Tagged Questions

Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations.

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3
votes
0answers
71 views

Question about a length inequality in algebraic dynamics

Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of ...
2
votes
1answer
209 views

Open dynamical system of doubling map

I want to prove the following simple lemma: Let $T(x)=2x\mod 1$, be defined on $[0\,,1)$ to itself,suppose for any $k\ge 0$, $T^{k}(a)\notin(a\,,b)$, then we have $\Omega=\{x\in[0\,,1):\mbox{for ...
5
votes
0answers
543 views

A question on “The weakened Hilbert 16th problem”

In this question we are interested in the number of limit cycles which appear in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } ...
3
votes
1answer
169 views

Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$? (Possibly by perturbing a rotation in the real-analytic topology?)
2
votes
0answers
93 views

Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?

I would be interested to know if a certain set of periodic solutions for the two-dimensional Navier-Stokes equations is closed generically. Many similar (yet not identical) set-ups can be found in the ...
2
votes
2answers
202 views

Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...
2
votes
1answer
106 views

Power series expansion of the Koenigs function

Given a non-zero holomorphic function $f$ fixing $0$ which isn't a Mobius transform, the Koenigs function of $f$, which we'll call $h$, is the function which linearizes $f$ in the sense that $$ ...
2
votes
0answers
93 views

Existence and uniqueness of heteroclinic orbits

I am looking for conditions on a nonlinear dynamical system $\frac{d\vec{x}}{dt} = \vec{F}(\vec{x})$ that guarantee the existence of a unique heteroclinic orbit between a stable attractor of this ...
7
votes
2answers
280 views

Iteration of a 2D map involving absolute value: phase transition?

I was looking at this map: $f(x,y) \mapsto (|x-y|,x)$, starting from some point with coordinates $(x,y) \in [0,1]^2$, and iterating: $(x,y),\, f(x,y), \, f^2(x,y), \,f^3(x,y), \ldots$. It displays ...
3
votes
1answer
106 views

How is the Fractal Dimension of a Parametric Curve Related to the Fractal Dimensions of its Coordinate Functions?

Question: Suppose the fractal dimension $1\le d_c\le2$ of a planar parametric curve $c(t) := (x(t),y(t))$ is given; can any nontrivial estimates for the fractal dimensions $d_x$ of $x(t)$ and $d_y$ of ...
2
votes
1answer
117 views

Does conjugacy preserve the set of synchronizing blocks?

A synchronized system is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then ...
2
votes
2answers
148 views

Mixing coded systems and period of their graph presentations

A coded system [see F. Blanchard, G. Hansel, Systèmes codés, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8. ...
5
votes
2answers
263 views

Lebesgue entropy zero and positive topological entropy

I am looking for examples of volume preserving $C^{\infty}$ diffeomorphisms $f$ of a surface, which have positive topological entropy ($h(f) > 0$), but that the Lebesgue measure entropy (metric ...
3
votes
1answer
128 views

Automorphisms of strictly ergodic shift spaces

Let $X$ be a strictly ergodic shift space, and $\omega_1$, $\omega_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega_1)=\omega_2$? By an automorphism I ...
1
vote
1answer
158 views

There is a horseshoe with positive measure

Here is a theorem by Bowen : My question is about the highlighted part in the picture. why there such a function $g$ exist?
1
vote
0answers
44 views

Id monodromy in hamiltonian dynamics

In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t-1)q^2$, $a,b$ - parameters). From numerical ...
12
votes
2answers
339 views

Are rounded rectangle billiard dynamics ergodic?

Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.            (Image from this link.) Q. Is it known that the ...
13
votes
0answers
396 views

Blocking light with mirrored convex objects

There is a long-unsolved problem posed by Janos Pach, sometimes known as the enchanted forest problem, which asks if it is possible to block a point light source in the plane from reaching infinity by ...
71
votes
2answers
105k views

Perfectly centered break of a perfectly aligned pool ball rack

Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a ...
0
votes
1answer
118 views

An absolutely continuous foliation, which is not transversely absolutely continuous

Currently I'm studying "Introduction to dyamical systems" by Stuck and Brin. In chapter 6 they define absolutely continuous and transversely absolutely continuous foliations. By proposition 6.2.2 if ...
2
votes
0answers
86 views

random maass waveforms

Let $H$ be the upper half complex plane and $\Gamma$ a discrete subgroup of $SL_2(\mathbb{Z})$ such that the volume of of $\Gamma \backslash H$ is finite. There is a conjecture of Berry that Maass ...
0
votes
1answer
88 views

Looking for methods/results for explicitly bounding iterations of rational functions

In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series. That is, suppose that $$ ...
9
votes
1answer
292 views

An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part. 1.Is there a polynomial Hamiltonian ...
2
votes
1answer
210 views

Angle between two subspaces

Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space $T_xM=E^s(x)\oplus ...
1
vote
0answers
193 views

Question on measure zero set of initial conditions in dynamical systems

[Update] Let $S \subseteq \mathbb{R}^n$ be a closed, bounded, convex set with measure $m(S)>0$ and let an autonomous dynamical system (system of ODEs) be given by $$\frac{dx}{dt} = f(x),$$ where ...
2
votes
1answer
111 views

the union of local stable manifolds along local unstable manifolds

Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), ...
3
votes
1answer
175 views

A question about ergodicity

Let $X$ be a compact metric space, $T:X\rightarrow X$ a homeomorphism and $\mu$ be a $T$-invariant probability measure on $X$ such that the set of points with dense orbit in $supp(\mu)$ has full ...
3
votes
2answers
136 views

Decay of Correlation, references for a non-standard way

In ergodic theory is common to use the decay of correlation property to deduce properties analogues to those of i.i.d. random variables. Call $X\doteq [0,1].$ Examples of decay of correlation ...
2
votes
0answers
83 views

uniquely ergodic hyperbolic invariant set

The question is to classify uniformly hyperbolic invariant sets supporting uniquely ergodic invariant measure. The only examples that I expect are: fixed points, periodic orbits and Cantori(Denjoy ...
4
votes
1answer
98 views

For a linear dynamic system, what can we learn from its singluar value and rank?

Given a linear system $\frac{dx}{dt}=Mx$, what's the relationship between the dynamic's property and the singular value decomposition/rank of $M$ ?
19
votes
5answers
820 views

Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given point $p$ such that it gets "stuck" reflecting between two congruent mirror-disks. For why there is such a ray, see the (amazing!) answer ...
0
votes
0answers
133 views

Express measurable entropy in terms of Fourier coefficients of the measure

Let $S^1$ be the unit circle , $\mu$ be a Borel probability measure on $S^1$ and $T:S^1\to S^1$ is a measure-preserving map (not necessarily invertible) with respect to $\mu$. The Fourier ...
2
votes
1answer
117 views

expansive continuous flow

I encounter with two definitions for expansive continuous flows and their equivalence is unclear for me. Could anyone can explain for me please? Thanks in advance. I cite below these two definitions. ...
3
votes
1answer
216 views

Julia sets without Montel's theorem

Let $J(c)$ be the Julia set of $f(z)=z^2 +c$ defined as the closure of repelling periodic orbits. Is there a way to prove that $J(c)$ is the boundary of the basin of attraction of attractive fix ...
5
votes
1answer
172 views

Topological classification of Morse-Smale flows

Does anyone know of papers that mention the classification of non-singular Morse-Smale (NMS) flows up to topological equivalency? I am particularly interested in the flows on manifolds of dimension 3. ...
5
votes
1answer
134 views

Is there a similar theorem in the partially hyperbolic case?

Theorem 5.10.3 from Introduction to dynamical systems, by Brin & Stuck: Let $f:M\rightarrow M$ be an Anosov diffeomorphism. Then the following are equivalent: $NW(f)=M$, every unstable manifold ...
3
votes
1answer
114 views

Convergence of trajectories and asymptotic stability

Say that an autonomous system $\dot{u} = f(u)$ in $\mathbb{R}^{m}$ has the property that for any two solutions $x(t), y(t)$ corresponding to initial conditions $x(0)$ and $y(0)$ the trajectories are ...
1
vote
0answers
177 views

Examples of amenable groups acting on the real line

In the literature, there are several examples of solvable groups acting by order preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth acting in the same ...
7
votes
2answers
167 views

Well-definedness of single-particle smooth billiards flow

Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well ...
6
votes
1answer
189 views

Approximating an iteratively defined function

Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows: $$f_0(x) =1+2x$$ $$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } ...
3
votes
2answers
226 views

Systems similar to Erdős numbers?

As many mathematicians know, each person has an Erdős number (see: http://en.wikipedia.org/wiki/Erd%C5%91s_number). That is, Erdős himself has Erdős number zero, each person who published anything ...
2
votes
1answer
152 views

The centralizer of Lienard equation

Consider the lienard vector field $\cases{ x'=y -F(x) \\ y'=-x } $ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
15
votes
1answer
737 views

Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...
4
votes
1answer
192 views

Volume-preserving mappings in the torus $T^n$

Let $T^n$ be the $n$-dimensional torus and let $F$ be the set of all volume preserving continuous mappings $f:T^n\to T^n$. I would like to know if $F$ is connected in the sense that for any $f\in F$ ...
8
votes
2answers
530 views

Getting unique ergodicity from minimality

It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask whether the implication holds in following particular situation: Suppose $X$ is a compact space, $f:X \to ...
2
votes
0answers
104 views

Pointwise ergodic theorem for amenable semigroups

Using tempered Folner sequences one may show a pointwise ergodic theorem for amenable groups. (see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full) Is there a similar ...
3
votes
1answer
207 views

Uniform convergence of Birkhoff averages and unique ergodicity

I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET): Let $T$ be a ...
0
votes
0answers
127 views

Prove that origin is globally exponentially stable with Lyapunov Indirect Method

I'm wondering, if we have a nonlinear system governed by $\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz how can we show that the origin is globally exponentially ...
1
vote
1answer
173 views

Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations $$ \sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n, $$ has ...
2
votes
1answer
301 views

Dynamics of Master Equation

I'm going to do research on dynamics of master equation of $n$ states $$\dot p_i=A_{ij}p_j\qquad i=1\ldots n$$ where $p_i$ is the $i$-th component of probability vector and $A_{ij}$ is transition rate ...