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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Does a substitution tiling being FLC depend on starting seed?
I've been trying to understand more on "geometric" substitutions rather than just symbolic ones. As symbolic substitutions always yield FLC tilings, I wanted to know whether a tiling coming ...
- 633
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64
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Physical measure of a dynamical system in terms of its density
Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
In ergodic theory, the occupation measure is
$$\mu_{x, T}(...
- 247
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154
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The space of ergodic elements of a topological or Lie group
Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
- 356
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The boundedness of dynamical systems discretized from Hamiltonian systems
Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e.,
\begin{align}
&\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\
&\frac{dq}{dt}...
- 47
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165
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Rotation number for homeomorphisms of a Lie group other than $S^1$
Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$.
For what kind of Lie group $G$ the standard process of definition of rotation number ...
- 356
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178
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Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?
Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region?
Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
- 356
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89
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Pre-images of the critical point of $3.83 x(1-x)$
This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below.
Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
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55
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Periodic Orbit without Complex Eigenvalues
I am studying the following ODE system, representing a simple excitable circuit:
$$
\dot{V}_m = I_{app} - (V_m - \alpha_f PL(V_m) + \alpha_s PL(V_s))
$$
$$
\tau_s \dot{V}_s = V_m - V_s
$$
where
$$
PL(...
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0
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Convex combination of positive mean-ergodic operators
Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that:
For every $h:[0,1]\to \mathbb{R}_+$ we have that
$$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
- 333
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93
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Time-inhomogeneous Krylov-Bogoliubov Existence Theorem
I am interested in what is known about the application of the Krylov-Bogoliubov existence theorem to the time-inhomogeneous case, especially as it relates to an underlying random dynamical system (...
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
Using the following definition of Dirichlet L-function
$$
L(1,\chi)=\begin{cases}
\dfrac{2\pi h}{w\sqrt{m}} & \textit{if}\ \chi(-1)=-1 \\\\
\dfrac{2 h \log{|\epsilon|}}{w\sqrt{m}} & \textit{...
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Repelling invariant manifold of a discrete dynamical system
Given a $C^\infty$ map $Q: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following properties
$Q$ fixes the $x_1$-axis, i.e. $Q(x_1,0,\dotsc,0) = (x_1,0,\dotsc,0)$.
For $x_1$ in a neighborhood of $...
- 511
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102
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Homothety vector fields generating a foliation of $S^3$
Inspired by this question on homothety vector fields we realize that non homotheticity is some how an intrinsic property of the foliation associated to the vector field. See the comment by Prof. ...
- 356
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Reason behind the names of sub and supercritical bifurcations
What is the reasoning behind the names sub- and super-critical bifurcations that occur in the context of pitchfork and Hopf bifurcations? Textbooks seem to introduce this terminology without any ...
- 3,322
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100
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Generalizing an application of the Poincaré-Bendixson theorem
I was looking for applications of the Poincaré-Bendixson theorem and on this site I have found several examples almost all similar to this post. So I tried to make a quite natural generalization
$$
\...
- 143
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156
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Time-scale calculus (an similar approaches - measure chains) on more general "time" sets
Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
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35
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Construction of a homogeneous Moran set
Fix a positive integer $N\ge 2$, for $n \in \mathbb{N}$, denote
$$\Sigma=\{0,1,\dots,N-1\},\\
\Sigma^n=\{(\omega_1,\dots,\omega_n):\omega\in\Sigma, i=1,\dots,n\}.$$
Let $p>2$ be a positive integer. ...
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Reference for rigidity of higher rank action
I heard some results about the rigidity of higher rank action and it looks very interesting. I would like to know if there are any good survey of paper to get started in this field. Thank you in ...
- 307
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48
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Smoothness of unstable manifold near (non?)-hyperbolic fixed point w.r.t. generator of the flow
Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can ...
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177
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Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
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96
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Building random homeomorphisms of the circle
Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...
- 101
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70
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What *piecewise* smooth curves/surfaces/hypersurfaces give rise to forward-invariant regions of dynamical systems?
Consider a set $\mathcal{B}\subset \mathbb{R}^n$ that is homeomorphic to a closed n-dimensional ball, and denote its boundary by $\mathcal{H}$. Assume that $\mathcal{H}$ is a "piecewise smooth&...
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53
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Basin of attraction comparative statics* using local energy functions?
Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
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0
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180
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Denjoy example in the Poincaré–Bendixson theorem
I have already finished understanding the Poincaré-Bendixson theorem as a consequence of Schwartz's theorem, but I also want to analyze the example that Denjoy gave in $C^1$ that is not within the ...
- 143
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157
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Motivation for Ionescu Tulcea-Marinescu (Lasota-Yorke inequality)
I wonder about motivations of a work of Ionescu Tulcea-Marinescu. In order to establish the decomposition of the operator $T$ they assume (condition (1.3)) this operator satisfies the inequality
$$\|...
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99
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Limit cycles or stable solutions for k-dimensional piece-wise linear ODEs
As a branch of reinforcement learning, restless multi-armed bandits have been shown PSPACE-HARD but Whittle has offered an implementable solution called the Whittle Index Policy. Weber and Weiss ...
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97
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Bounds and repulsion domains for the Dirichlet eta function $\eta(\sigma+it)$, for fixed $\sigma$
Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T
=\Omega_T(\sigma)$ as the ...
- 3,098
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53
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Unique solution to nonlinear optimization through gradient descent
I am trying to estimate the path of a random walk described by the following SSM
$$
\begin{align}
x_{t+1} &= x_{t} + q_{t+1} \newline
y_{t+1} &= h(x_{t+1}) + r_{t+1}
\end{align}
$...
- 11
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117
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Relation between the distance projective maps and their angles
Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
- 1,043
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98
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Clarification on the proof of Lyapunov-Razumikhin asymptotic stability theorem for delayed differential equations
this is my first question here, hope I am in the right place :)
Recently I have been looking at the proof of theorem 4.2 on Razumikhin stability for RFDEs in the book by Jack Hale and Lunel Verduyn: ...
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56
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Minimising risk in dynamical systems
I have been reading the paper of Goerner and Ulancowicz - "Quantifying economic sustainability" in which it is suggested that there is a tradeoff between sustainability and efficiency. ...
user478214
1
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1
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165
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Multidimensional intersection property
Consider the multidimensional annulus $\{(p,\theta)\} = \mathbb R^n\times\mathbb T^n$ endowed by the $1$-form $\omega=p\,d\theta$. A diffeomorphism $A$ of this annulus onlo itself is said to be exact ...
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Eigenvalue assignment via state feedback: existence proof
Consider the linear time invariant system:
$$\tag{1}\label{eq1}
\dot{x}(t) = Ax(t) + Bu(t), \ \ x(0)=x_0\in\mathbb{R}^n,
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$. Let $p_M(s)...
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Minimal sets of foliations in the plane (generalisation of Poincaré-Bendixson)
Let $F$ be a $1$-dimensional foliation of an open subset of the plane defined by a
locally Lipschitz line field. Suppose $C$ is a compact minimal set of $F$ (i.e. $C$ is non-empty, compact, a union of ...
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91
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Random matrix heuristics for Koopman operators
Consider a nice hyperbolic dynamical system $(X, T)$, for instance a $\mathcal{C}^\infty$ Anosov map. The action of the Koopman operator
$$\mathcal{K} : \ f \mapsto f \circ T$$
has a nice spectrum ...
- 581
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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 $\kappa,\epsilon\in\mathbb{R}$, $t\in\...
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$L^p$-continuity for discrete linear causal systems
Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by:
\begin{...
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Equivalence between smoothly regular and analytically regular
I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...
- 803
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59
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Reference of the fact that Hoelder cocycles are associated to Hoelder potentials in Ledrappier's correspondence
Let $\tilde{M}$ be the universal cover of a compact pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
...
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Under reasonable assumptions, is a closed invariant graph with only negative Lyapunov exponents necessarily stable?
Let $\Omega$ and $M$ be compact $C^\infty$ manifolds, let $\theta \colon \Omega \to \Omega$ be a $C^\infty$ diffeomorphism, and let $\Theta \colon \Omega \times M \to \Omega \times M$ be a $C^\infty$ ...
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Selecting a suitable Lyapunov function for the following systems?
i) SI MODEL
Consider
\begin{align}
\frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex]
\frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I
\end{align}
Where $N=S+I$ is the total population.
If ...
- 185
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88
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Sequences generated from commuted quaternions and general commuted linear transformations
Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...
- 1,547
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61
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Holomorphic dynamical systems defined on a contractible bounded open subset of $\Bbb{C}^n$
Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$:
Attracting Case: There is an ...
- 3,599
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answers
117
views
Is a "global period" similar to a "local period"?
Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the ...
- 449
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0
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47
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Hypercylic operators have very typical cyclic vectors
Let $W$ be the Wiener measure on $C_0(\mathbb{R})$ and let $T\in L(C_0(\mathbb{R}),C_0(\mathbb{R}))$ be a hypercylic operator; i.e. there exists some $f \in C_0(\mathbb{R})$ such that $\{T^n(f)\}_{n=1}...
- 5,405
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80
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Dynamics of composition of reflections
Let $C$ be a curve defined by $y = f(x)$, and define the vertical reflection over $C$ to be the map $(x,y) \mapsto (x,y')$, where $y' = 2 f(x) - y$. In other words, the vertical distance from $(x,y)$ ...
- 213
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0
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210
views
Is there a condition for a subshift of finite type to be uniquely ergodic?
Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic?
- 111
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61
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Number of permitted words up to permutation in a subshift
Let $A$ be a finite set and let $X \subseteq A^{\mathbb{N}}$ be a subshift. Let $\mathcal{L}_n$ denote the set of words of length $n$ appearing in $X$. For a word $w \in \mathcal{L}_n$, one can ...
- 323
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0
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61
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Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?
Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
- 356
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0
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61
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Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?
Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics.
In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...
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