Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,397
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Example of a measure-preserving system with dense orbits that is not ergodic
Let $X$ be a Borel probability space (i.e. equipped with a measure $\mu$ on the Borel $\sigma$-algebra such that $\mu(X) = 1$) with a measure-preserving transformation $T$ such that every point has a ...
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4
votes
1
answer
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All rational periodic points
I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes ...
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votes
0
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Surreal numbers and the Collatz iteration as a game?
Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$.
Each number $n$ represents a game played by left $L$ and right $R$:
$$n = \{L_n | R_n \}$$
The rules ...
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0
votes
1
answer
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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 = \...
- 16.7k
6
votes
0
answers
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views
2D quadrant sandpile: emergent highway structure
Consider the top-right quadrant of the plane divided into unit cells, each cell containing some number of chips. A cell containing at least two chips can fire two chips, one to the cell above it and ...
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votes
12
answers
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Importance of Poincaré recurrence theorem? Any example?
Recently I am learning ergodic theory and reading several books about it.
Usually Poincaré recurrence theorem is stated and proved before ergodicity and ergodic theorems. But ergodic theorem does not ...
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votes
0
answers
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Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
This question, comes out of a question in MSE and I hope it is ok to ask it here:
Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
...
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votes
3
answers
543
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Can an "almost injective'' function exist between compact connected metric spaces?
Let $\pi: X\to Y$ be a surjective continuous function between the compact, metric and connected spaces $X$, $Y$, and $Y_0 = \{y\in Y: \#\pi^{-1}(y)>1\}$. Suppose that:
$Y_0$ is dense in $Y$,
$Y\...
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votes
1
answer
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Generating functions of Collatz iterates?
Let $C(n) = n/2$ if $n$ is even and $3n+1$ otherwise be the Collatz function.
We look at the generating function $f_n(x) = \sum_{k=0}^\infty C^{(k)}(n)x^k$ of the iterates of the Collatz function.
The ...
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3
votes
3
answers
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Free ergodic probability measure-preserving actions of the free group
Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group.
An action of $\Gamma$ on $X$ is:
essentially free if for all $g \in \Gamma \setminus \{e \}$,...
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votes
0
answers
111
views
Can non wandering sets be connected?
I know that the alpha and omega limit sets of a flow on a compact connected invariant subset of a manifold must be connected and these limit sets are contained in the non wandering set.
My question is ...
4
votes
2
answers
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Inclusion of infinite intersection
Let $E$ be a Banach space, $T:E\rightarrow E$ a continuous bounded nonlinear mapping., and $\{x_n\}_{n\in\mathbb N}$ such that $$x_{n+1}=T(x_n),\:\forall n\in \mathbb{N}:=\{0,1,\cdots\}.$$
Let $$X_n=\...
- 1,423
1
vote
1
answer
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Starting vector in Lyapunov exponents evaluation
Let us consider the equation:
$$
\dot{x}_i = F_i(x)
$$
with $x\in \mathbb{R}^n$ and $i=1\dots n$, and the equation for small displacements:
$$
\dot{\delta x} = \sum_j \frac{\partial}{\partial x_j} F_i(...
- 16.7k
7
votes
2
answers
676
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A second order nonlinear ODE
In my research (in differential geometry) I recently came across the following nonlinear second order ode:
$$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$
It ...
- 106k
1
vote
0
answers
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Vandermonde shift
I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let
$$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\
0 & 0 & 1 & 0 & \...
- 364
0
votes
0
answers
79
views
Distribution of the values of the product $\prod_{k=1}^n |1-e(k\alpha)|$ for an irrational number $\alpha$
For an irrational number $\alpha$, let $e(k\alpha):=\exp(2k\pi i\alpha)$. It was indicated in this thread that
$$\limsup_{n \to \infty} \prod_{k=1}^n |1-e(k\alpha)|=\infty$$
(actually a weaker result ...
- 1,553
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votes
0
answers
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views
Uniform stability of linear operators - reference request
Let $T$ be a bounded linear operator on a complex Banach space $X$. I am looking for a reference for the following result:
Theorem 1. Let $p \in [1,\infty]$. The following assertions are equivalent:
(...
- 11.6k
2
votes
0
answers
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Rotation set vs existence of rotation number
Let $f\colon \mathbb{S}^{1}\to\mathbb{S}^{1}$ be a continuous function of degree 1 and $F\colon \mathbb{R}\to \mathbb{R}$ a lift of $f.$ One can define, for each $x\in \mathbb{R}$, the rotation number ...
2
votes
0
answers
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Ergodicity of a dynamical system on the $n$-sphere
Let $v$ be continuous and nowhere-vanishing vector field tangent to the $n$-sphere $\mathbb{S}^n$ (hence $n$ is odd, w.r.t the Hairy-Ball Theorem). Let $x$ be a trajectory on $\mathbb{S}^n$, defined ...
- 557
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Gradient flows on Hilbert manifolds
I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded.
To be more precise, a ...
5
votes
0
answers
93
views
Area preserving diffeomorphisms of surfaces with only hyperbolic periodic points
This is a (probably very naive question) about area-preserving maps of surfaces.
Does there exist a Hamiltonian diffeomorphism
$$ f: \Sigma \to \Sigma $$
of a symplectic surface (real dimension $2$), ...
- 28.8k
9
votes
1
answer
781
views
Why the least action principle is always (?) used in this particular form?
The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...
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4
votes
1
answer
498
views
Is the series $\sum_{n=1}^{\infty} \sin(n^4)\sin(4^n)$ convergent or divergent?
Is the series $$ \sum_{n=1}^{\infty} \sin(n^4)\sin(4^n) $$ convergent or divergent?
I tried expanding the sine functions and got no clue, and any test that I know of isn't helping me with this series. ...
- 22.5k
3
votes
2
answers
385
views
Classification of Lagrangians with given Euler-Lagrange equations
In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of ...
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Entropy of flow and time-1 map
Let $\Phi=(\phi_t)_{t\in \mathbb{R}}$ be a continuous flow on a compact metric space $X$. Let $\mu$ be a $\phi_1$-invariant measure. Then it is not hard to verify tht $\int_{0}^{1} \phi_t\mu dt$ is $\...
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What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?
When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
- 183
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votes
2
answers
142
views
Torque term coming from added mass effects
I'm studying a quasi-steady force model (for a 2D problem) published in a fluids journal, and one of the torque terms is a bit perplexing: the term is a product of a difference of added mass ...
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votes
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answers
210
views
Foliations and locally free action of $\mathbb{R}^{n-1}$
Let $M$ be a $n$-dimensional closed manifold endowed with a foliation ${\cal F}$ suppose that the leaves of ${\cal F}$ are diffeomorphic to $\mathbb{R}^{n-1}$ are the leaves of ${\cal F}$ defined by a ...
- 3,644
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0
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views
When are all average trajectories of $w_{k+1}=Aw_k+b$ bounded?
Below is an open-problem in my field, and I'm wondering if someone has insights I'm missing. (cross-posted on math.se)
Suppose observation $x$ is drawn from some distribution $\mathcal{D}$, $w_0\in \...
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votes
1
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views
A unique equilibrium state which does not have Gibbs property
Let $T:\Sigma \rightarrow \Sigma$ be a topologically mixing subshift of finite type and let $f:\Sigma \rightarrow \mathbb{R}$ be a continuous functions over $(T, \Sigma)$. Assume that there is a ...
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Difficulty of homeomorphism of effective Cantor dynamics
Let $X = \{0,1\}^{\mathbb{N}}$ with the product topology. Given a Turing machine $M$ and $x \in X$, define $M(x) \in \{0,1\}^* \cup X$ as the sequence of bits output by $M$ when given an oracle for $x$...
- 60.2k
8
votes
2
answers
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Topological objects associated to Steinerberger's 4-regular graphs
Very recently, in arXiv:2008.01153, Steinerberger has associated to any sequence $(x_n)_{n\in\mathbb{N}}$ of distinct real numbers a 4-regular graph.
In the case irrational multiples, like $x_n=n\sqrt{...
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votes
3
answers
752
views
What are some foundational authors/papers in dynamical systems?
I have just begun my first dynamical systems class, and I would like to try out the advice in the top answer here. To summarize, the answer suggests that when studying a new field, one should look at ...
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votes
0
answers
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views
Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?
Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...
- 381
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votes
0
answers
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views
Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?
My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider,
$\dot{x}=Ax$, where $x$ is the infinite dimensional ...
- 2,271
12
votes
4
answers
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Undecidability in Conway's Game of Life
I strongly believe that - given the rules of Conway's Game of Life and an initial configuration - it is not decidable by a Turing Machine whether a given pattern will emerge, let alone as a stable ...
- 1
5
votes
2
answers
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Local cross-sections for free actions of finite groups
Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...
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9
votes
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213
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Rigorous results on chaos in a driven damped pendulum
The harmonically driven damped pendulum is often used as a simple example of a chaotic system, the equation is just $\ddot{\phi}+\frac1q\dot{\phi}+\sin\phi=A\cos(\omega t)$. As long as $A$ and $\omega$...
- 1,599
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votes
1
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views
The graph of Rule 110 and vertices degree
Consider the elementary cellular automaton called Rule 110 (famous for being Turing complete):
It induces a map $R: \mathbb{N} \to \mathbb{N}$ such that the binary representation of $R(n)$ is ...
- 6,337
0
votes
0
answers
215
views
Proving positive invariance
I need to prove that set $D$(A picture for Set $D$) given by
$$D=\{(x,y):0\leq x\leq L_0,~0\leq y\leq X_0,~0\leq x+y \leq R_0\}\subseteq \mathbb{R}_+^2$$ of the system:
$$\dot{x}=k_1(R_0-x-y)(L_0-x)-...
- 1
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votes
5
answers
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Computing the centers of Apollonian circle packings
The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to ...
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Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?
It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form
$$
x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)}
$$
for ...
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votes
3
answers
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Classes of dynamical systems
A consequence of Birkhoff ergodic theorem tells us that ergodicity is equivalent to:
$\forall A,B \in \mathcal{B} \quad \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to \infty}{\...
- 985
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vote
1
answer
164
views
Ratner's orbit closure for a unipotent semigroup
For Ratner's orbit closure theorem, one may refer to the following Wikipedia page.
Let $\{u_t\mathrel: t\in \mathbb{R} \}\subset G$ be a unipotent one-parameter subgroup of a connected Lie group. Let $...
- 2,379
1
vote
1
answer
128
views
Balls in minimal systems
If $(X,T)$ is a minimal system uniquely ergodic with $\mu$, is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$?
- 11.4k
6
votes
2
answers
235
views
Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing
The following result is on page 26 of this paper by Ferenczi [PDF].
Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., $\mu(T^nA \cap B) \...
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5
votes
1
answer
202
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Noncommutative symmetric spaces
I am recently studying ergodic actions of Lie groups acting on Riemannian symmetric spaces. Since I am also interested in operator algebras, it makes me wonder if there are some very natural ...
- 1,144
3
votes
0
answers
42
views
Approximation of vector field by vector fields with all trajectories closed
Let $\vec j$ be a smooth compactly supported vector field in $\mathbb{R}^3$ such that $div(\vec j)=0$. Are there known either necessary or sufficient conditions such that there exists a sequence of ...
- 21.1k
3
votes
1
answer
513
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Lagrange’s interpolation formula: Theoreme and Example [closed]
I would like to know where they come up with the formula of Lagrange interpolation (Lagrange’s interpolation formula),Lagrange_polynomial because I did some research, but I find a different definition ...
- 30k
1
vote
1
answer
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What are the hypotheses we should add for the generalizations of Furstenberg recurrence theorem?
In my question here I suggest a possibility for generalization of Furstenberg recurrence theorem needing some hypothesis for that generalization to be hold in the side of convergence of the below ...
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