Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
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Proven chaos in logistic maps
For a logistic map using $f_r(x)=rx(1-x)$, what values of $r$ and starting $x$ are guaranteed (i.e., with an accepted proof) to be chaotic? I mean "chaotic" in the loosest sense: The ...
- 1,547
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A foliation version of S.Husseini counter example in fixed point theory
In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977
"The Products of Manifolds with the f.p.p. Need Not have the f.p.p"
who gave an example of two ...
- 356
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1
answer
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Entropy maximising ergodic transformation
Let $(\Omega, \mathcal F, \mu)$ be a standard probability space.
Question: For each $f \in L^\infty (\Omega)$, does there exist an ergodic measure preserving transformation $T: \Omega \to \Omega$ such ...
- 6,215
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Periodic orbits of generalized cat map near the origin
Let $M\in SL(2,\mathbb{Z})$ have eigenvalues $\lambda, \lambda^{-1}$ with $\lambda>1$., and suppose $M$ is diagonalized as $Q\Lambda Q^{-1}$ with $\det Q=1$. (Note that his doesn't determine $Q$, ...
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How to approach this semilinear system of PDEs?
This question is cross-posted from Math StackExchange (link). I'm not sure it qualifies as research-level mathematics (although the application is to research) but it has been on MSE for several days, ...
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Why all the coefficients of the center manifold of this system are zeros?
I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...
- 159
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2-ball billiards in a circle
Consider a 2D circular billiards table with diameter 1m containing two
balls with diameter 0.25m. Let each ball start with a speed of 1m/s.
In general, this speed could change after the balls hit ...
- 1,547
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Is $\mathbb{Q}$ the orbit of a continuous function that is computable when restricted to $\mathbb{Q}$?
In the previous post What is the smallest set of real continuous functions generating all rational numbers by iteration? I asked for the smallest set of continuous real functions that could generate $\...
- 4,862
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1
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SRB measure and Gibbs u-state
I have been reading the famous paper of Alves, Bonatti, and Viana where they proved that there is an SRB measure for partially hyperbolic systems. Since I am new to this field, I have some basic ...
- 1,043
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An example of deterministic sequence from Terence Tao's blog
The following is taken from a post by Terence Tao on the Chowla conjecture and the Sarnak conjecture
:
Given a bounded sequence ${f: {\bf N} \rightarrow {\bf C}}$, define the topological entropy of ...
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An algebraic foliation of $\mathbb{C}^2$ which admits a non-algebraic complex limit cycle $L$ with $L\cap \mathbb{R}^2=\emptyset$
Is there a polynomial vector field on $\mathbb{R}^2$ whose corresponding singular complex foliation of $\mathbb{C}^2 $ admits a complex limit cycle $L$ such that $L$ does not intersect the real ...
- 356
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Maximal Lyapunov exponent of Schrödinger-Newton equation
I am trying to determine the sign of the maximal Lyapunov exponent of the Schrödinger-Newton equation
$$
\partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|...
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Convergence of ODE solutions almost everywhere to a stable equilibrium point
Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\...
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Reference request for a general-to-specific text(book) on abstract dynamical systems
In all references on dynamical systems---encyclopedias, textbooks and articles---I have so far consulted, either
there is from the beginning an emphasis on a certain class of dynamical system being ...
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An algebraic foliation of $\mathbb{C}^2$ with real coefficients whose all complex limit cycles intersect the real plane $\mathbb{R}^2$
Is there a polynomial vector field $X$ on $\mathbb{R}^2$ such that every complex limit cycle
of the corresponding foliation of $\mathbb{C}^2$ must necessarily intersect the real plane $\mathbb{R}^2$.
...
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Eigenvalue assignment via state feedback: existence proof
Consider the linear time invariant system:
$$\tag{1}\label{eq1}
\dot{x}(t) = Ax(t) + Bu(t), \ \ x(0)=x_0\in\mathbb{R}^n,
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$. Let $p_M(s)...
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The baker 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 \...
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Decidability of choosing delay in Takens' theorem
In Dynamical systems theory, Takens' embedding theorem is as follows:
Suppose that a measured time series $y(1), y(2), \ldots, y(N)$ lies on a $D$-dimensional attractor of an $n$th-order ...
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Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
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3-periodic point implies positive topological entropy
When I learn some basic ergodic theory, I encounter an interesting exercise. As we all know, 3-periodic point often means chaos. Therefore, when a continuous map has a 3-periodic point, it may have ...
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Fixpoints of $m\longmapsto \mathrm{rad}(\phi(m^2))$ under iteration
Given a strictly positive integer $m$ let $\alpha(m)=\mathrm{rad}(m\phi(m))$
be the radical (product of all distinct prime divisors) of the product of $m$ and of Euler's totient function $\phi(m)=m\...
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Bound on the period of the identity (in a free group) for an automorphism followed by left-multiplication
Let $F$ be a finite-rank free group, $g$ an element of $F$ and $\Phi\colon F \to F$ an automorphism. Consider the dynamical system $\psi_g\colon F \to F$ defined by $x \mapsto g\Phi(x)$. Say that $g$ ...
- 1,996
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1
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Subshifts with special property
I am looking how to prove the following fact:
If $ X \subseteq A^\mathbb{Z}$ is an infinite minimal subshift, then for any $N\ge 1$, $X$ is conjugate to a minimal subshift $Y\subseteq B^\mathbb{Z}$ ...
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Does iterating the derivative infinitely many times give a smooth function whenever it converges?
I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...
- 1,763
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Properties of the orbit $Abx_0$ when $b$ is upper or lower triangular but not diagonal
Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $A$ denote the subgroup of $G$ ...
- 1,565
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Measure of non-commutativity of two invertible functions
I need to estimate $|x - f^{-1}(g^{-1}(f(g(x))))|$ for various values of $x$ for two smooth invertible functions $f$ and $g$ on $\mathbb{R}$ (actually some other spaces, but $\mathbb{R}$ will do.) Are ...
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Well-behaved trajectories
Call trajectory any continuous function $f: \mathbb{R}_{\geq 0} \to \mathbb{R}^n$ (here, $\mathbb{R}_{\geq 0}$ is interpreted as time).
A polyhedral partition of $\mathbb{R}^n$ is a finite set of ...
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The closure of the orbit of an irrational grid contains the fiber
Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $Y_d$ denote the space of unimodular ...
- 1,565
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Minimal sets of foliations in the plane (generalisation of Poincaré-Bendixson)
Let $F$ be a $1$-dimensional foliation of an open subset of the plane defined by a
locally Lipschitz line field. Suppose $C$ is a compact minimal set of $F$ (i.e. $C$ is non-empty, compact, a union of ...
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Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$
The motivation for the following is to convert the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=-kx+\beta\int_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds,
\end{equation}
into a ...
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Compressible Ebin-Marsden?
In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of ...
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$\mathbb{R}^n$-flow, cross-section and Whitney theorem
For a $\mathbb{R}$-flow (X, $\Phi_{\mathbb{R}}$), the (local) cross-section is well defined (recall that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]}(x)=\{x\...
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An example of an SRB measure which is not a physical measure
Let $f:M \to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold with a normalized Riemannian volume $\mathrm{Leb}$. Given an $f$-invariant Borel probability $\mu$ in $M$, we call the ...
- 1,043
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vote
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Random matrix heuristics for Koopman operators
Consider a nice hyperbolic dynamical system $(X, T)$, for instance a $\mathcal{C}^\infty$ Anosov map. The action of the Koopman operator
$$\mathcal{K} : \ f \mapsto f \circ T$$
has a nice spectrum ...
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Lie group flows [closed]
I recently came across a way to think of an ordinary differential equation on a smooth manifold $M$ is as a Lie group homomorphism $\phi : (\mathbb{R}, +) \rightarrow \operatorname{Diff}(M)$ where $\...
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On the solvability of a nonlinear differential system
A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system ...
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Birkhoff ergodic theorem for ergodic Markov processes
This question was previously posted on MSE.
This question might be easy but I am really stuck on it.
Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
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Symplectic structure on moduli space of holomorphic Abelian differentials
I've heard a "symplectic structure" referred to on the moduli space of holomorphic Abelian differentials by numerous people / sources. I do not know how to interpret this - I am looking for ...
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Finding the eigenvalues and eigenvectors of Jacobian at equilibrium point of nonlinear ODEs
Consider the vector field $V:\mathbb{R}^4\rightarrow\mathbb{R}^4$, defined by
\begin{equation}
V(x,v,M_0,M_1)=(v,\kappa^{-1}(\beta M_0-v-kx),-M_0+v M_1,-M_1+1-vM_0),
\end{equation}
such that $\...
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Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)
I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the ...
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Are $C^1$ vector fields generating an ergodic flow $C^0$ dense?
Note: This is a concrete case of the following question: Are almost all measure-preserving flows on compact manifolds ergodic?
Let $M$ be a Riemannian manifold with its natural Riemannian measure, and ...
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Are almost all measure-preserving flows on compact manifolds ergodic?
This may be a naive question, but I have been unable to find a reference that answers it directly, at least at a level that I can understand. My intuition from physics is that non-ergodicity is ...
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Local decomposition of semisimple Lie groups at the identity into the product of centralizer, unstable and stable horospherical subgroups
First let us look at an example. Let $G=\text{SL}(d,\mathbb R)$ and $A$ be its subgroup consisting of diagonal matrices with positive entries.
Let $a\in A$ be an element. We define the stable ...
- 1,565
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Metric entropy and topological entropy
It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words,
if
$$M(X,T) := ...
2
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0
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A.e. global existence of solution to 'encased' n-body problem already somewhere in the literature?
In the study of the Newtonian n-body problem, it seems that Von Zeipel's theorem and Saari's theorem concerning the improbability of collision singularities ought to lend themselves to a nice ...
- 1,461
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71
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Domain of definition of a certain mapping
Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold.
I am studying the mapping
$$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...
1
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0
answers
67
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Solution to recurrence relation from integro-differential dynamical system?
Consider the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1}
\end{equation}
such that $\kappa,\epsilon\in\mathbb{R}$, $t\in\...
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0
answers
70
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Partially hyperbolic systems and specification
Let $f: M \rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism on a Riemannian compact manifold. Suppose that $f$ admits a dominated splitting $T M=E \oplus F$ with $E\ll F$, where $E$ is uniformly ...
- 1,043
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votes
0
answers
153
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Stability test for LTV systems by differential Lyapunov inequalities
Consider a linear time-varying system:
\begin{equation}
\dot x(t) = A(t) x(t), \tag{$*$}
\end{equation}
where $A(t)$ is a time-varying block matrix defined as
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
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votes
3
answers
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Ergodic measures for the logistic map
$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$This is mostly curiosity on my part and I hope that the MO community might be able to help.
For $c\in (0,4]$ consider the logistic ...
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