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
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Global first integral for certain $3$ dimensional system
A physicist colleague asks me the following question. I have no idea to answer him. Your answer is very appreciated.
Is there a global first integral on $\mathbb{R}^3$ for the following vector field?
...
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0
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Run-away Volterra operator
For a continuous function $k:[0,1]^2\to \mathbb{R}$, let $A$ be the generalized Volterra integral operator on $C([0,1],\mathbb{R})$ defined by
$$
A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds, \quad t \in [...
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Integrability/regularity of Lyapunov exponents
My question is about some basic properties of Lyapunov exponents. Sorry if this stuff is basic, I just don't know where to find the statements that I'm looking for.
Preliminaries. Let $X$ be a closed ...
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The mysterious numbers $ \frac{13}{20} $ and $20$?
Let $g(x) = x^6 - 30 x $
Let $h(x) = x^6 $
Let $f(x) = x^2 - 2 $
Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$
Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...
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Countable-to-one factors of measure preserving systems do not change entropy
It is known that if $\psi$ is a factor map between probability measure preserving systems $(X,\mathscr{X},\mu,T)$ and $(Y,\mathscr{Y},\nu,S)$ is countable-to-one almost everywhere, then $h(\mu,T)=h(\...
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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.
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Understand the condition of transcritical bifurcation (Crandall-Robinowitz) geometrically
Consider the dynamical system $\dot{x}=F(x,\lambda), x\in\mathbb{R^n}$, and let $F(0,\lambda)=0$ for some neighborhood of $\lambda_{0}$, the transcritical bifurcation arises if we have $w\frac{\...
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Piecewise expanding map
I think that piecewise expanding maps on the unit interval have been studied. Is there something analogous in dimension 2 with the same good properties? In other words, maybe I want some good results ...
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Is there a generalization of Furstenberg theorem from SL(2,R) to SL(2,C) matrices?
I learnt from a talk that consider a random product of i.i.d. matrices, randomly chosen from SL(2,R): $T_n=A_n \cdots A_2 A_1$, where the random matrices $A_i$ are i.i.d.
A classical Furstenberg ...
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Existence and uniqueness of a stationary measure
This same question was also posted on MSE https://math.stackexchange.com/questions/3327007/existence-and-uniqueness-of-a-stationary-measure.
Recently I have posted the following question on MO ...
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What is the current status on methods to find limit cycles?
What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form:
$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$
Where $...
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On the first sequence without collinear triple
Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
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On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
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15
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Unconventional examples of mathematical modelling
I'll soon be teaching a (basic) course on mathematical control theory. The first part of the course will focus on mathematical modelling of dynamical systems. More precisely, I will present examples ...
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Trapping lightrays under nonstandard reflections and/or paths
Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open:
"It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
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Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra
Let $H$ be a Hilbert space and $X$ be a compact Hausdorff space with a homeomorphism $\alpha: X \to X$. Assume that $C(X)$ is a commutative sub algebra of $B(H)$, namely $C(X)$ is embedded in $B(H)$...
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The number of minimal components of a dynamical system via certain invariants of corresponding cross product $C^*$ algebra, some precise examples
Let $X$ be a compact Hausdorff space and $\alpha$ be a homeomorphism of $X$.
So we have a natural action of $\mathbb{Z}$ on $C(X)$ which generates the cross product algebra $C^*(X,\alpha)$...
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5
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A non-geodesible foliation of $S^3$ or $S^2\times S^1$
Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?
If the answer is ...
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Existence of the eigenvalue of the dual operator of the transfer operator
In the passage that I marked in green apparently the author uses a relationship between fixed point and eigenvalues. The result that I know of to ensure the existence of this eigenvalue requires that ...
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The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1
We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1?
I ...
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Orbit-based metric
Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be continuous. Then, is there any meaning/research done on the metric
$$
D(x,y)\triangleq \sum_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y));
$$
...
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3
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1
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Attractors in random dynamics
Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
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124
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On invariant cones of the Katok map
I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
- 151
13
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1
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452
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Is the set of escaping endpoints for $e^z-2$ completely metrizable?
Let $f:\mathbb C \to \mathbb C$ be the complex exponential $$f(z)=e^z-2.$$ It is known that $J(f)$, the Julia set of $f$, is a uncountable collection of disjoint rays (one-to-one continuous images ...
- 3,317
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3
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532
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Ruelle-Perron-Frobenius theorem for shift of finite type
I know a version of Ruelle's theorem for expansive transformations in a compact metric space that says there is a single equilibrium state for a potential holder. In this Ruelle-Perron-Frobenius ...
- 677
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0
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87
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Semi-conjugacies between interval and circle maps
There are examples of self-maps of the circle which are semi-conjugate to self-maps of a compact interval. A famous one is the covering map $z\mapsto z^2$ of the unit circle which is semi-conjugate to ...
- 3,599
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135
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Example shows that entropy is not upper semi continuous
Let $(X, \beta, \mu)$ be probabilty space of compact space $X$. Let $T:X \rightarrow X$ be continuous function, and expansive. It is well known that entropy $\mu \mapsto h_{\mu}$ is upper semi ...
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Reduced master equation for a multistable Hamiltonian dynamical system
I am looking for rigorous results on the derivation of a reduced master equation for a (possibly stochastic) Hamiltonian dynamical system with a coercive potential energy term with multiple local ...
- 3,873
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110
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Example of overtwisted contact manifold with finitely many periodic Reeb orbits
Are there examples of overtwisted manifolds with only a finite number of periodic Reeb orbits?
An example is given by the irrational ellipsoid in $(\mathbb{R}^4,\omega_\text{st})$, which is not ...
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Existence of topologically mixing (discrete) dynamical system on manifold
If $M$ is a connected $(d\geq 2)$-dimensional smooth closed manifold, then does there exist a class $C^1$-diffeomorphism $\phi$ from $M$ onto itself, such that $(M,\phi)$ is a topologically mixing (...
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77
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Equivalence between Gibbs measures and conformal measures
I was reading an article about Gibbs measures, but the author defines Gibbs measures in a different way than the usual (which is done by using conditional expectations). The way that he defines I have ...
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3
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160
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Non-existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$
Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question:
Is there a complete vector field $X$ on $\mathbb{R}^2\...
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5
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1
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An entire function all whose forward orbits are bounded
Edit: I revise the question according to the comment of Gabe Conant.
What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?:
For every $...
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Quantitive and computational improvement of the Oseledets multiplicative ergodic theorem for irrational rotation
Consider irrational rotation $T:S^1\to S^1, T(x) = x + \alpha$ where $\alpha\notin \mathbb{Q}$ (you may assume additional number theoretic properties of $\alpha$, say $\alpha = \sqrt{2}$ is already ...
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1
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Is there a name for a "stable" physical measure?
Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support.
Definition. The ...
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A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)
Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
- 356
4
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1
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When entropy SRB measure is zero
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures.
Let $f:M \rightarrow ...
- 1,043
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An analytic vector field on $S^3$ whose all orbits are dense (à la Seifert conjecture, 2)
Does there exist a real analytic vector field on $S^3$ all of whose orbits are dense? The second paragraph of page 285
Of this paper says that there is a vector field whose almost all orbits are dense:...
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1
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Keeping track of limit cycles via certain second order differential operator
Inspired by the two posts which are linked bellow we ask the following question:
Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
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3
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151
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Anosov flows on non-compact manifold
So, when defining Anosov flows on a compact manifold specifying the Riemannian metric is not necessary as any two are equivalent. So, my question is:
Given a non compact manifold $M$ and a proper ...
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1
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258
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Does Anosov geodesic flow imply asphericity?
Let $(M, g)$ be a closed smooth Riemannian manifolds with Anosov geodesic flow, does it implies that $M$ is an aspherical manifold?
I am thinking it is not known yet?
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2
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278
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Periodic orbit for certain vector field on $S^3$ (à la Seifert conjecture)
The standard frame for $S^3$ consists of $X_i,X_j,X_k$ with $X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$ where $i,j,k$ are standard quaternion numbers, $a\in S^3$, and the multiplication is the quaternion ...
- 356
7
votes
2
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572
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Smooth Julia set for quadratic polynomials
This question is related to a classification of rational maps in terms of properties of their Julia set.
Let $f= z^2 + c$, with $c\in \mathbb{C}$ be a quadratic polynomial such that its Julia set $J(...
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6
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2
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625
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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 ...
- 356
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282
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A cohomology associated to a vector field on a Riemannian manifold
Edit: Accoring to the comment of Asura Path I revise the question.
Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...
- 356
2
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0
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100
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Homomorphism or derivation conserving irreducibility
Let $R$ be a integral domain and $\phi$ be an automorphism of $R$. For a given element $x \in R$, we consider a sequence $(\phi^n(x))_{n=0}^{\infty}$.
I wonder if there is any related theory to ...
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802
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Hilbert 16th problem and dynamical Lefschetz trace formula
I would like to apply the known version of the conjectural formula (11) page 10 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...
- 356
5
votes
1
answer
636
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Leafwise de Rham cohomology (A true definition of differential forms along leaves)
For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of ...
- 356
9
votes
2
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472
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Orbit of an irreducible representation of a surface group under that action of the mapping class group
Let $F$ be a closed oriented surface of negative Euler characteristic. Let $X^i(F)$ be the subset of the $SL_2\mathbb{C}$-character variety of the fundamental group of $F$ corresponding to irreducible ...
- 3,474
2
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2
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584
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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 ...
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