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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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(Some possible obstructions to ) Limit cycles as closed geodesics(3)

First we explain our Motivation: Motivation: First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ ...
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0answers
106 views

If the sum of everywhere linearly independent vector fields are periodic, are the component vector fields periodic?

I feel like the above must be true but embarrassingly cannot seem to prove it. Take linearly independent, commuting vector fields $X$ and $Y$ on a manifold and corresponding flows $\Phi^t_X$, $\Phi^...
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2answers
140 views

special flows and Rudolph's theorem

The Rudolph's theorem confirm the existence of a special representation of an ergodic flow on the Lebesgue space. (In the book of I.P.Cornfeld entitled Ergodic theory). My question is: what is the ...
2
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1answer
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 ...
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2answers
147 views

Handel's Theorem for surfaces with boundary

Handel's Theorem(Entropy and semi-conjugacy in dimension two, 1987): let $M$ denote a closed surface. Let $\vartheta$ be a pseudo-Anosov (orientation-presrv.) homeomorphism of $M$ and $g$ be an (...
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127 views

Average of irrational flow on the torus

Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$ defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that $$ \lim_{T\...
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161 views

Is this “stretched eigenvector” studied? (If so, what are its properties?)

An eigenvector is defined by $$ \lambda \mathbf{v} = A\mathbf{v}.\tag{1} $$ But suppose I change this to $$ \lambda \mathbf{v} = A\mathbf{v}^\alpha,\tag{2} $$ for real $\alpha\ne 1$, where $\mathbf{v}^...
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46 views

Change of polynomial eigenvalues between polynomials

Given the polynomial eigenvalue problem $$ p_t(z) = det ( P(z) + Q(t) ) = 0, $$ where $P(z) = \sum_{i=0}^k P_i z^i$ with $P_i \in \mathbb{C}^{n \times n}$ and $Q(t) \in \mathbb{C}^{n \times n}$. The ...
2
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1answer
106 views

Differential operators and rules Ore polynomial

(I have posed this question over at math.se but since there were no answers I hope it's okay to post here.) When dealing with (nonlinear) dynamical systems, one often deals with state space ...
3
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1answer
103 views

Measures maximizing entropy in a set of measures with fixed average for some observable

Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$. Consider $0<\alpha<1$ and let $$K_\alpha=\{...
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61 views

Perturbed trajectory of non-autonomous ode

Does the existence of strict Liapunov function guarantee that limit point of perturbed trajectory will be close to an isolated equilibrium for non-autonomous ode ? Can someone help me with some ...
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100 views

Smoothing properties of convolutions of $P^1(\mathbb{R})$ by $SL(2,\mathbb{R})$

Consider the action of $SL_2(\mathbb R)$ on real projective space $P^1(\mathbb R)$; given $A \in SL_2(\mathbb R)$ and $\alpha \in P^1(\mathbb R)$ we write $A . \alpha \in P^1(\mathbb R)$ for this ...
3
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1answer
144 views

An elliptic operator whose corresponding symbol Hamiltonian vector field has an isolated periodic orbit

Let $D$ be a differential operator on the space of smooth functions on a manifold $M$. The symbol of $D$ can be considered as a Hamiltonian on the cotangent bundle $T^*M$. We call ...
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63 views

Numerical and computational approaches to limit cycle theory

I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach. I would like to ask ...
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1answer
204 views

Riccati-type recurrence: infinitely many sign changes?

Suppose $b_1, b_2, b_3, \dots \in \Bbb{R}$ satisfy the Riccati-type recurrence $$b_{k+1}=\frac{1+kb_k}{k-b_k},\quad k\ge 1.$$ Is it true that such a sequence reaches infinitely many positive as ...
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166 views

Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems

I have a query regarding two equalities in the lemma in the book. But first I'll provide two definitions that one needs for this lemma. Definition 4.2.4: Consider the vector field $f(x,t)$ with $f:\...
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1answer
77 views

The continuity of the the stable and unstable in definition of hyperbolic sets for flows

I would like to know whether the continuity of the stable and unstable subbundles $E^{s}$ and $E^{u}$ follows from the growth conditions as in the discrete case, or must be hypothesized, in the ...
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1answer
119 views

Ergodicity of a measure preserving Anosov flow

Let $M$ be a Riemannian manifold and $\phi^t$ an Anosov flow on $M$. If $\phi^t$ is measure preserving (with respect to any Borel-measure on $M$), it is ergodic. Does anybody have a proof of that ...
2
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1answer
118 views

Li-Yorke chaos: the non compact case

1) Is there any notion of Li-Yorke chaos for non compact (metric) spaces $X$ and non continuous transformation $f:X \rightarrow X$? Could you bring some references? 2) I mean, why are so important ...
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1answer
279 views

Unclear construction in a paper of Ornstein and Weiss

I originally posted this on math.stack, but no one answered, so im posting here: I need help understanding the following construction (Taken from the paper "Entropy and isomorphism theorems for ...
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1answer
218 views

Compact manifolds which do not admit a diffeomorphism with a dense orbit

What is an example of a compact manifold which does not admit a diffeomorphism with at least one dense orbit? Moreover, is it true to say that every isometry of $\mathbb{C}P^n$ with the Fubini-Study ...
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1answer
526 views

Status of the three-body problem

I find many numerical results on the three-body problem, but what is rigorously proved? Especially I would be interested in the parameter domains for which we have rigorous lower bounds on the ...
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0answers
52 views

Stability of ODEs with exponentials in the vector field

What is known about fine stability properties of ODEs of the following kind? $$ \dot{x} = Ax + b + \phi(x),\quad x\in \mathbb{R}^d ,$$ where $d\geq 1$; $A$ is a constant matrix with all e.v. having ...
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0answers
75 views

How to find the best convergence rate of a dynamical system $x_{n+1} = g(x_n),\ n\ge 0,\ x_0\in \mathbb{R}$?

Let $\{x_n\}_{n\ge 0}$ be a sequence of reals such that $x_{n+1}=g(x_n)$, where $g:\mathbb{R}\to \mathbb{R}$ is a continuous function such that $0$ is a fixed point of $g$. My question is the ...
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3answers
292 views

Reference Request: KAM Theory

I intend to learn KAM Theory. Could you please suggest me a good book on KAM Theory to begin with, where main results are discussed with complete proofs. Thank you.
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1answer
92 views

Dynamics for approximating harmonic functions on graphs

A harmonic function on a graph is a function on its vertices such that the value at every vertex is the average of the values at its neighbors. Consider the following method for approximating a ...
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0answers
61 views

Can this construction generate bounded aperiodic functions?

This question is based on this old MathOverflow question: How this set of functions is ordered? In that question, Vladimir Reshetnikov asked about a class $S$ of functions $f:\mathbb{N}\to\mathbb{N}$ ...
5
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1answer
292 views

Dynamical system and omega limit set

Can Omega limit sets of dynamical systems be connected but not road connected? In the process of reading Wiggins, we have encountered the definition and properties of Omega limit sets for autonomous ...
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3answers
375 views

Proving convergence of sum over $\mathbb{Z}^n$

In my research, I am trying to use the following construction by Benson Farb and John Franks, which proves that for all $n$, the group of $n\times n$ matrices with 1's on the diagonal, 0's above the ...
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81 views

Adjoint of differential equation

Motivation: Consider the ODE $$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation $$y'(t)=A^*y(t).$$ I ...
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3answers
248 views

fixed points of quadratic iteration

Consider the well-known iteration $f:z\to z^2 + c,$ and consider the values of $c$ for which $0$ is a periodic point. Experiment shows that most such values of $c$ (about $480$ out of $512$ for period ...
4
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1answer
174 views

symplectic topology of (perturbed) KAM tori

Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies ...
2
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2answers
149 views

Gradient of a proper function is integrable

In one paper the author uses the statement without citation: Let $(M,g)$ be a Riemannian manifold. The gradient $\nabla F$ of a proper function $F: M\rightarrow \mathbb{R}$ is integrable vector ...
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2answers
196 views

Behavior of a non-linear differential equation

Let us consider the following differential equation $$ \dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}. $$ My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the ...
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2answers
291 views

A possible dynamical approach to the “Invariant Subspace Problem”

In the literature, is there any paper or research investigating the invariant subspace problem with consideration of differential operators acting on an appropriate Sobolev space?In particular is ...
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2answers
114 views

ODE with Holder drift - Cauchy-Peano theorem

Consider the following ODE: $$ x′(t)=b(x(t)),\quad x(0)=x_0. $$ If $b$ is bounded and Holder continuous, then the Cauchy-Peano theorem ensures that there exists a solution to the above equation (but ...
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1answer
74 views

Topological full groups and minimal orbit closures

Let $X$ be the Cantor set, and let $g$ be a minimal homeomorphism of $X$. Let $h$ be a homeomorphism in the topological full group of $g$, that is, for every $x \in X$, there is a neighbourhood of $x$...
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0answers
62 views

Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by $\dot{x} = M D(x) P x - \frac{c}{2}x$ $P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
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3answers
342 views

What does the flow of the principal symbol of the differential operator tell us about the PDE?

Disclaimer: Let me apologize in advance for asking this slightly vague question Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
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1answer
129 views

Example of a prime action on a compact Hausdorff Space

Suppose that a discrete group $\Gamma$ acts on a compact Hausdorff space $X$ via homeomorphisms. This action induces an action on $C(X)$, the space of all continuous functions from $X$ to $\mathbb{C}$,...
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1answer
273 views

Can we estimate the error $\left| \frac{1}{N^2} \sum f ( \{ \sqrt{2} m + \sqrt{3} n \} ) - \int_0^1 f(x) \, dx \right|$?

As a computer experiment I did a few Riemannian sums to see if I could quantify the density statement $\overline{\mathbb{Q}(\sqrt{2}, \sqrt{3})} = \mathbb{R}$ : $$ \Big| \frac{1}{N^2} \sum_{0 \leq m,...
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74 views

Lyapunov exponents for a $C^1$ diffeomorphism of a disk for which the boundary is not attracting

Let $D$ be an $n$-dimensional disk (considered a $C^{\infty}$ manifold-with-boundary), and let $\partial D$ denote its boundary. Assume that $f \colon D \to D$ is a $C^1$ diffeomorphism having the ...
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Finite translation surfaces with Veech groups that are non-elementary Fuchsian groups of the second kind?

I know that all Veech surfaces have Veech groups which are Fuchsian groups of the first kind and that there exist finite translation surfaces with Veech groups that are elementary Fuchsian groups of ...
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1answer
182 views

Action of homeomorphism on real line

An element $f ∈ Homeo^+(\mathbb{R})$ is said to be of : (1) type A, if it has a trivial germ at ∞ and it does not fix any point in an interval of the form (−∞, r). (2) type B, if it has a trivial ...
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1answer
318 views

Orbits of the function f(x)=2x (mod 1)

I am currently studying the dynamics associated with the function $f(x)=2x$ (mod 1). In particular, if we define the orbit of an element $y \in [0,1]$ $$ orb(y)= \{ f^m(y): m \in \mathbb{Z}\}$$ it ...
2
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1answer
111 views

On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(...
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0answers
111 views

Flows commuting with Anosov flows and further reference request

Hello respected members of Mathoverflow. I was reading the paper "Flots d’Anosov dont les feuilletages stables sont différentiables" by Etienne Ghys and there was a statement which he remarked was ...
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1answer
103 views

Importance of the $2^{\tau(G)}\leqslant A(n,g(G))$ conjecture

During a course about finite dynamical systems the following conjecture was presented to us : Let G be a directed graph of order n. Let $\tau(G)$ be the minimum size of a subset of $V(G)$, $I$ ...
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0answers
81 views

A singular foliation analogy of the Riemann Hilbert problem

Note: In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$. ...
2
votes
1answer
195 views

Confusion about Teichmuller curves and $SL_2$ action

Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $...