Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
276
questions
23
votes
1
answer
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The Dedekind eta function in physics
This interesting little fellow (a nice introduction is the video "Mock Modular Forms are Everywhere" by Cheng and Felder) popped up in some operator algebra (Witt / Virasoro Lie algebra) I ...
22
votes
3
answers
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A Collatz-like function that bifurcates on primes
This is likely piling one mystery on another, but ...
I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...
20
votes
2
answers
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views
Applications of number theory in dynamical systems
I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics.
...
19
votes
2
answers
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views
Existence of continuous map on real numbers with dense orbit?
Does there exist a continuous map $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the forward orbit of 0 is dense in $\mathbb{R}$?
18
votes
2
answers
953
views
"Derived" polyhedra and polytopes
The notion of derived polygon is natural and leads to remarkable convergence.
Start with a polygon, and replace it by locating a point on every edge
a fraction $\alpha$ between the two endpoints. For ...
17
votes
3
answers
1k
views
Codimension zero immersions
Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?
Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...
15
votes
1
answer
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views
Nonperiodic points of homeomorphisms of a ball
Suppose $B$ is a $d$-dimensional ball (for some $d \geq 1$) and $T$ is a homeomorphism from $B$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that $...
14
votes
7
answers
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Furstenberg $\times 2 \times 3$ conjecture, bibliography
Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a ...
13
votes
2
answers
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Connectedness of space of ergodic measures
Let $X = \Sigma_p^+ = \{1,\dots,p\}^\mathbb{N}$ and let $f=\sigma\colon X\to X$ be the shift map. Let $\mathcal{M}$ be the space of Borel $f$-invariant probability measures on $X$ endowed with the ...
12
votes
3
answers
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Vector field with holomorphic flow
Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms.
I know one reference ...
11
votes
3
answers
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Integer dynamics hitting infinitely many primes
I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). To make it specific, ...
10
votes
1
answer
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views
is there a diffeomorphism with only finite orbits but of infinite order?
I asked this in stackexchange, but got no answer, so I am trying here.
Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties:
All its orbits are ...
9
votes
2
answers
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views
An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)
Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question.
Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ(...
8
votes
4
answers
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Uniform convergence of Birkhoff averages and unique ergodicity
I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET):
Let $T$ be a ...
6
votes
2
answers
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What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
6
votes
2
answers
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Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?
The irrational rotation on the circle is both a homeomorphism and minimal but is not topologically mixing. The argument-doubling transformation on the circle is topologically mixing but is neither a ...
6
votes
0
answers
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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 ...
6
votes
1
answer
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A generalization of Gradient vector fields and Curl of vector fields
Let $M$ be a smooth Riemannian manifold. The Riemannian metric enables us to equip the tangent bundle $TM$ with a symplectic structure $\omega$, which is the pullback of the standard symplectic $2$ ...
6
votes
2
answers
581
views
On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
5
votes
1
answer
284
views
The spectral radius of a binary matrix - polynomial growth?
(This is a follow-up to The spectral radius of a binary matrix)
Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$.
QUESTION. Is there a $\delta\in\bigl(0,\frac12\...
4
votes
0
answers
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The Moyal action of a planar vector field
Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$:
$\tilde{D}_{X}(f)=...
4
votes
2
answers
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A Stochastic Dynamical Billiard
Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t))$ be the position at time $t$ of a point which moves in the rectangle ...
2
votes
1
answer
315
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The study of dynamics of a polynomial vector field via Green's function methods
In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is ...
2
votes
1
answer
210
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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 ...
2
votes
2
answers
563
views
A non-vanishing vector field on $S^3$ whose flow does not preserve any transversal foliation
Is there a non-vanishing vector field $X$ on $S^3$ which does not admit a transversal $2$-dimensional foliation? if the answer is negative, is there a non-vanishing vector field $X$ on $S^3$ which ...
1
vote
1
answer
263
views
Is there an entire solution for the Van der pol equation?
Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation?
$$\begin{cases}\dot{x}=y-x^{3}\\\dot y=-...
1
vote
2
answers
214
views
Behavior of a non-linear differential equation
Let us consider the following differential equation
$$
\dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}.
$$
My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the ...
0
votes
1
answer
100
views
Lyapunov vectors along a trajectory
I have the equation:
$$
\dot{x}_i = F_i(x)
\tag{1}
$$
with $x\in \mathbb{R}^n$. To deal with the Lyapunov exponents, we write the equation for small displacements $\delta x_i$:
$$
\dot{\delta x}_i = \...
127
votes
2
answers
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What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
65
votes
4
answers
4k
views
Perron number distribution
A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...
44
votes
3
answers
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views
Is there an elementary proof that distal maps are invertible?
Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.
Then it is true that $T$ is bijective.
Question: Is there an ...
41
votes
5
answers
3k
views
Which polynomial's roots are its coefficients?
Start with any polynomial of degree $n$ with complex coefficients, e.g.,
$$z^3+z^2+2 z+3 \;.$$
Find its $n$ roots, and list them in order of their modulus:
$$-1.28, (0.14\pm 1.53 i)$$
Now form a new ...
37
votes
1
answer
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Is the area of the Mandelbrot provably computable?
Recall the Mandelbrot set $M$ is the set of points $c$ in the complex plane such that the sequence $z_0 = 0, z_{n+1} = z_n^2 + c$ is bounded. It is well-known that $M$ is a compact set of positive ...
33
votes
4
answers
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Does there exist a shot in ideal pocket billiards?
Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard ...
33
votes
4
answers
5k
views
Is there a categorical treatment of dynamical systems?
Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a category of $T$-dynamical systems on $X$ which yields useful information about $X$ and $T$?
More precisely, is there a category whose ...
32
votes
2
answers
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A Collatz-like problem on prime numbers
Consider the function $f$ on the prime numbers defined by $$ f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle $$(3,7,5,11,23,...
31
votes
1
answer
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Vanishing line on Conway's game of life
If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$.
...
29
votes
3
answers
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views
Rational functions with a common iterate
Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...
27
votes
10
answers
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Book recommendation for ergodic theory and/or topological dynamics?
Hello,
I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...
26
votes
5
answers
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Do complex iterates of functions have any meaning?
Using a method explained in this answer to How to solve $f(f(x)) = \cos(x)$?, it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th ...
25
votes
6
answers
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Parametrization of the boundary of the Mandelbrot set
Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.
The ...
25
votes
4
answers
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views
Hamiltonian, Lagrangian and Newton formalism of mechanics
If my thinking is wrong please let me know. I have little knowledge on beyond-college physics.
For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,...
23
votes
1
answer
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Time for Langton's ant to cover a "square" torus
Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel in
...
22
votes
5
answers
2k
views
When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?
Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ...
22
votes
13
answers
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Is there a "crash-course" book on Abelian varieties (e.g., an introduction for physicists)?
Hello,
In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...
22
votes
3
answers
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Cyclic action on Kreweras walks
A Kreweras walk of length $3n$ is a word consisting of $n$ $A$'s, $n$ $B$'s, and $n$ $C$'s such that in any prefix there are at least as many $A$'s as $B$'s, and at least as many $A$'s as $C$'s. For ...
22
votes
6
answers
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views
Quantitative versions of ergodic theorem
Are there any general theorems similar to Birkhoff's ergodic theorem, but giving quantitative estimates on the rate of convergence or average time of recurrence (perhaps with additional assumptions)? ...
21
votes
2
answers
1k
views
How does it End?
A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context.
Let $\mathcal{C}$ be a category. ...
21
votes
4
answers
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views
Prime factorization "demoted" leads to function whose fixed points are primes
Let $n$ be a natural number whose prime factorization is
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$
Define a function $g(n)$ as follows
$$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \;,$$
i.e., exponentiation is "...
19
votes
5
answers
999
views
Lightray trapped between two mirror disks: Computation formulation?
I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...