**5**

votes

**0**answers

74 views

### What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?

By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...

**5**

votes

**0**answers

95 views

### A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$
Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...

**5**

votes

**0**answers

63 views

### Is there a name for a bifurcation where a stable fixed point bifurcates into three fixed points: stable, unstable and saddle?

Consider the differential equation on $\mathbb{R}^2$ whose polar-coordinate representation is given by
$$ \begin{array}{r c l} \dot{r} & = & r(\alpha - r^2) \\ \dot{\theta} & = & -sin(\...

**4**

votes

**2**answers

270 views

### Entropy equals zero?

Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$
in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define
$\mathcal{P} = \{[0],[1]\}$;
$\mathcal{P}^{n} = \mathcal{P}\...

**2**

votes

**0**answers

122 views

### Flow on invariant Lagrangian tori

The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

**2**

votes

**0**answers

88 views

### What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).
Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...

**1**

vote

**0**answers

221 views

### A Perron-Frobenius problem

Let $A$ be an irreducible nonnegative matrix with column sums equal to 1.
Let $b\in R^n$ have components summing to 0, and let $u$ be the solution of $u=Au+b$ with components summing to 1 (unique ...

**2**

votes

**1**answer

124 views

### Difference between Hopf-Turing bifurcation and traveling-waves bifurcation in reaction-diffusion systems

I am perturbing a dynamical system with a perturbation that looks like $$\vec u e^{-i k x} e^{\sigma(p;k)t}$$ where $\sigma$ is a function of the parameters of the dynamical system and the wavenumber, ...

**5**

votes

**0**answers

111 views

### cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...

**1**

vote

**1**answer

89 views

### Definition o branched 1-manifold [closed]

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia:
"A finite graph whose edges are smoothly embedded arcs in a surface, ...

**6**

votes

**1**answer

309 views

### electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...

**4**

votes

**1**answer

63 views

### Problem on differential inclusion

For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the ...

**3**

votes

**0**answers

154 views

### Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...

**7**

votes

**0**answers

127 views

### Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...

**1**

vote

**0**answers

55 views

### Ordering periodic orbits

I want to prove the proposition:
Proposition- Let $f:I \to I$ be continuos, and let f have a (2n+1)- periodic orbit {$x_{k}=f^{k}(x_{0})$, $k=0,1,\dots,2n$}, but no (2m+1)-periodic orbit for $1\...

**2**

votes

**1**answer

126 views

### Multi dimensional symbolic dynamics

I want to learn Multi dimensional symbolic dynamics. can you point to any recent thesis containing a good exposition or lecture notes?

**1**

vote

**0**answers

46 views

### Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?

I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...

**4**

votes

**1**answer

214 views

### The converse of von Neumann's mean ergodic theorem

Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following.
Let $\{F_n\}_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ ...

**39**

votes

**5**answers

2k 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 ...

**3**

votes

**0**answers

91 views

### Perturbations to a vector field

I ran into some problems while working through a proof of the Poincare-Hopf theorem that essentially boiled down to the following question: given a smooth vector field $V$ on a (compact Riemannian) ...

**2**

votes

**0**answers

50 views

### When do positively invariant subset contain a given set?

Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't ...

**1**

vote

**0**answers

123 views

### Discrete group action on the sphere

Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$
be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...

**3**

votes

**2**answers

123 views

### Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...

**4**

votes

**1**answer

191 views

### Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...

**5**

votes

**0**answers

227 views

### The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...

**0**

votes

**1**answer

87 views

### Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...

**5**

votes

**1**answer

139 views

### When is the time one map of a suspension flow ergodic?

I'm sure the answer to the following question is well known but I couldn't find the answer I needed.
Let $(\Sigma,\sigma)$ denote the full shift on $k$ symbols and let $\mu$ be an invariant measure ...

**1**

vote

**2**answers

154 views

### Do ergodicity, minimality and equicontinuity on a compact space imply total ergodicity?

Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ?
According to some results I found in some books, a rotation on a ...

**3**

votes

**1**answer

139 views

### Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...

**1**

vote

**0**answers

48 views

### Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...

**7**

votes

**2**answers

244 views

### Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...

**3**

votes

**1**answer

177 views

### Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.
For any $x \in E$, there is a ...

**2**

votes

**1**answer

111 views

### Determining whether $k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$

My REU partner and I are working on a problem involving iterations of quadratic rational maps over an algebraically closed field $K$ that is complete with respect to a non-trivial non-archimedean ...

**5**

votes

**2**answers

212 views

### Are periodic billiard trajectories stable on a manifold with strictly convex boundary?

Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary.
Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reflects specularly at the boundary).
...

**2**

votes

**2**answers

311 views

### Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...

**2**

votes

**1**answer

324 views

### Hilbert 16th problem, distribution of Limit cycles

Edit: Can one help for translation of the link in Russian(comment by Dimitry Todorov)(Or at least a summary of it)?
It seems that the second part of the Hilbert 16th problem is solved or is going to ...

**1**

vote

**2**answers

225 views

### Growth of the size of iterated polynomials

I have been working independently on a project but now I am stuck and need to seek an expert's wisdom for a part of it. I am basically looking for theorems related to growth of the size of polynomials....

**3**

votes

**3**answers

116 views

### Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} ...

**5**

votes

**0**answers

102 views

### Quantitative approximation of invariant measures by periodic ones

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. Is there some knowledge ...

**1**

vote

**1**answer

238 views

### Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...

**2**

votes

**0**answers

59 views

### Any minimal WAP dynamical system is distal

I'm trying to show that any minimal WAP dynamical system $(X, G)$ is almost periodic. By Ellis's joint continuity theorem, it suffices to show that any minimal WAP system is distal. There are many ...

**1**

vote

**0**answers

87 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

**10**

votes

**1**answer

211 views

### Is an explicit $c$ known to lead to a noncomputable Julia set?

Braverman & Yampolsky have shown that there exist noncomputable Julia sets,
i.e., there exist $c \in \mathbb{C}$ such that the Julia set of $f(z) = c + z^2$
is not computable.
"A set is computable,...

**26**

votes

**4**answers

1k views

### Stability of the Solar System

Is the Solar System stable?
You can see this Wikipedia page.
In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and ...

**1**

vote

**1**answer

145 views

### Convergence to equilibrium via gradient descent

J. B. Rosen proved that in concave games of n players (which assumes that Cartesian product of strategy profiles is convex) if the game satisfies the condition of diagonally strictly concave then ...

**20**

votes

**0**answers

752 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**1**

vote

**0**answers

22 views

### Bifurcations in flows on 2-dimensional torus

I am doing research on bifurcations which appear in flows on the
2-dimensional torus, in particular on such which do not appear in flows
on $\mathbb{R}^2$.
Can anyone provide some references on ...

**0**

votes

**0**answers

76 views

### Two vector fields are cojugate but not take orbits

Let $X$ and $Y$ be $C^1$ vector feilds on $R^m$. Suppose that $0$ is an attracting hyperbolic singularity for $X$ and $Y$. Show that there exists a homemorphism $h$ of a neighborhood of origin which ...

**3**

votes

**0**answers

40 views

### Limit Behavior of Iterated Curvature-Function

What can happen, if one defines an infinite sequence of functions as follows
$f_0\in C^\infty: x\in\mathbb{R}\mapsto y\in\mathbb{R}$
$f_{n+1}: \int_0^x \sqrt{1+(f_n'(t))^2}dt\mapsto\frac{f_n''(x)}{\...

**6**

votes

**0**answers

261 views

### Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...