to read the paper [1] before meeting a would-be supervisor, but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, not even one ...
5
votes
0answers
172 views
Absolute oscillator in Langton's Ant
We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...
6
votes
0answers
226 views
A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory
The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
1
vote
1answer
77 views
Does the following percolation model have a name?
Consider the following model for percolation in an infinite graph: each vertex has a certain region (set of vertices) associated with it, which at the beginning contains only the vertex itself, and ...
6
votes
0answers
237 views
Geometric bang-bang theorem for nonlinear optimal control
The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems ...
1
vote
1answer
85 views
Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive
Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.
Let $S : X → X$ and ...
4
votes
0answers
126 views
mixing time of Young tower
Given Young tower $(\Delta, m, F)$ with base $\Delta_0$, return map $R$, distortion of $F^R$, g.c.d($R$)$=1$.
This system is exact, so mixing, and implies:
given fixed $\epsilon$, exists $N$, s.t. $...
2
votes
2answers
203 views
Ergodic theorem and products
If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that
$$ \lim_{n \rightarrow \infty} \frac{f_n}{...
0
votes
1answer
111 views
Probabilistic approach for cellular automata
Few months ago my scientific adviser asked me to use probabilistic ideas in such problem :
Consider a matrix NxN. Each element of matrix is a number 1 or 0. We may change all elements of this matrix ...
4
votes
0answers
67 views
Weighted distribution of irrational rotation
Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
2
votes
0answers
73 views
Random Young Tower
the following setting is the same as Baladi's paper "ALMOST SURE RATES OF MIXING FOR I.I.D. UNIMODAL MAPS" (2002), define random Young tower ($\Delta_{\omega})_{\omega\in \Omega}$ sharing the same ...
2
votes
0answers
112 views
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 $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
