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
4
votes
0
answers
60
views
A geometric quantity associated to a vector field on a surface
Let $(M, g)$ be a $2$ dimensional Riemannian manifold.
Then we consider the Riemannian metric on TM described here.
Assume that $X:M\to TM$ is a vector field. For every $p\...
- 356
4
votes
0
answers
187
views
Asymptotic formula, polynomial, irrational number and uniformly distribution
Problem 1
Given a irrational number $\alpha$ and two polynomials with positive integer coefficients $P(n),Q(n)$, is it possible to get the asymptotic estimate and reasonable error term for:
$$\...
- 697
4
votes
0
answers
141
views
Asymptotic behavior of gradient descent on a smooth, convex, non-negative function with no finite minimum - part II
Let $f: \mathbb{R}^d\rightarrow\mathbb{R}_{\geq 0}$ be a function which is convex and smooth (i.e., in $C^{\infty}$). If $x^* \in \mathbb{R}^d$ is the (global) minimum of $f$, it is well known that ...
- 968
4
votes
0
answers
142
views
Equivalent Idempotents in the Ellis Semigroup
Let $(X,T)$ be a dynamical system where $T$ is a (at least countably infinite) group acting on a compact Hausdorf space $X$, and let $E(X)$ be the Ellis semigroup of this system (if we abuse notation ...
- 297
4
votes
0
answers
177
views
Explicit symbolic codings
The short version of my question is that I need examples of explicit continuous symbolic codings of invertible dynamical systems.
Here's a longer version. Suppose $(\Omega,\mu,T)$ is an invertible ...
- 2,135
4
votes
0
answers
495
views
Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)
This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.
Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
- 356
4
votes
0
answers
96
views
On decidability of an infinite dimensional dynamic system
Consider two infinite dimensional Toeplitz integer matrices $A,B$ where $B$ is just a shift operator and an infinite dimensional vector $V_0$.
Given a prime $p$ and an infinite dimensional integer ...
- 13.9k
4
votes
0
answers
142
views
KAM stable orbits are smooth
I'm in my final year of my undergraduate studies doing work on modelling the n-body problem numerically and I also have some interest in theoretical guarantees. Now, I've been looking for a theorem ...
- 3,871
4
votes
0
answers
116
views
Dynamics of pairwise distances in the $n$-body problem
Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...
4
votes
0
answers
149
views
Connection between cardiac equations and untangling knots?
I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe. ...
- 151k
4
votes
0
answers
104
views
Why can every twist map be realized as the time-1 map of a time-dependent Hamiltonian?
if have problems getting my head around the following claim made
by Moser in "Monotone twist mappings and the calculus of variations" and Gole in "Symplectic twist maps".
Setting:
Let $F : \mathbb{R}...
- 41
4
votes
0
answers
197
views
Dynamics of an inequality
The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...
- 128k
4
votes
0
answers
111
views
Integrability of Continuous Tangent Subbundles
Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?
More specifically given a smooth manifold of $M$ and a ...
- 419
4
votes
0
answers
466
views
Lorenz attractor power spectrum
If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
- 41
4
votes
0
answers
119
views
Conjectural growth rate for ergodic sums of logarithms
Let $\theta, \phi \in [0,1)$, and consider the sums
$$
S_n(\theta,\phi)=\sum_{k=0}^n \log|e^{2\pi i (k\theta+\phi)}-1|.
$$
The possible boundedness from above of such sums plays a key role in ...
- 2,758
4
votes
0
answers
327
views
The Moyal action of a planar vector field
Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$:
$\tilde{D}_{X}(f)=...
- 356
4
votes
0
answers
168
views
Automorphisms of Nilmanifolds
Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...
- 419
4
votes
0
answers
327
views
Amenable groups acting on the real line, that are not subexponentially-amenable
In the literature, there are several examples of solvable groups acting faithfully by order-preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth with ...
4
votes
0
answers
423
views
The $\Omega$-Stability Theorem
I'm currently studying the $\Omega$-Stability Theorem:
Theorem: If $\mathcal{R}(f)$ has a hyperbolic structure then $f$ is $\mathcal{R}$-stable.
Some explanations about the statement: $f$ is a $C^1$ ...
user11178
4
votes
0
answers
83
views
Do identical orbit tiles imply identical combinatorial types?
Given a periodic trajectory on a triangle, we can associate to this trajectory a sequence of integers $1,2$ and $3$ by labeling the edges of the triangle and taking the sequence of edges the ...
- 881
4
votes
0
answers
355
views
Perturbation of Morse functions at critical points leaving stable manifolds invariant
Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858).
Is there a way ...
- 53
4
votes
0
answers
153
views
Minimality of time-t minimal flows
This question is mainly motivated by the question Transitivity of a flow and its time-1 map
Let $M$ be a closed smooth manifold and $\Phi\colon\mathbb{R}\times M\to M$ be a smooth minimal flow, i.e. ...
- 1,060
4
votes
0
answers
335
views
Algebraic Dynamics over separated schemes
I have a few questions regarding the current status of research on algebraic dynamics over separated schemes. In what follows $\varphi:X\rightarrow X$ will be a finite self-morphism of a noetherian ...
- 3,101
4
votes
0
answers
795
views
Almost linear ODE: how node becomes a spiral
Most introductory ODE books contain a discussion of almost linear systems, and there are two cases when the behavior of an almost linear system near an equilbrium point can differ from the behaviour ...
- 29.1k
3
votes
0
answers
66
views
Borel complexity of the set of generic points for an invariant measure in a minimal system
I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is ...
- 1,658
3
votes
0
answers
107
views
Stability of a nonlinear dynamical system with non-elementary dynamics
I am trying to prove stability and get a non-asymptotic upper bound on the convergence rate of a nonlinear discrete-time dynamical system, whose dynamics are stated in terms of the (non-elementary) ...
- 31
3
votes
0
answers
183
views
Bounded solutions of nonlinear third-order ODEs
I am interested in understanding the behavior of solutions to certain nonlinear third-order ODEs. Specifically, I am curious about conditions that guarantee all solutions remain bounded for $t \in [0, ...
- 807
3
votes
0
answers
119
views
The topological entropy of potential space filling curves on the unit interval
By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1$...
- 356
3
votes
0
answers
56
views
Perturbation method for time-periodic singular system of ODEs
I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
- 131
3
votes
0
answers
50
views
Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
3
votes
0
answers
73
views
Name for a product of actions / dynamical systems
Suppose $G \curvearrowright X, H \curvearrowright Y$ are group (or monoid) actions, or dynamical systems. Then $X \times Y$ is a $G \times H$-system of the same type in the obvious way by $(g, h) \...
- 6,652
3
votes
0
answers
157
views
Has this metric been considered anywhere?
I posted this on math stack exchange some 10 days ago, but received no answers (https://math.stackexchange.com/q/4773194/1223994).
Let $X$ be a compact metric space and denote by $d$ the metric on $X$....
- 339
3
votes
0
answers
59
views
Maximal number of aperiodic Wang tiles
I was wondering whether there is an analogue result to the minimality of Wang tiling, in the direction of maximality.
I think that the paper by Jeandel and Rao, shows that the minimal number of Wang ...
- 633
3
votes
0
answers
135
views
the second largest eigenvalue of transfer operators
A Gauss map $T$ is mixing and satisfies Lasota-York inequalities. By Henon's theorem, we know that the transfer operator $\hat{T}$ associated with $T$ has a spectral gap. This means there exists a ...
user509763
3
votes
0
answers
71
views
Trapped vs. nonwandering points
For a continuous flow on a topological space the concept of a nonwandering point (in forward/backward time) is well-known. Another useful concept (for example from the point of view of scattering ...
- 1,942
3
votes
0
answers
210
views
Jacobi equation and conjugate points on solution curves of the Van der Pol vector field
Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
- 356
3
votes
0
answers
74
views
The most general (but useful) definition of "attractor" for dynamical systems
Consider J. Milnor's paper: On the concept of attractor.
There he writes: "A less restrictive definition" [than some of the previous ones he had considered] of the concept of an attract is &...
- 429
3
votes
0
answers
148
views
General references on dynamics of continuous, piecewise linear interval maps
I want to find a general reference on topological and measurable dynamics of continuous piecewise linear interval maps. I am particularly interested in cases with only three pieces. Even ...
- 339
3
votes
0
answers
134
views
Asymptotic behaviors of equilibrium points of a switching SDE with Levy jumps?
Consider the following paper titled: Stochastic regime switching SIR model driven by Lévy noise, authored by Yingjia Guo.
Link: https://www.sciencedirect.com/science/article/pii/S0378437117302145
The ...
- 185
3
votes
0
answers
101
views
Turing reaction diffusion equations and neural networks
Suppose you have a Turing-type reaction-diffusion system
$$
\begin{cases}
\partial_t \phi = & f(\phi, \psi) + D_\phi \nabla^2\phi \\
\partial_t \psi = & g(\phi, \psi) + D_\psi \nabla^2\psi
\...
3
votes
0
answers
160
views
Ergodic diffeomorphisms of the circle
From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...
- 101
3
votes
0
answers
93
views
Regularity of center manifold
Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$.
Suppose that the spectrum of $\mathrm{D}f(\...
- 31
3
votes
0
answers
115
views
Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 101
3
votes
0
answers
145
views
Naturality of geodesic flow
Let $\texttt{Man}$ be the category of smooth manifolds with local diffeomorphisms as morphisms, and $ \texttt{Bun}$ --- the category of bundles (affine bundles or just fibre bundles, if necessary) and ...
- 2,934
3
votes
0
answers
92
views
What dynamical properties should we expect from systems satisfying statistical ones?
Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example:
the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...
3
votes
0
answers
257
views
Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
- 136
3
votes
0
answers
187
views
Exponential map of cotangent bundle and Morse theory on based loop space
Consider $M$ to be a compact manifold and $q_0,q_1\in M$. Let $L_t$ be a lagrangian and $\mathcal{E}_L$ the lagrangian action functional on the based loop space $\Omega(M,q_0,q_1)$ defined as $\...
- 791
3
votes
0
answers
74
views
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
3
votes
0
answers
145
views
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
3
votes
0
answers
105
views
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