Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,398
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coboundary in Dynamical system
a question about the definition:
given measurable dynamic system $ ( X, \mathcal{B}, T, \mu)$, $ \mu \circ T^{-1}=\mu$ ergodic.
$\phi \in L^{\infty}$ is coboundary with $\int \phi d\mu =0 $, means ...
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3
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Repeatedly halve and twist a planar shape: Limiting shape?
Consider the following iterative process.
Start with a planar region $R=R_0$ of $\mathbb{R}^2$.
I am thinking of $R$ as connected,
but it may become disconnected.
In the example below, $R$ starts as ...
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Determinant as a Hamiltonian
Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...
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(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
\begin{eqnarray*}
\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)...
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A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature
Is there a 2 dimensional Riemannian manifold $M$ whose curvature is not negative but its geodesic flow is an ergodic flow?
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89
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On the measure of regular and chaotic regions in a phase space
Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...
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1
answer
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Ergodicity of geodesic flow in negative curvatutre as a possible obstruction for consideration of limit cycles as closed geodesics(4)
Does the ergodicity of geodesic flow of compact surfaces with negative curvature stile hold for non compact case?
Is not the ergocity theorems of geodesic flow an obstruction to have a ...
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89
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Counting orbits of the standard map
Consider the standard map. Might it happen that for some nonzero parameter value $K$ and for some positive integer $q$ that there exist an infinite number of periodic orbits having period $q $ I ...
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212
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Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$
I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...
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Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$?
Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\...
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answer
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Symplectic forms and sign of eigenvalues
This question has come out while reading J. Moser "New Aspects in the Theory of Stability
of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ...
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1
answer
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A Lie algebra associated to a foliation(A kind of saturation of foliations)
Inspired by the Lie algebra discussed in this answer, we consider the following Lie agebra associated to a given foliation:
Let $\mathcal{F}$ be a nontrivial foliation of a ...
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1
answer
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Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action
Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $...
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On Mathematical Foundations of Football
Football (soccer) is arguably one of the most unpredictable sports. Countless variables play a role in determining the outcome of a certain football match. Due to the high complexity of the entire set ...
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Does there exist a leaf of this holomorphic foliation with non trivial holonomy?
Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$.
Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, ...
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Transitive homeomorphisms of Erdős spaces
A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense.
Does either of the Erdös spaces $\...
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Has this type of pathwise (S)DE been studied before?
I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before.
Let $(G,\ast)$ be an abelian $C^1$ Lie group....
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Two questions on "foliation by geodesics"
I would appreciate if you consider the following two questions on $1$ dimensional foliations whose leaves are geodesic.
1)Assume that $M$ is a Riemannian manifold which is either an open ...
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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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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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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 ...
3
votes
1
answer
320
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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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votes
2
answers
236
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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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
$$ \lim_{T\...
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Is this "stretched eigenvector" studied? (If so, what are its properties?)
An eigenvector is defined by
$$
\lambda \mathbf{v} = A\mathbf{v}.\tag{1}
$$
But suppose I change this to
$$
\lambda \mathbf{v} = A\mathbf{v}^\alpha,\tag{2}
$$
for real $\alpha\ne 1$, where $\mathbf{v}^...
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Differential operators and rules Ore polynomial
(I have posed this question over at math.se but since there were no answers I hope it's okay to post here.)
When dealing with (nonlinear) dynamical systems, one often deals with state space ...
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answer
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Measures maximizing entropy in a set of measures with fixed average for some observable
Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$.
Consider $0<\alpha<1$ and let $$K_\alpha=\{...
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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 ...
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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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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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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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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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1
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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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answer
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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 ...
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1
answer
157
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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 ...
3
votes
1
answer
360
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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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1
answer
282
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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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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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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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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3
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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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answer
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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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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 $f:\mathbb{N}\to\mathbb{N}$ ...
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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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3
answers
522
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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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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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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 ...
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1
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139
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Separation property for non-injective flows
Let us consider a non-injective flow $X$ on $\mathbb{R}^d$, i.e. a continuous map $X:\mathbb{R}_+\times \mathbb{R}^d \to \mathbb{R}^d$ with $X(0,\cdot)=\mathrm{id}$ and satisfying the semigroup ...
4
votes
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answer
289
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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 ...
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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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