Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations.
2
votes
1answer
27 views
Centre manifold theory for a curve of equilibrium points
I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001).
The part which is known
Let's consider a differential equation in higher-dimensional ...
0
votes
1answer
66 views
Non-hyperbolic fixed points in multidimensional systems
Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start ...
2
votes
1answer
41 views
Lyapunov Exponents for independent-nonidentically distributed matrices?
My question is highlighted in bold at the end.
$\mathrm{\underline{Background}}$
Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
2
votes
0answers
83 views
Deterministic shifts
We consider (topological) dynamical systems $(\Omega, S)$, where $S$ is the shift $(Sx)_n=x_{n+1}$, and $\Omega$ is a compact, shift invariant subspace of $[0,1]^{\mathbb Z}$. I call $(\Omega, S)$ ...
4
votes
1answer
106 views
Graph presentation of Lexicographic shifts
Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma_n = \mathop{\prod}\limits_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic ...
3
votes
2answers
134 views
Nielsen-Thurston classification of homeomorphisms for open surfaces?
In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
3
votes
1answer
118 views
Homeomorphisms that admit a decomposition
Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$.
If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ...
2
votes
0answers
74 views
Quick estimate of attractor of non-linear dynamical system [closed]
Say I have a system of form
$$
\frac{dy}{dt} = f(y),
$$
and it is know this system has an attractor. Can I quickly for given $\varepsilon$ guess some point, such in its $\varepsilon$- neighbourhood ...
2
votes
1answer
97 views
Omit each vertex in turn of convex polygon: Iterative limit?
Let $P=P_0$ be a convex polygon of $n$ vertices $v_k$.
Let $P_{i+1}$ be the convex polygon obtained by intersecting the halfplanes
determined by the lines through every other vertex.
Below, $P_0$ is ...
6
votes
6answers
455 views
Furstenberg $\times 2 \times 3$ conjecture, bibliography
Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a ...
4
votes
1answer
94 views
Automorphism group of compact abelian group
I am looking for references on the automorphism group $\mathrm{Aut}(X)$ of a compact abelian group $X$. By automorphisms I mean topological group automorphisms. Some particular questions are as ...
1
vote
0answers
53 views
Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic
Consider mechanical Hamiltonian system of the form
$$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$
Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian ...
4
votes
0answers
75 views
Example for a dynamical system which is not point-distal
Let $(X, d)$ be a compact metric space, let $T$ be a group of actions on $X$. Then $(X,T)$ is a topological dynamical system with transformation group $T$, and we denote it by $(X,T)$. We say points ...
1
vote
0answers
55 views
Periodic solution of first order ODE
There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system
$$
\left\{
\begin{array}{l}
\dot{x}=f(x), \\
x(t_0)=x_0,
...
3
votes
0answers
90 views
Nonexistence of Limit Cycle
Consider a planar dynamical system described in polar coordinates as
$$
\left\{
\begin{array}{ll}
\dot{\theta}=\Delta - r \sin \theta,\\
\dot{r} = - r + 1 + \cos \theta,
\end{array}
\right.
$$
where ...
3
votes
2answers
100 views
Mixing property of first return map
Let $(X,\mathcal{X},\mu)$ be a probability measure system, $T:X\to X$ be a $\mu$-preserving isomorphism on $X$. Let $A\in \mathcal{X}$ such that $\mu(\bigcup_{n\ge 0}T^nA)=1$, and $\mu_A$ be the ...
4
votes
0answers
306 views
Limit cycles as closed geodesics(geodesiable flow)
The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...
1
vote
1answer
139 views
Analytic vector fields on surfaces which have infinite number of singularities
Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$.
A local question
Is there an analytic vector ...
2
votes
1answer
80 views
Global Solutions of Ordinary Differential Equations
Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...
7
votes
1answer
287 views
Raphael Douady's thesis: Applications du théorème des tores invariants
Raphael Douady's thesis, Applications du théorème des tores invariants, has been cited in numerous papers by many experts.
According Wikipedia, he proves of the equivalence of KAM ...
2
votes
1answer
77 views
Constructing an interval exchange given a prescribed trajectory
Given a prescribed trajectory, is it possible to construct an interval exchange having this trajectory?
For example, given a 3-letter word (like aaabbbccabcaaa ), is it possible to construct a 3- ...
14
votes
2answers
447 views
Spearing rolling hula hoops
Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, ...
2
votes
0answers
66 views
Link between presence of attracting random fixed points and synchronisation - is this an open question?
This is a question in the theory of random dynamical systems.
Let $(X,d)$ be a compact metric space, let $(I,\mathcal{I},\nu)$ be a probability space, and let $(f_\alpha)_{\alpha \in I}$ be an ...
0
votes
0answers
38 views
Question on center-stable manifold
Assume that you have a gradient system smooth enough and a fixed point $x_{0}$. Is it true that if $x_{0} \in \omega(x)$ then $\gamma^{+}(x)$ must intersect the local center-stable manifold of ...
23
votes
2answers
921 views
Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$
Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ ...
0
votes
1answer
63 views
Find a sequence with uniform frequencies and recurrent property
Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$
such that this sequence has uniform ...
0
votes
0answers
21 views
n-body systems and bifurcations
From what I understand, in bifurcation theory, one definition of the equivalence of two dynamical systems is that they are topologically equivalent. However if say proteins A, B start out as straight ...
1
vote
1answer
124 views
Physical Measure Vs. SRB measures
Anybody can help me to have an idea about an example showing the difference of a Physical measur with compare to an SRB measure?
By a Physical measure i mean in the sense of $\nu$ a ...
11
votes
2answers
420 views
Invariant subsets of $z \mapsto z^2$
Where can I find an explicit construction of closed invariant subsets of the map $z \mapsto z^2$ on the unit circle? Furstenberg mentions that there are continuum of such disjoint minimal sets but ...
4
votes
2answers
160 views
Invariant measures on a compact metric space
I'm dealing with a continuous flow on a compact metric space $X$, and $\mu$, $\nu$ are two invariant Borel probability measures on $X$. If I know that $\mu(A)=\nu(A)$ for all the invariant Borel ...
0
votes
1answer
204 views
A variation of the Banach fixed-point theorem
Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem ...
6
votes
1answer
174 views
Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture
Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...
1
vote
0answers
90 views
Gradient-like systems and limit sets
Suppose we have an autonomous ode $$\dot{x} = f(x)$$ which corresponds to a $\underline{gradient-like}$ dynamical system, $f: \mathbb{R}^n \to \mathbb{R}^n$. Is it true that the set of initial ...
0
votes
0answers
98 views
excplicit formula of iterates of an interval exchange
Let $f$ be an interval exchange transformation of $[0,1]$. Is there an explicit formula giving $f^k(0)$ in function of $k$?
If not, are there particular cases where this formula is simple? (except ...
3
votes
2answers
107 views
Markov Partitions for toral automorphisms
I know that my question is more practical than theoretical. But, I do know where to look for the theoretical sources.
I want to find a program in the case that it exists (does it?), or to program it. ...
3
votes
0answers
67 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
169 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 ...
2
votes
0answers
223 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
146 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
73 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
1answer
123 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
76 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
62 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 ...
3
votes
0answers
148 views
What is a pure algebraic interpretation for this dynamical property?
According to comments of Yves Cornulier to the previous version of this question I revise the question as follows:
To what extent the following types of Lie algebras $A$ are classified? And what is ...
7
votes
2answers
259 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
95 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
106 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
1answer
112 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
206 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
109 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 ...
