# 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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votes

**1**answer

356 views

### A curvature description for center condition for quadratic vector field

We consider the quadratic vector field $V$ $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y)
\end {cases}\;\;\;\;(V)$$
where $P,Q \in \mathbb{R}[x,y]$ are polynomials of degree $2$ with $P(0,0)=Q(0,0)=...

**2**

votes

**1**answer

198 views

### continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.
Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...

**7**

votes

**1**answer

622 views

### (In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations
$$
\dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t) = -\gamma x_2(t) - \cos(\...

**6**

votes

**0**answers

58 views

### Pursuit-evasion with many slow pursuers

Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...

**0**

votes

**0**answers

92 views

### Book recommendations beyond an introduction [on hold]

So I’ve scraped the surface of many topics, but I would like to go further. Can anyone recommend some continuations to the following introductory books? It’s okay if necessarily it needs to be a ...

**55**

votes

**6**answers

4k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...

**2**

votes

**1**answer

263 views

### Linear difference inequality

It is well known how to find a solution for the following linear difference equation
$$h_{m} = h_{m-1} + a \cdot h_{m-2}$$
Finding the roots $r_1$ and $r_2$ of $r^2 - r - a$, we have that the ...

**2**

votes

**0**answers

70 views

### From quenched to annealed decay of correlation

If we have quenched decay of correlation, can we transfer it to annealed decay of correlation? To be precise, let us consider following setting:
Given transformations $T_{\omega}: (S^1, dm) \to (S^1, ...

**2**

votes

**1**answer

197 views

### Isochronization of quadratic vector fields with center

What is a classification of all quadratic vector fields
$$\begin{cases}
x'=P(x,y)\\
y'=Q(x,y)
\end{cases}\qquad (V)$$
with a center at origin such that $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\...

**2**

votes

**1**answer

57 views

### Intersection property of a composition of annular mappings

It's easy to construct mappings $R,S:A\to A$ of an annulus $A$ on the plane such that $R$ is a rigid rotation, $S$ possesses the intersection property, but $R\circ S$ does not. However, I'd like to ...

**4**

votes

**3**answers

369 views

### Reference request: Dynamical systems

I’m currently reading Brin and Stuck’s Introduction to Dynamical Systems, and I think I like the field a lot so far. I haven’t finished it quite yet, but what are some other good textbooks I can read ...

**5**

votes

**0**answers

123 views

### Looking for a counterexample for Ruelle's inequality on compact manifold

Let $M$ be a compact differentiable manifold, and $f:M\to M$ be a $C^1$- smooth diffeomorphism.
If Assume that $\mu$ be a $f$-invariant probability measure on $M$.
Then D.Ruelle proved that
$$
h_\...

**0**

votes

**0**answers

87 views

### A complex limit cycle not intersecting the real plane(2)

Inspired by this question and the counter example provided in its answer we ask:
Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the ...

**1**

vote

**1**answer

126 views

### A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf

Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.

**2**

votes

**0**answers

68 views

### Two mixing rates of random dynamical system

Given random dynamical system $(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$ where $(\Omega, \mathbb{P})$ is probability space with ergodic transformation $\sigma: \Omega \to \Omega$. Define ...

**1**

vote

**1**answer

312 views

### A complex limit cycle not intersecting the real plane

Edit: This is a real coefficient version of the current post.
Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow?
There is ...

**4**

votes

**0**answers

48 views

### Obstructions for finding a volume form for a given flow

Let $X$ be a non-singular $C^\infty$ vector field on a three manifold $M$.
There are some obvious obstructions for finding a volume form that is preserved under the flow given by $X$:
If $X$ is ...

**1**

vote

**1**answer

112 views

### Trajectory leaving a set

Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...

**1**

vote

**0**answers

36 views

### Regularity of the pdf of partial Birkhoff sums

Suppose that $T: X \to X$ is some measurable map on a Riemannian manifold $X$ (possibly with boundary). Let $\mu$ denote the Riemannian measure on $X$. For measurable, real-valued $g$ we may consider ...

**0**

votes

**1**answer

225 views

### Center-localized oscillating modes with exponential decay tails, solved from coupled ODE

Two coupled non-linear differential equations in a radial $r$-direction in the region $r \in [0, \infty)$:
$$
-a\big(\partial_r^2+\frac{\partial_r}{r}-\frac{n^2}{r^2}+c\big) U(r)+
B(r) (\partial_r-...

**1**

vote

**1**answer

36 views

### Measuring how suboptimal control is

Suppose I have a linear dynamical system to control. I use PMP to find necessary conditions for the optimal control of the system wrt to some objective function. Now, suppose that the trajectory I ...

**3**

votes

**0**answers

46 views

### Types of triangles admitting periodic billiard orbits

It is an open problem in dynamical systems if every triangle has a periodic billiard orbit. So far it has been proven that equilateral triangles, isosceles triangles, right triangles, and obtuse ...

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vote

**0**answers

217 views

### Functions holomorphic on a region minus a Cantor set - pt.2: Iterated function systems

This post is a follow up to my previous question enquiring whether it is always possible to extend a homeomorphism conformal on a region $R$ minus a Cantor set to the whole of $R$. From the answers I ...

**1**

vote

**1**answer

105 views

### A special oscillatory orbit in space

Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...

**12**

votes

**2**answers

645 views

### Periodic lightray paths trapped between two nested mirror circles

I wonder if the periodic paths of a lightray trapped between two nonconcentric circles,
each perfectly reflecting, are known.
The behavior of such rays seems chaotically complicated. For example, ...

**5**

votes

**0**answers

172 views

### Absolute oscillator in Langton's Ant

We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...

**3**

votes

**0**answers

48 views

### Flow lines of a real analytic vector field convergent to a point

Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...

**3**

votes

**1**answer

107 views

### Non-injective continuous maps that appear quasiconformal

Suppose that I have a continuous surjection $f: U \rightarrow V$ between two open subsets of the plane. Suppose that $f$ appears to be quasiconformal in the sense that there is a uniform constant $K \...

**12**

votes

**1**answer

225 views

### Poincaré on analytic dependence on parameters of solutions of linear differential equations

There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...

**0**

votes

**0**answers

60 views

### A problem in trying to show that a system is ergodic

If $f:[0,1]\to [0,1]$ is given by
$$
f(x)=
\begin{cases}
2x & \mbox{ if } x\in [0,1/3)\\
& \\
2x-\frac{2}{3} &...

**5**

votes

**0**answers

106 views

### Extension of Vector Field in the $\mathcal{C}^r$ topology

This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...

**0**

votes

**1**answer

34 views

### Nonsmooth dynamical system (DAE) - systematic way to calculate period numerically?

What I have in mind is a mechanical system that is described by an implicit system of ODEs or a system of DAEs (differential algebraic equations). The system is asymptotically stable, meaning that ...

**1**

vote

**0**answers

224 views

### A differential operator associated with a vector field on the torus

Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$.
We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows:
$T(f)=...

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votes

**0**answers

66 views

### Does singularity confinement imply a fixed pattern of irreducible factors?

Consider a rational map
$f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...

**0**

votes

**0**answers

37 views

### Characterization of Time-homogeneous flows for conditional expectation

Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...

**8**

votes

**0**answers

188 views

### Structural Stability on Compact $2$-Manifolds with Boundary

I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...

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vote

**1**answer

121 views

### Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...

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votes

**0**answers

226 views

### A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...

**3**

votes

**1**answer

117 views

### Symmetries for Julia sets of perturbations of polynomial maps

This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...

**3**

votes

**0**answers

61 views

### Piecewise linear expanding maps

Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose ...

**16**

votes

**3**answers

1k views

### Do complex iterates of functions have any meaning?

Using a method explained in this answer it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th iterate, where $i=\sqrt{-1}$. Here ...

**16**

votes

**5**answers

2k views

### Arithmetic dynamics and dynamics on moduli spaces

The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces.
In my dissertation, I have been ...

**9**

votes

**1**answer

561 views

### Lorenz attractor path-connected?

Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.
EDIT: The answer below is unsatisfactory, and possibly ...

**2**

votes

**1**answer

344 views

### Formal group law and Koenigs function conjecture?

Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). \tag{1}$$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$
This equation has many ...

**1**

vote

**0**answers

62 views

### Nearly eventually almost periodic functions

Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges.
Suppose $f: ...

**14**

votes

**9**answers

6k views

### Book recommendation for ergodic theory and/or topological dynamics?

Hello,
I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...

**2**

votes

**1**answer

81 views

### Discrete dynamical system and bound on norm

Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...

**2**

votes

**1**answer

95 views

### 6-periodic billiards trajectory in acute triangle

We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards?

**2**

votes

**0**answers

134 views

### Baker map-like problem

Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...

**1**

vote

**0**answers

44 views

### About strange invariant set of the Lozi mappings

Consider the Tent map: $f_{μ}(x)=μx$, if $x<0.5$ and $f_{μ}(x)=μ(1-x)$ if $x≥0.5$. In this page (https://en.wikipedia.org/wiki/Tent_map) it was stated that:
If $μ$ is greater than $2$ the map's ...