Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$

Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...
0 votes
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13 views

Time-scale calculus (an similar approaches - measure chains) on more general "time" sets

Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
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2 votes
1 answer
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Lyapunov exponents of convergent sequence of matrices

Let $A_n$ be a sequence of $d \times d$ matrices converging to a matrix $A$, all invertible and diagonalizable. We can define the Lyapunov spectrum of the corresponding dynamical system: $$ \chi = \...
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0 answers
19 views

Construction of a homogeneous Moran set

Fix a positive integer $N\ge 2$, for $n \in \mathbb{N}$, denote $$\Sigma=\{0,1,\dots,N-1\},\\ \Sigma^n=\{(\omega_1,\dots,\omega_n):\omega\in\Sigma, i=1,\dots,n\}.$$ Let $p>2$ be a positive integer. ...
0 votes
0 answers
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Hopf normal form of 2D system – How to calculate parameter $\beta$?

In the Scholarpedia article on the Hopf bifurcation, it is discussed that any generic (nondegeneracy and transversality) 2D system exhibiting a Hopf bifurcation has a locally topologically equivalent ...
2 votes
0 answers
117 views

Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/\operatorname{SL}(d,\mathbb Z)$ is ergodic

I wonder if there are any direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/\operatorname{SL}(d,\mathbb Z)$ is ergodic (or even stronger, mixing). ...
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12 votes
1 answer
386 views

Does locally nilpotent imply nilpotent for continuous self-maps of intervals?

Let $f\in C([0,1],[0,1])$ be such that: $$\forall x\in [0,1], \; \exists k\in \mathbb N, \; f^{\circ k}(x)=0.$$ Is it true that $f$ is nilpotent (i.e., that there is some $k$ such that $f^{\circ k}=0$)...
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2 votes
0 answers
74 views

Ergodic diffeomorphisms of the circle

From the paper Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a ...
4 votes
1 answer
291 views

A process of repeated convolution and conditioning and the resulting sequence of probability distributions

I am interested in the following procedure that yields a sequence $D_1,D_2,\ldots,$ of probability distributions over $\mathbb{R}^n$. Let $D_1$ be the $n$-dimensional Gaussian distribution with ...
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0 answers
36 views

Reference for rigidity of higher rank action

I heard some results about the rigidity of higher rank action and it looks very interesting. I would like to know if there are any good survey of paper to get started in this field. Thank you in ...
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1 vote
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Smoothness of unstable manifold near (non?)-hyperbolic fixed point w.r.t. generator of the flow

Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can ...
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Properties of the quadratic form for the sum of the Jacobian and its transpose

Consider the Jacobian matrix of $$ \begin{bmatrix} f_{1}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))\\ \vdots \\ f_{N_2}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t)) \end{bmatrix} $$ Let the jacobian ...
7 votes
1 answer
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State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye : "If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...
3 votes
0 answers
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Regularity of center manifold

Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$. Suppose that the spectrum of $\mathrm{D}f(\...
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A measure on the group of homeomorphisms of $\mathbb T^2$

Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
1 vote
0 answers
151 views

Building random homeomorphisms of the torus $\mathbb T^2$

In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
1 vote
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80 views

Building random homeomorphisms of the circle

Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \...
2 votes
0 answers
79 views

Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?

The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
2 votes
1 answer
254 views

Examples of ODEs with complex constant coefficients and applications to physics?

This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics? but received no answers. I am reposting it here on the hope that it catches ...
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2 votes
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Weakly mixing diffeomorphism

From Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...
6 votes
2 answers
1k views

Poincaré recurrence and its implications for statistical physics and the arrow of time

A very important theorem in mathematical physics is Poincaré’s recurrence theorem. As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for ...
4 votes
1 answer
214 views

What is the name for a point that is periodic to within $\varepsilon$?

Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$. Now suppose that $X$ is a topological space and $f$ is ...
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2 votes
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Examples of chaotic self-similar blowup in PDEs

When the Cauchy problem to a PDE blows up, it can often be analyzed using self-similar variables. In the reference: Eggers, J., & Fontelos, M. A. (2008). The role of self-similarity in ...
6 votes
1 answer
395 views

Equidistribution modulo 1

We know that the time spent by the sequence $na \mod 1$, $n$ ranging from $1$ up to $x$ and $a$ irrational, at any interval of length $\delta$ is approximately $\delta x$. There are known results when ...
3 votes
0 answers
43 views

Stable sets for gradient flow of functions with singularities

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a real-analytic function, and let $F_t$ denote the gradient flow of $f$ with respect to some background metric. Suppose that $df = 0$ at a point $p$. In the ...
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1 vote
1 answer
101 views

Local rule for the product of two cellular automata

Consider two one-dimensional cellular automata $(A^{\mathbb Z},F)$ and $(B^{\mathbb Z},G)$ with alphabets $A$ and $B$ and global rules $F: A^{\mathbb Z} \to A^{\mathbb Z}$ and $G: B^{\mathbb Z} \to B^{...
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6 votes
0 answers
192 views

Examples of expansive homeomorphisms with the specification property that are neither symbolic nor factors of mixing SFT nor product of thereof

I am looking for nontrivial examples of expansive homeomorphisms with the specification property on compact metric spaces. Here, by a ``trivial'' example I understand a subshift with the specification ...
4 votes
0 answers
90 views

The Logistic map have subexponential decay of correlation?

I was looking for information about the correlation decay of the logistic map, more precisely if there is any parameter for which its decay is subexponential, in which case I would like to know if it ...
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2 votes
0 answers
43 views

Search for period N logistic map

The logistic map is a period doubling bifurcation system. Are there known dynamical maps, which oscillate between $N$ points where $N$ is a prime number, like 2, 3, or 5, or 7... , where each ...
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3 votes
1 answer
214 views

Solution to a Sylvester equation with positive definite coefficients

Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$ \begin{align*} C = A^TXA + B^TXB. ...
0 votes
0 answers
30 views

How would you approach unknown states/variables in steady-state or equilibrium?

Consider a linear state space system $ \dot{x} = Ax + B$, with $x$ being a $n\times 1$ vector of $n$ state variables, and $A$ and $B$ being known matrices with dimensions $n\times n$ and $n\times m$, ...
2 votes
0 answers
93 views

Entropy of a sequence

I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following, It would be interesting to rule out the possibility of $\lambda_f(n)$ ...
1 vote
0 answers
31 views

What *piecewise* smooth curves/surfaces/hypersurfaces give rise to forward-invariant regions of dynamical systems?

Consider a set $\mathcal{B}\subset \mathbb{R}^n$ that is homeomorphic to a closed n-dimensional ball, and denote its boundary by $\mathcal{H}$. Assume that $\mathcal{H}$ is a "piecewise smooth&...
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1 vote
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Oseledets splitting of $f_{0}$ is either dominated with index 1 or trivial at $x$

There is a well-known result by Bochi and Viana that said Let $f_{0} \in \operatorname{Diff}_{\mu}^{1}(M)$ be such that the map $$ f \in \operatorname{Diff}_{\mu}^{1}(M) \mapsto\left(\mathrm{LE}_{1}(...
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4 votes
2 answers
276 views

Borel summation and the Abel function of $e^z-1$

This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
6 votes
1 answer
222 views

Are all quasi-regular points on Polish spaces generic points?

Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
1 vote
0 answers
36 views

Lipschitz continuous paths on manifolds which accumulate on non-injective paths

This question is concerned with a dynamical system given by a shift flow on a metrisable topological space of functions from $\mathbb{R}$ to a closed manifold. The choice of manifold affects the ...
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1 vote
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Basin of attraction comparative statics* using local energy functions?

Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
2 votes
0 answers
87 views

Uniformly weak mixing transformations

Let $(X, T, \mathcal F, \mu)$ be a nonatomic standard probability space equipped with a measure preserving transformation $T$. We say $T$ is uniformly weak mixing if for every $\varepsilon > 0$, ...
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1 vote
0 answers
106 views

Denjoy example in the Poincaré–Bendixson theorem

I have already finished understanding the Poincaré-Bendixson theorem as a consequence of Schwartz's theorem, but I also want to analyze the example that Denjoy gave in $C^1$ that is not within the ...
3 votes
0 answers
112 views

Naturality of geodesic flow

Let $\texttt{Man}$ be the category of smooth manifolds with local diffeomorphisms as morphisms, and $ \texttt{Bun}$ --- the category of bundles (affine bundles or just fibre bundles, if necessary) and ...
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1 vote
1 answer
43 views

Analytically characterizing basins of attraction boundaries and sizes

While I understand that doing the above is not possible in general, I would like to know more about how to proceed when it is possible. That is, what are the common methods people use to analytically ...
6 votes
2 answers
214 views

Topological dynamical systems with only zero-entropy factors

Suppose the dynamical system $(X,T)$ has only proper factors (i.e. not $(X,T)$ itself) of zero topological entropy. Does the system $(X,T)$ also have zero entropy?
44 votes
3 answers
6k views

Does Conway's game of life admit a notion of energy?

(I am not sure if this is a mathematics or physics question so I am not sure where to post it. I am posting it here because the chief subject is an unreal universe that is purely a subject of ...
1 vote
0 answers
41 views

Motivation for Ionescu Tulcea-Marinescu (Lasota-Yorke inequality)

I wonder about motivations of a work of Ionescu Tulcea-Marinescu. In order to establish the decomposition of the operator $T$ they assume (condition (1.3)) this operator satisfies the inequality $$\|...
3 votes
1 answer
110 views

Does uniform recurrence imply uniform convergence of the Birkhoff sums?

Let $(X, T, \mathcal F, \mu)$ be an ergodic measure preserving system with finite measure. Suppose $T$ is uniformly recurrent, in the following sense: For every $A \in \mathcal F$, there exists an $M \...
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4 votes
1 answer
132 views

Ergodic decomposition of the action of a subgroup

Let $G$ be a countable abelian group and let $H \le G$ be a subgroup. Let $G \curvearrowright (X,\mu)$ be an ergodic measure preserving action on some probability space $(X,\mu)$. Now we know that the ...
3 votes
1 answer
87 views

the definition of the topological pressure for matrices

Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma_n$ the set of addmisible words with the ...
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2 votes
1 answer
100 views

Finding the repelling fixed point of an exponential, knowing only its attracting one

This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...
3 votes
0 answers
83 views

What dynamical properties should we expect from systems satisfying statistical ones?

Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example: the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...

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