Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
854 questions with no upvoted or accepted answers
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Subset of the domain of attraction
Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
- 61
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74
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When is the unstable direction map $x\mapsto e^{u}(x)$ injective?
Let $f:M \to M$ be a $C^{2}$-Anosov diffeomorphism. Therefore, there exists an invariant splitting of the tangent bundle $T_{x}M = E^s(x) \oplus E^u(x)$ into a stable and an unstable directions, that ...
- 1,043
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193
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Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map
Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...
- 3,098
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55
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Continuity of Kneading invariants of generalised $\beta$-trasformations
For $\beta \in (1,2]$ and $\alpha \in [0,2-\beta]$ consider the generalised $\beta$-transformation $T_{\alpha,\beta}:[0,1] \to [0,1]$ to be $$T_{\alpha, \beta}(x) = \beta x + \alpha \mod 1.$$ It is a ...
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342
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Are the orbits of this discrete dynamical system bounded?
Somehow I believe this should be true and easy to prove but cannot nail it down. A reference, proof, or counterexample will suffice. Didn't get any help over at MSE even with a bounty so I came here.
...
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85
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Dynamical obstructions for a vector field $X$ whose adjoint operator $ad_X$ sends a global orthonormal frame to a set of mutually orthogonal vectors
Let $X$ be a vector field on a parallelizable manifold $M$.
Can we equipe $M$ with a Riemannian metric such that we have at least one global orthonormal frame $\{V_1,V_2,\ldots,V_n \} $ such that $[...
- 356
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175
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Example of topologically transitive dynamical system with invariant non-ergodic Borel measure
Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which
$f : \Lambda \to \...
- 151
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62
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Lyapunov theory in coupled nonlinear dynamic system with input
Suppose I have the following nonlinear coupled dynamic system
\begin{align*}
&\dot{x}_1 = f_1(x_1,x_2)\\
&\dot{x}_2 = f_2(x_2) + u
\end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...
- 345
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87
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How fast will it converge to equilibrium？
$\alpha \in \mathbb{R} \backslash \mathbb{Q}$, given $\varepsilon>0 \quad \lambda>0$.
given $n_{1}, \cdots, n_{k} \in N^{*}$ satisfied $(1-\varepsilon) \lambda<n_{1}<\cdots<n_{k}<(1+...
- 543
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Diophantine approximation and the Euclidean algorithm
My question is whether something I've noticed is well-known. It seems like it must be, but I've been unable to find any references that describe what is outlined below.
Given real $x$ and irrational $...
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99
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Long-term behavior of asynchronous, stochastic, numerical solution to a dynamical system
I am simulating the behavior of a dynamical system, say $$\dot{x} = f(Ax; \lambda), $$
with an Euler update, where $x\in \mathbb{R}^n$ and $\lambda$ are some parameters. In my scenario, $A\in \mathbb{...
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Is there a term for a linear operator on an $L^p$ space that "locally respects boundedness"?
Let $X$ be a Polish space, and $\mu$ a locally finite measure. Take any $p \in \{0\} \cup [1,\infty)$. We will say that a linear operator $T \colon L^p(\mu) \to L^p(\mu)$ has property $(\ast)$ if ...
- 708
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45
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Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces
Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm
$$
\|f\|_M:=\inf\left\{\lambda &...
- 5,405
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Is a set over which dynamics are topologically conjugate to a shift map on two symbols always repelling?
Consider the one-sided full shift map $\sigma$ and the associated shift space of infinite sequences in two letters $\{0,1\}^\mathbb{N}$ on which the shift map acts, equipped with the usual metric. ...
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189
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Vandermonde shift
I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let
$$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\
0 & 0 & 1 & 0 & \...
- 384
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77
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What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?
When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
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When are all average trajectories of $w_{k+1}=Aw_k+b$ bounded?
Below is an open-problem in my field, and I'm wondering if someone has insights I'm missing. (cross-posted on math.se)
Suppose observation $x$ is drawn from some distribution $\mathcal{D}$, $w_0\in \...
- 2,242
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Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system
Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
- 81
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115
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Algorithm/iterative procedure for constructing hypercyclic vectors?
Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
- 5,405
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80
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Entropy of Markov measure using marginal distribution
Let $S$ be a finite set and let $\mathcal{X}$ be the set of all bi-infinite sequences over $S$. Let $\eta_1,\eta_2$ be two shift invariant 1-step Markov measures over $\mathcal{X}$. For a finite word $...
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72
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Is there a name for and/or reasonably nice characterisation of "mixingly physical" measures?
Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a diffeomorphism, and let $\mu$ be a probability measure on $M$ with compact support.
As stated in the ...
- 708
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Hypercylic operators with sets of hypercyclic vectors almost covering the space
Let $\{T_i\}_{i \in I}$ be a family of hypercylic operators on a separable Banach space $X$. From the transitivity theorem, we know that $HC(T_i)$, the set of vectors $x \in X$ with $\{T_i^n(x):n \in ...
- 5,405
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271
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Is my ansatz for finding $n$-periodic-points of the exponential-function exhaustive?
The following is about getting help for a proof on existence and indexability of periodic points of the exponential-function, here with base $e:=\exp(1)$.
Update The question is a complete rewriting ...
- 5,342
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52
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On smooth extensions of functions
Let $f(x) = \left(I - \hat{n}\hat{n}^T \cdot\textbf{1}_{\vec{n}^TAx \geq 0}\right)Ax$, where $I$ is the identity matrix, $A$ is a (symmetric) $d\times d$ positive definite matrix, $\hat{n}$ is an ...
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97
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Determining the behavior of a contraction mapping with undefined points
Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
- 1,087
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36
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Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
- 5,405
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67
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Stability of a continuous piecewise linear map
I am studying random perturbation of a system that is continuous and piecewise linear. More precisely: I am given a map $\Phi_1:\mathbb{R}^d\to \mathbb{R}^d$ such that
$$ \Phi_1(x) = \left\{\begin{...
- 562
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68
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Alternate characterization of floquet multipliers: Floquet theory
Given an autonomous ode $\dot{x}=f(x)$ in $\mathbb{R}^n$ possessing a period-p time-periodic solution $\bar x(t)$, one can use the so-called variational equation about $\bar x$ to study its stability. ...
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Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
- 5,405
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89
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Lower bounds of kappa class functions
I saw in the paper "Smooth Satabilization Implies Coprime Factorization" of
Eduardo Sontag the following argument: Given a smooth map $a:\mathbb{R}
^{n}\rightarrow\mathbb{R}^{+}$, let $\rho$ be any ...
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106
views
Existence of large first return times
Let $(X,T,\mu)$ be a measure preserving system, with $\mu$ a probability measure. Let $E \subset X$ of positive measure and $\tau_E$ be the first return time to $E$. Then the Kac Lemma asserts that $$\...
- 2,283
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45
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Nonlinear fixed-point equation with linear solutions?
Let $S$ be an $N\times N$ row-stochastic matrix and let $w'$ be the left Perron eigenvector of $S$ (i.e., $w$ is the stationary distribution of the Markov chain represented by $S$). Let $T$ be the ...
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76
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Decomposition into distal and proximal
For a topological group $G$ and a bounded real- or complex-valued function $f$ on $G$, the orbit closure of $f$ is the pointwise closure in the space of all bounded functions on $G$ of the orbit of $f$...
- 1,074
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45
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On the center-stable manifold theorem for sets
Suppose I have a dynamical system $f:S \to S$ where $S \subset \mathbb{R}^n$ and $S$ is compact and $f$ is twice differentiable. Assume there exists a function $V$ such that $V(f(x)) < V(x)$ unless ...
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72
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Distance between value function of deterministic and stochastic control problems
Suppose that one wants to control a diffusion process
$$
dX_t^u = \mu(X_t^u,u)dt + \sigma dW_t; \qquad X_0^u=x
$$
in order to optimize a stochastic control problem with value function
$$
V_T(u)=\...
- 5,405
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A generalization of competitive systems
We consider the following standard partial order relation on $\mathbb{R}^n$:
We say $X=(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots,y_n)=Y$ iff $\sum_{i=1}^k x_i \leq \sum_{i=1}^k y_i,\quad \forall k: 1\...
- 356
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64
views
Finding monochromatic subset of triangular lattice
Let
$$ \Lambda = \{(x,y)\in\mathbb{N}^2:y\geq x\} $$
the upper triangular lattice and $d:\Lambda\to\{1,\dots,c\}$ a coloring (i.e. an arbitrary function) on $c$ colors. Let $k\geq2$. I am looking for $...
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65
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Closure of the periodic points in the logistic family
My question is about the closure of the periodic point for the logistic family $f_\lambda(x) = \lambda x(1-x)$ of maps of the interval $[0,1]$.
Is there an explicit parameter $\lambda$ for which the ...
- 18.7k
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77
views
Are there existing approaches to estimate inherent dynamics of an unknown system?
Consider a set of measured discrete-time signals $y_t \in \mathbb{R}^L$, which are captured from a dynamic physical system with underlying states $x_t \in \mathbb{R}^N$. Let's assume we have more ...
- 111
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A linear map on $\chi^{\infty}(\mathbb{R}^2)$ arising from the Cauchy integral formula
The space of smooth vector fields on $\mathbb{R}^2$ and open unit disc $\mathbb{D}$ are denoted by $\chi^{\infty} (\mathbb{R}^2)$ and $\chi^{\infty}(\mathbb{D})$, respectively. A vector field on $\...
- 356
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209
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Volume-preserving flows with cross section
Let $M$ be an orientable closed smooth manifold of dimension n. Let $\Omega$ be a volume form for $M$, i.e., a nowhere-zero smooth n-form. A smooth $\Phi_t$ flow defined on $M$ is volume-preserving ...
- 365
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103
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Is there a research direction within dynamical systems theory / ergodic theory that concerns conjugability to a two-point motion?
Let $X$ be a set equipped with some structure (e.g. topological space, measurable space, probability space, etc.). We say that two endomorphisms $f,g \colon X \to X$ are conjugate to each other if ...
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56
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Are there multiple conjugacy classes of order 2 elements in the smooth automorphism group of $\mathbb{R}$?
Consider the group $\text{Aut}\mathbb{R}$ of smooth invertible maps from $\mathbb{R}$ to $\mathbb{R}$. If $f\in\text{Aut}\mathbb{R}$ has order 2 ($f$ is an involution), is $f$ conjugate to $g(x)=-x$?
...
- 11
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37
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Attractivity of a system with state-dependent transitions
Let $A\in\mathbb{R}^{n\times n}$ and consider the following dynamical system:
$$
\frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n,
$$
where $\max\{\cdot\}$ acts ...
- 2,712
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472
views
The mysterious numbers $ \frac{13}{20} $ and $20$?
Let $g(x) = x^6 - 30 x $
Let $h(x) = x^6 $
Let $f(x) = x^2 - 2 $
Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$
Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...
- 763
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84
views
Trapping lightrays under nonstandard reflections and/or paths
Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open:
"It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
- 151k
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67
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Reduced master equation for a multistable Hamiltonian dynamical system
I am looking for rigorous results on the derivation of a reduced master equation for a (possibly stochastic) Hamiltonian dynamical system with a coercive potential energy term with multiple local ...
- 3,873
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answers
82
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Foliations with algebraic foliation chart
An algebraic foliation chart for a foliated manifold is a foliation chart for which the transition maps are polynomial maps.
What is an example of an analytic foliation of the Euclidean space $\...
- 356
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0
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49
views
Example of a "very noisy" SDE on a compact manifold with zero maximal Lyapunov exponent
Setting:
Let $M$ be a compact connected $C^\infty$ Riemannian manifold of dimension $D \geq 2$, with $\lambda$ the normalised Riemannian volume measure.
Write $T_{\neq 0}M \subset TM$ for the non-...
- 708
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69
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Equidistribution of linear forms over euclidean ball
Given a vector $v\in \mathbb{Z}^d\setminus\{0\}$, an irrational number $\eta$ and some big $M>0$ what type of bound can one get on $$\sum_{w\in \mathbb{Z}^d\cap B(0, M)}\exp(2\pi i \eta \cdot \...
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