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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Probabilistic approach for cellular automata
Few months ago my scientific adviser asked me to use probabilistic ideas in such problem :
Consider a matrix NxN. Each element of matrix is a number 1 or 0. We may change all elements of this matrix ...
- 137
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Does differentiating an integro-differential equation results in equivalent stability of the solution?
I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation:
...
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Closed orbit for vector field $f(\bar{z})$ where $f$ is holomorphic function
Edit : According to the comments of Michael Renardy and Christian Remling I revise the question as follows:
Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such ...
- 356
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434
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Long term behavior of a certain discrete time dynamical system on graphs
Consider the graph $(V,E)$ with vertex set $V=\{v_1,...,v_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v_i\in V, (v_i,v_i)\in E$.
Assume that each vertex has an $\textit{...
- 2,441
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underdamped oscillation with quadratic decay
I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form:
...
- 169
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114
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Fit a system of linear ODEs from several experiments
Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...
- 749
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Quadratic stability of linear time varying system
(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \...
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Recurrence and transience of cocycle over a dynamical system
Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and $\phi_n(x)=\phi(x)+\cdots+\phi(T^{n-...
- 2,244
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323
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Logistic map periodic point
$x_{n+1}=4x_n(1-x_n)$ I already proved that for $x_n\subset [0,1]$, $x_n=sin^2(2\pi y_n)$
with $y_{n+1}=\begin{cases}2y_n & 0 \le y_n < 0.5 \\vee 2y_n -1 & 0.5 \le y_n < 1 \end{cases}$...
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Hölder continuity of uniform limit of piecewise constant functions
Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,...
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Modulo dynamics on [0,1)
For $T: \mathbb{R} \mapsto \mathbb{{R}_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of $T(\...
- 2,619
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Basic question on minimal flows
I know that minimal flows are actions for which no proper closed invariant subsets exist, but I am unclear how to understand this concept.
If a coset flow on a quotient space Gamma/S is ergodic, ...
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Deeper meanings of Phase Space -- any books? [closed]
I often think about the phase space with quite deep interpretations. For example, contraction of phase space means losing energy. But, some of the energy is easily restored (free energy?) while some ...
- 175
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Commuting Nonlinear Vector Fields
I am working on stability of nonlinear switched systems and recently, I have proven that switched systems with homogeneous, cooperative, Irreducible and commuting vector fields , i.e., vector fields ...
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Conditions required for the orbit of a set of positive measure to cover state space?
Suppose $(X, \mathcal{M}, \mu, T)$ is a measure-preserving dynamical system with $T$ invertible.
I am wondering what properties the dynamical system would need to have in order for the following to be ...
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Equivalence of Wind Forces: Intensity vs. Duration [closed]
The strongest tornado in the world happened recently in Greenfield Iowa with winds over 318 mph: https://www.facebook.com/watch/?v=2176728102678237&vanity=reedtimmer2.0
I am curious, are less ...
user531026
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Same occupation measure $\Rightarrow$ same trajectory
Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
The occupation ...
- 247
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Invariance of the Kronecker factor
Let $(X,\mathcal{F},\mu,T)$ be a measure preserving system and $U_T$ Koopman operator on $L^2(X)$, i.e. $U_T f = f\circ T$. Note that, for the moment, I am not imposing any further assumptions on $X$, ...
- 141
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Non-asymptotic convergence rates for gradient descent
I'd like to know how the number of steps needed for gradient descent depend on properties of the Hessian in non-asymptotic regime.
More specifically, number of gradient descent steps needed to obtain ...
- 2,442
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282
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Limiting distribution in $M_t/M_t/1$ queue
Consider a $M/M/1$ queue with a constant arrival rate $\lambda$ and service rate $\mu$ with $\lambda < \mu$. We know that in this case the limiting distribution exists and it is a geometric ...
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Non-minimal system in which every point is a full entropy point
Is there a discrete topological dynamical system $(X,f)$, where $X$ is a compact metric space (with distance $d$), which is transitive but not minimal, such that $h(f)>0$ and every point is a full ...
- 4,459
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Why does bounded distortion imply the following inequality?
Let $f: I \to I$ be a one-dimensional differentiable function of bounded distortion with distortion constant $M$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such ...
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Can the identity function be approximated by compositions of a "uniformly monotone-and-convex" set of functions?
Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties?
For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$.
There exist $0&...
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Lyapunov vectors along a trajectory
I have the equation:
$$
\dot{x}_i = F_i(x)
\tag{1}
$$
with $x\in \mathbb{R}^n$. To deal with the Lyapunov exponents, we write the equation for small displacements $\delta x_i$:
$$
\dot{\delta x}_i = \...
- 469
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140
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On proving the absence of limit cycles in a dynamical system
I'm studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I'm stuck at this two-variable problem which I've struggled with for too long now.
$$ \dot M ...
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734
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Infinite composition of continuous functions
Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F_n:= f_n\circ \dots\circ f_1$. Then $F_n$ is continuous. However, the pointwise limit need not be (consider Mateusz'...
- 5,405
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Difference of hypercyclic operator and identity
Let $B$ be a separable Banach space, $L:B \to B$ be a hypercyclic operator, $k>0$, $I_B$ the identity on $B$, and define $L_k: =k (I_B - L)$. When is $L_k$ hypercyclic on $B$? Can anything else ...
- 5,405
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Can the immediate basin of attraction of super-attracting fixed point at 0 of a polynomial contain non-zero roots?
Let $f$ be a polynomial with a super attracting fixed point at $x=0$. Can the immediate basin of attraction of the fixed point contain other roots? If so, please provide a specific example with the ...
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Topologically transitive dynamical system mapping space into ball
Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily ...
- 5,405
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Question about graph Ginzburg-Landau equation
I am reading the following paper:
http://ieeexplore.ieee.org/document/7798534/
For the equation (5):
$$\dot{a}=(1-a_i^2)a_i-\beta\sum_{k-i}(a_k\cos(\theta_k-\theta_i)-a_i)$$
$$\dot{\theta}_i=\beta\...
- 345
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Calculate of Lyapunov Exponents of a sequence of random matrices
Let $(\Omega,\mathcal{F},\mathbb{P}):=(M^{\mathbb{N}_{0}},\mathcal{M}^{\mathbb{N}_{0}},\mathbb{P})$ be a probability space where $M=\left\{0,1,2,3,4\right\}$, $\mathcal{M}^{\mathbb{N}_{0}}$ is product ...
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A one dimensional fractal like set with the same line width within a bounded area? [closed]
Say that we have a line $\left(0,0.5\right)$. I want a process that can split that line and half and move that half a bit, and then take half of that half and moved it and so on, so that by the end ...
- 143
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What is the relationship between solutions for the parameterised second order differential equations
Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for
$$
u''+u'-i\lambda V(x)u=0, \, x\in [0,1],
$$
What is the ...
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Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$
This question has been asked here but there is no answer:
https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
Consider autonomous ODE $y' = ...
- 103
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Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?
I am reading a blog which talks about a $C^1$ diffeomorphism $f: \mathbb{D}\{ x^2 + y^2 < 1\} \to \mathbb{R}^2$ and estimates the Hausdorff dimension of its image $\mathcal{H}_\sqrt{2}^d (f(\mathbb{...
- 22.8k
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Help with notation for the state of a dynamical system defined by a PDE
Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of ...
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173
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Stability analysis of ODE
My questions concerns the stability analysis of the following dynamical system :
$\dfrac{d}{dt} a_{i}(t) = D_{i} + \displaystyle{\sum_{j=1}^{n}L_{ij}a_{j}(t) + \sum_{j=1}^{n}\sum_{k=1}^{n} C_{ijk} a_{...
user40957
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860
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Sierpinski Triangle and the Chaos Game
The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
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Entropy of inverse map for endomorphism case on surfaces
Hi,
I know that in the diffeomorphism case the measure entropy of the T:M^{2}-->M^{2} (M smooth Rimannian surface) will be the same as the measure entropy of T^{-1}. But i need to know about the C^{2}...
- 1
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396
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Dynamics of polynomial roots
Are there any good tools to understand the movement of roots of polynomials in single variable with real or rational coefficients? That is say the coefficients are of the form $a_{i} + M b_{i}$ where $...
- 13.9k
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intersection partition as an orbital partition
Let $X=\{0,1\}^{\mathbb{N}}$ and $\xi_n$ be the partition of $X$ defined by the equivalence relation $x \sim_n x' \Leftrightarrow (x_{n}, x_{n+1}, \ldots) = (x_{n}', x_{n+1}', \ldots)$. The sequence ...
- 2,560
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Symplectic submanifolds and first integrals
I was working with symplectic submanifolds when I posed the following question:
Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard ...
- 3
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739
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Simple system of ODEs with periodic coefficients
I am stuck with a little problem that I cannot solve mith the standard methods I learn at university. I have a system of coupled ODEs:
$f'(t) = P \cos(k t + \Phi_1) g(t)$
$g'(t) = Q \cos(k t + \...
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169
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Refining ladders and orbit segments - with a picture
I am trying to understand the following paragraph from The Classification of Non-Singular Actions, Revisited, page 5 paragraph 2.
Remember that $S \in [T]$ so that for every $x\in X, S(x) = T^{n(x)...
- 718
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761
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When is convergence transitive?
Suppose I have a discrete dynamical system with a finite set X of states, and suppose I want to prove that every state of X ends up, sooner or later, in a subset Z under the dynamics of the system. ...
- 531
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189
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Difference Equations & Possible Limits
The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.
If we look at the behaviour of a point in R n under matrix multiplication, we ...
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34
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Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term
The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
- 395
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79
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Alternative proof of parabolic implosion
I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation.
Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
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For which values of $\mu$ is the Standard Map $f_{\mu}(x,y)=(x+y+\frac{\mu}{2\pi} \sin (2\pi x),y+\frac{\mu}{2\pi}\sin (2\pi x))$ non wandering?
Let $f_{\mu}(x,y)=(x+y+\frac{\mu}{2\pi} \sin (2\pi x),y+\frac{\mu}{2\pi}\sin (2\pi x))$, with $\mu > 0$ and considered in the cylinder $\mathbb{R}/\mathbb{Z} \times \mathbb{R} $. For which values ...
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Analysis of sensitivity to initial conditions in dynamic systems
Consider the iterative function defined by:
$$
x_{n+1} = f(x_n)
$$
where $x_0\in [0, 1]$ and
$$
f(x) = \sin\left(\pi \left(b^{rx(x-1)}\mod 1 \right)\right)
$$
with $b, r > 0$. We aim to demonstrate ...
- 11