Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
11
votes
2
answers
308
views
Cohomology of foliations and closed forms along the leaves
Let $M$ be a manifold equipped with a codimension one, transversely orientable, regular foliation $F \subset M$. Let $\alpha\in \Omega^k(M)$ be a differential form on $M$ that is not closed on $M$ ...
- 2,816
0
votes
0
answers
34
views
Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term
The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
- 395
4
votes
0
answers
179
views
Recognize this metric? Do you have a name for this metric on the product of spheres?
Take the product $S^2 \times S^2$ of two two-spheres,
but perturb the product metric as follows.
Think of each $S^2$ as the unit two-sphere in Euclidean 3-space
in the standard way
so that for $p ...
- 8,336
4
votes
0
answers
116
views
Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 41
1
vote
0
answers
38
views
Is there an equivalent to the logistic map for a nonlinear path through some of the other nodules of the Mandelbrot set?
The logistic map can be related to the real axis of the Mandelbrot set, looking at the different cycle lengths as you pass through all the various nodules along the real axis. But there are other ...
5
votes
1
answer
274
views
Why "no wandering domain" fails in parabolic basin?
Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$
I am familiar with the proof: spread around ...
0
votes
0
answers
79
views
Alternative proof of parabolic implosion
I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation.
Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
0
votes
1
answer
64
views
Conditions required for the orbit of a set of positive measure to cover state space?
Suppose $(X, \mathcal{M}, \mu, T)$ is a measure-preserving dynamical system with $T$ invertible.
I am wondering what properties the dynamical system would need to have in order for the following to be ...
- 1,087
0
votes
1
answer
64
views
Transitive map on a profinite group
Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
- 361
0
votes
0
answers
26
views
For which values of $\mu$ is the Standard Map $f_{\mu}(x,y)=(x+y+\frac{\mu}{2\pi} \sin (2\pi x),y+\frac{\mu}{2\pi}\sin (2\pi x))$ non wandering?
Let $f_{\mu}(x,y)=(x+y+\frac{\mu}{2\pi} \sin (2\pi x),y+\frac{\mu}{2\pi}\sin (2\pi x))$, with $\mu > 0$ and considered in the cylinder $\mathbb{R}/\mathbb{Z} \times \mathbb{R} $. For which values ...
1
vote
0
answers
59
views
Asymptotic behavior of positive solution to nonlinear scalar field equation
It is well-known that the radial positive solution
$u=u(r)$ to nonlinear scalar field equation $$-\Delta u+u=u^p\text{ in } ~\mathbb{R}^d, 1<p<\frac{d+2}{d-2}$$ has the following asymptotic ...
- 705
4
votes
1
answer
130
views
Restrict sigma algebra in measure-preserving system
Consider a measure space $(X,\mathcal{A},\mu)$ and a measure-preserving transformation $\phi \colon X\rightarrow X$, that is, $\phi$ is measurable and $\phi_*\mu = \mu$.
My intuition tells me that we ...
- 257
0
votes
1
answer
89
views
Singular continuous ergodic measures for the map $z \to z^2$
Where can I find the details of constructing singular continuous ergodic measures for the map $z \to z^2$ on the unit circle? I know that it was done by Furstenberg, but I could not find it explicitly ...
4
votes
1
answer
270
views
Examples of discrete-space continuous-time dynamical systems
Something that I see occur repeatedly in my work is the need for formal notions of discrete-space continuous-time dynamics — these are generally realized as digital oscillators that are interact using ...
- 41
7
votes
1
answer
211
views
Existence of asymptotic sequence in ergodic measure-preserving transformations
Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \...
- 113
1
vote
0
answers
63
views
Stability Problem in a Nonlinear Dynamical System
Consider the nonlinear dynamical system given by the following differential equations
\begin{cases}
\dot{x} = y, \\
\dot{y} = x - x^3 - \gamma y + \delta x^2 y.
\end{cases}
I want to demonstrate that, ...
- 31
4
votes
2
answers
165
views
Convergence of the Cesàro mean of iterated continuous functions
Does anyone have a counter-example of the following statement :
Let $f : [0;1] \to [0;1]$ a continuous function w.r.t. the usual topology. Let $A_n(x) = \frac{1}{n} \sum_{k=0}^{n-1} f^k(x)$ for $n \ge ...
- 43
1
vote
1
answer
151
views
Does this sequence of Blaschke Product have rescaling limit $z-1$?
Background: The conformal conjugacy class of parabolic isometry of upper half plane $\mathbb{H}$ consists of $f(z) = z+1$ and $g(z)=z-1$.
Consider surjective proper holomorphic $F_n: \mathbb{H} \...
6
votes
1
answer
165
views
Number of periodic points of subshift of finite type
Let $(X, \sigma)\subset (\{0, 1, 2, 3\}^\mathbb{N},\sigma)$ be a subshift of finite type. Let $P_n$ be the set of $n$-periodic points. If $|P_n|=2^n$ for all $n\ge 1$, then it is true that $(X, \sigma)...
- 675
3
votes
0
answers
66
views
Borel complexity of the set of generic points for an invariant measure in a minimal system
I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is ...
- 1,658
5
votes
1
answer
389
views
Is a random circle rotation weak mixing almost surely?
Consider the random circle rotation $x \to x + Z \text{ mod 1}$ on $([0, 1], \text{Lebesgue})$ where at each rotation, $Z$ is uniformly distributed on $[0, 1]$ and independent of previous rotations.
...
- 6,215
4
votes
2
answers
376
views
Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
- 807
3
votes
1
answer
130
views
Do sets of big returns contain sets of returns?
We say a subset $E$ of $\mathbb{N}$ is a set of returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B}$ with $\mu(A)>0$ such that $E=\{n\in\mathbb{N};\...
- 10.6k
2
votes
2
answers
175
views
Great literature on discrete dynamical systems and/or qualitative theory of difference equations
I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some ...
5
votes
1
answer
212
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
3
votes
0
answers
107
views
Stability of a nonlinear dynamical system with non-elementary dynamics
I am trying to prove stability and get a non-asymptotic upper bound on the convergence rate of a nonlinear discrete-time dynamical system, whose dynamics are stated in terms of the (non-elementary) ...
- 31
2
votes
0
answers
78
views
On reproducing the Poincare section figure in a paper by Sato, Akiyama and Doyne Farmer [closed]
I am trying to reproduce Figure 1 in the paper "Chaos in learning a simple two-person game" (English) Proc. Natl. Acad. Sci. USA 99, No. 7, 4748-4751 (2002) (MR1895748, Zbl 1015.91014), by ...
1
vote
0
answers
60
views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
1
vote
1
answer
74
views
On the maximal difference between points in orbit
Let $(X, T, \mu)$ be an ergodic measure preserving system with finite measure, and $f \in L^{\infty} (X)$.
Define the maximal orbit deviation function $D_f: X \to \mathbb R$ by
$$D_f := \sup_{n, m \...
- 6,215
0
votes
0
answers
41
views
Analysis of sensitivity to initial conditions in dynamic systems
Consider the iterative function defined by:
$$
x_{n+1} = f(x_n)
$$
where $x_0\in [0, 1]$ and
$$
f(x) = \sin\left(\pi \left(b^{rx(x-1)}\mod 1 \right)\right)
$$
with $b, r > 0$. We aim to demonstrate ...
- 11
5
votes
0
answers
156
views
What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
- 6,215
13
votes
2
answers
800
views
For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,215
0
votes
0
answers
63
views
Convergence of the trajectory of ODE
Consider a $C^\infty$ smooth nonconvex function $f$, and ODE
$$
\begin{cases}
\dot{x}=-x\circ\nabla f(x),\\
x(0)\in\mathbb{R}^d_{++}.
\end{cases}
$$
Here $\circ$ is elementwise product, $\mathbb{R}^d_{...
- 43
1
vote
0
answers
82
views
Dynamical properties of cellular automata of small diameter
Let $f\colon\{0,1\}^k\to\{0,1\}$ be a function, $j$ an integer, and define the cellular automaton $F\colon\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ by $F(x)_i=f(x_{i+j},\dotsc,x_{i+j+k-1})$.
I wonder ...
- 29
1
vote
0
answers
84
views
Coarse well-distributedness/equidistribution of Pell sequence prefixes
I am interested in the distributedness or "mixing" behavior of certain
linear recurrences modulo powers of $2$.
In particular, consider the Pell sequence (https://oeis.org/A000129),
modulo $...
- 11
0
votes
0
answers
81
views
Replacing the sequence in Chowla's conjecture and positiveness of the entropy
For any fixed integer $m>0$ and not-all-even $(a_1,\ldots,a_m)\in\mathbb N^m$, one version of Chowla's conjecture states that
$$
\lim_{x\rightarrow\infty}\frac{1}{x}\sum_{n\leq x}\mu(n+1)^{a_1}\...
- 457
5
votes
1
answer
478
views
Status of infinitesimal Hilbert's sixteenth problem
What is the status of the "infinitesimal Hilbert's sixteenth problem" (aka "Hilbert-Arnold Problem")? According to the Russian Wikipedia article, it was still open in 2009. I am ...
- 161
2
votes
0
answers
92
views
Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
- 307
2
votes
0
answers
54
views
Ashkin-Teller Model
Consider the two-dimensional Ashkin-Teller model on the square lattice $\mathbb{Z}^2$ with Hamiltonian:
$$ H = - \sum_{\langle i,j \rangle} \left[ K \sigma_i \sigma_j + K \tau_i \tau_j + k \sigma_i \...
- 21
1
vote
0
answers
47
views
Computing the language of an $S$-adic shift
I have been looking online for how or if one can compute the language of an $S$-adic subshift generated by finitely many substitutions. I know that one can compute the language of a substitution ...
- 633
3
votes
2
answers
271
views
Orbits under the automorphism group of projective space
Let $\mathbb{P}^d_K$ be projective space of dimension $d\geq 1$ over an infinite field $K$. Let $x\in\mathbb{P}^d_K$ with $\dim\overline{\lbrace x\rbrace}=n\leq d-1$.
My question: is the set $\lbrace ...
- 133
0
votes
0
answers
42
views
Geometric alignment of adaptive models on evolving manifolds
Let $(M_t)_{t\in[0,T]}$ be a smooth family of compact $d$-dimensional Riemannian submanifolds of $\mathbb{R}^n$. Consider a function $f_t : \mathbb{R}^n \to \mathbb{R}$ evolving over time $t \in [0,T]$...
1
vote
0
answers
262
views
Is every self homeomorphism of the open disk conjugate to a homeomorphism extendable to the boundary?
Let $\mathbb{D}=\{z\in \mathbb{R}^2\mid |z|<1\}$
Is it true to say that every homeomorphism of $\mathbb{D}$ is conjugate to a self homeomrphism of the disk extendable to a homeomorphism of $\bar{\...
- 356
0
votes
0
answers
112
views
Vector field connecting two points
I'm now working on somehow an inverse problem of an ODE:
Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t).
Now there is a ...
- 1
3
votes
1
answer
127
views
Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
6
votes
1
answer
170
views
Bounding proportion of phase space which is chaotic
There are dynamical systems which have regions of phase space that are both chaotic and integrable, e.g. small perturbations of integrable systems as in KAM theory. Are there any tools for bounding ...
- 509
3
votes
1
answer
75
views
A uniform upper bound for the linking number of periodic orbits of algebraic vector fields
Inspired by these two posts on knots orbits of polynomial vector fields on $\mathbb{R}^3$(A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit) and (Are total curvature and the ...
- 356
0
votes
0
answers
51
views
A reference for an equation of evolution for a probability measure
I assume that there exist a family of probability measures $(d\mu_{t})_{t\geq 0}$ over the circle $\mathbb{R}_{|2\pi\mathbb{Z}}$ satisfying the following equation of evolution: for every continuous ...
- 449
4
votes
3
answers
288
views
A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit
Is there a polynomial vector field
$$P(x,y,z)\partial_x+Q(x,y,z)\partial_y+R(x,y,z)\partial_z$$
which has a closed orbit $K$ such that $K$ is a non trivial knot?
- 356
1
vote
0
answers
54
views
Are total curvature and the unknoting number of closed orbits of algebraic vector fields bounded uniformly by the degree of vector field?
I am interested in this question since 1999 when I heared the definition of a knot and I read the definition of unknoting and the total curvature of a knot.
To what extent can closed ...
- 356