Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,388
questions
12
votes
2
answers
689
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Periodic lightray paths trapped between two nested mirror circles
I wonder if the periodic paths of a lightray trapped between two nonconcentric circles,
each perfectly reflecting, are known.
The behavior of such rays seems chaotically complicated. For example, ...
4
votes
0
answers
100
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 $p$ ...
3
votes
1
answer
187
views
Non-injective continuous maps that appear quasiconformal
Suppose that I have a continuous surjection $f: U \rightarrow V$ between two open subsets of the plane. Suppose that $f$ appears to be quasiconformal in the sense that there is a uniform constant $K \...
13
votes
1
answer
592
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 $\...
5
votes
0
answers
207
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 $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...
0
votes
1
answer
67
views
Nonsmooth dynamical system (DAE) - systematic way to calculate period numerically?
What I have in mind is a mechanical system that is described by an implicit system of ODEs or a system of DAEs (differential algebraic equations). The system is asymptotically stable, meaning that ...
1
vote
0
answers
305
views
A differential operator associated with a vector field on the torus
Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$.
We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows:
$T(f)=...
3
votes
0
answers
97
views
Does singularity confinement imply a fixed pattern of irreducible factors?
Consider a rational map
$f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...
0
votes
0
answers
48
views
Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
8
votes
0
answers
251
views
Structural Stability on Compact $2$-Manifolds with Boundary
I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...
1
vote
1
answer
211
views
Complexity theory and closed form formulas in analysis
My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...
3
votes
1
answer
155
views
Symmetries for Julia sets of perturbations of polynomial maps
This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...
3
votes
0
answers
90
views
Piecewise linear expanding maps
Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose ...
9
votes
1
answer
773
views
Lorenz attractor path-connected?
Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.
EDIT: The answer below is unsatisfactory, and possibly ...
2
votes
1
answer
128
views
Discrete dynamical system and bound on norm
Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...
2
votes
1
answer
131
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?
2
votes
0
answers
152
views
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 \...
8
votes
2
answers
310
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, $K^G$, ...
3
votes
2
answers
636
views
Thom's gradient conjecture and analyticity
Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ is an open subset and $0 \in U$ is a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of ...
6
votes
1
answer
494
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 ...
2
votes
1
answer
269
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
$$f(0,0)=0,\ \frac{\partial f}{\...
3
votes
0
answers
1k
views
Matrix Operations Preserving Hurwitz Stability
I begin with terminology I use in the question. A real square matrix $A$ is
negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$;
$\ast$-negative-stable if for ...
1
vote
0
answers
83
views
Continuous self-maps of the plane are semiconjugate or conjugate?
Let $f : X → X$ and $g : Y → Y$ be continuous functions. We say that $f$ and $g$ are topologically conjugate if there exists a homeomorphism $α : X → Y$ such that $$f∘α=α∘g$$.
A related idea is the ...
2
votes
0
answers
139
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 ...
0
votes
1
answer
268
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 ...
3
votes
1
answer
214
views
An elliptic operator whose corresponding symbol Hamiltonian vector field has an isolated periodic orbit
Let $D$ be a differential operator on the space of smooth functions on a manifold $M$. The symbol of $D$ can be considered as a Hamiltonian on the cotangent bundle $T^*M$. We call ...
1
vote
1
answer
94
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 ...
1
vote
1
answer
160
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 ...
5
votes
1
answer
299
views
Rate of convergence in the Shannon-McMillan-Breiman theorem
The Shannon-McMillan-Breiman theorem, in its dynamical systems formulation, contains (or implies) the following statement: If $T$ is the map, $\mu$ is the ergodic $T$-invariant measure, $\mathcal{P} = ...
2
votes
2
answers
344
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
1
answer
163
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 ...
8
votes
1
answer
298
views
Under which conditions do ellipsoids have a focal property?
Given an ellipsoid $E$, we consider the trajectories of light inside $E$ assuming that $\partial E$ would be a mirror. In other words, let a light trajectory be piecewise linear path $\gamma:[0,\infty)...
2
votes
0
answers
149
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 $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
4
votes
0
answers
97
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
0
answers
105
views
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 ...
7
votes
2
answers
382
views
Is it possible to prove unboundedness of 3rd order ODE?
Consider the 3rd order ODE
$$\dddot{x}+A\ddot{x}-\dot{x}^{2}+x=0$$ where $\dot{x}\equiv \frac{dx}{dt},\ddot{x}\equiv \frac{d^{2}x}{dt^{2}}, etc$. $A$ is a constant.
If we multiply this equation by $\...
4
votes
1
answer
470
views
Invariant measures for $1$-dimensional discrete dynamical systems
The image below was created using the current release of the
visualization program 3D-XplorMath (available
by clicking here. )
It is an image of the Feigenbaum Tree, on
which is superimposed a ...
10
votes
1
answer
350
views
Trapping a particle
A particle starts a brownian walk in the middle of a long tunnel in the plane, at one end of the tunnel is a region Y of given area A.
Does the shape of region Y affect average time for the particle ...
4
votes
1
answer
138
views
Asymptotic colouring of edges and vertices, and untwisting cocycles
This question regards colourings on edges and vertices on countable directed multigraphs.
We start with an example. Let $G=\mathbb Z^2$. We define two functions $a_h$ and $a_v$ from $\mathbb Z^2$ to $\...
1
vote
1
answer
121
views
Irreducible subcontinuum of Lorenz attractor?
In my first question Lorenz attractor path-connected?, some are saying the Lorenz attractor $\mathscr L$ is not path-connected.
But suppose $x$ and $y$ are two points in different path components of ...
17
votes
2
answers
1k
views
Does 53 diverge to infinity in this Collatz-like sequence?
This function has been explored a bit at MSE (in June 2016):
\begin{eqnarray}
f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\
f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise}
\...
33
votes
4
answers
3k
views
Does there exist a shot in ideal pocket billiards?
Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard ...
4
votes
1
answer
273
views
Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$
If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...
11
votes
0
answers
248
views
Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?
Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
2
votes
2
answers
176
views
Quantitative bound on irrational rotation recurrence time
Given an irrational $a$, the sequence $b_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified.
Given a point $p$ in $\mathbb ...
2
votes
0
answers
133
views
$C^1$-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 $W$),...
1
vote
0
answers
37
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 ...
2
votes
2
answers
64
views
Lower bound of positive entropies of automorphisms on tori
Let $A$ be an automorphism on tori $\mathbb{T}^d$. It is well known that the topological entropy
$$
h(A)=\sum_{\lambda} \max\{0, \log|\lambda| \}
$$
where $\lambda$ goes through all eigenvalue of $A$ ...
3
votes
1
answer
258
views
On Krieger's Embedding Theorem
This is Theorem 10.1.1 of Lind & Marcus's book, An Introduction to Symbolic Dynamics and Coding. They say that is "straightfordward" to go from
Let $X$ a shift of finite type and $Y$ a mixing ...
15
votes
2
answers
2k
views
Are rounded rectangle billiard dynamics ergodic?
Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.
(Image from Microwave_billiards_and_quantum_chaos.)...