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
2
votes
0
answers
41
views
Why has the random Koopman matrix $ G_{xx}^{(-)} G_{yx} $ only eigenvalues on the complex unit circle?
Let U be a $\Bbb{R}^{(n+1)(n+1)} $ matrix with entries drawn from a independent normal distribution,
e.g.
$$ U_{i j} \sim N(0,1) \quad \quad i,j=1,...n+1$$
Let $ G=U U^* $ be a Gram matrix where $ U^* ...
- 253
2
votes
0
answers
126
views
Identification of Maharam extension
All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
- 307
2
votes
0
answers
59
views
Classification of real polynomial vector fields on R2, up to polynomial automorphisms?
A result of Brunella classifies complete complex polynomial vector fields on ${\mathbb C}^2$, up to polynomial automorphism, and relies heavily on an earlier work of Suzuki. I haven't fully digested ...
- 13.3k
2
votes
0
answers
74
views
Is it likely that gradient flow trajectories of a $G$-invariant function pass through degenerate points?
This question may be posed somewhat vaguely, but I'm interested to actually get an idea of what to expect, so I try to not target it at a specific result.
Assume that $G$ is a compact Lie group, ...
- 139
2
votes
1
answer
254
views
Chaotic dynamics of maps on unit square that are NOT Triangular
We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $...
- 271
2
votes
0
answers
61
views
Aligning frequencies
Let $\omega_1, \omega_2, \dots, \omega_n$ be frequencies between $1$ and $\log n$. I would like to find an upper bound for a point $t$ that align these frequencies up to a small error $\delta$, that ...
- 21
2
votes
0
answers
60
views
Identifying bifurcation
[![enter image description here]] 1]1I am trying to analyze the bifurcation of a 3D continuous model. For a certain range of parameter values, the origin is always an unstable point, whereas the ...
- 53
2
votes
0
answers
57
views
Regularization for Newtonian n-body collisions in $\mathbb{R}^3$
In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
- 46
2
votes
0
answers
71
views
Any solution of an evolution problem tends to a steady state in $L^2$?
I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
- 1,759
2
votes
0
answers
65
views
Where can I find resources for a paper "Stability analysis of a novel DDE of HIV CD4+ T-cells"?
I am currently working on a the paper [NND]:
Question:
On page 4, equation 6 introduces a concept related to the infection rate within the context of the HIV model. Unfortunately, the paper does not ...
- 232
2
votes
0
answers
136
views
Progess on conjectures of Palis
I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...
- 247
2
votes
0
answers
142
views
Unipotent closure in classical groups
Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...
- 81
2
votes
0
answers
319
views
A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
- 333
2
votes
0
answers
116
views
Aperiodic SFT equal to a substitution subshift
I was wondering whether there are primitive symbolic substitutions over $\mathbb{Z}^d$ and alphabet $\mathcal{A}$ whose associated subshift is equal to an aperiodic SFT. By SFT here I mean a subshift ...
- 633
2
votes
0
answers
469
views
Analogue of Margulis height function in non lattice subgroups
I have been reading this paper https://link.springer.com/article/10.1007/s11854-017-0033-4 on singular system of linear forms and non escape of mass in homogeneous spaces $G/\Gamma$ where $ G=SL(m+n,\...
- 337
2
votes
0
answers
257
views
When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?
Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$.
The pull back map $f^*$ is a symplectomorphism wrt the ...
- 356
2
votes
0
answers
153
views
Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
- 331
2
votes
0
answers
109
views
Proving light escapes mirrors via ergodic theory of billiards
There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems ...
- 509
2
votes
0
answers
313
views
Correlation decay rate
Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and
$\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
- 136
2
votes
0
answers
101
views
Persistence of KAM tori as a function of dimension
I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here.
In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
- 121
2
votes
0
answers
83
views
Question about stable manifold theorem and Frobenius integrability theorem
I have a question about Anosov diffeomorphism (Wikipedia: Anosov diffeomorphisms)
For hyperbolic fixed point $p$, $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the ...
- 39
2
votes
0
answers
92
views
A type of coupling problem II
This posting is related the following questions in MSE and in MO.
In general terms, suppose $(X,\mathscr{B},\mu)$ is a $\sigma$-finite measure space. If $\nu$ is another measure on $\mathscr{B}$, $\...
- 271
2
votes
0
answers
72
views
On bounded solutions of a given fourth-order linear ODE
Consider the fourth-order linear ODE
$$
\label{eq1}
v^{(4)} + \frac{-C_2 - 2\alpha \phi}{C_4}v'' + \frac{4\alpha \phi'}{C_4}v' + \frac{k_1 + 3k_3\phi^2 -2\alpha \phi''}{C_4}v = 0.
$$
Without getting ...
- 121
2
votes
0
answers
128
views
When is a composition of homeomorphisms topologically transitive provided one of the two is?
Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...
- 4,459
2
votes
0
answers
163
views
Weakly mixing diffeomorphism
From
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 measure ...
- 101
2
votes
0
answers
84
views
Examples of chaotic self-similar blowup in PDEs
When the Cauchy problem to a PDE blows up, it can often be analyzed using self-similar variables. In the reference:
Eggers, J., & Fontelos, M. A. (2008). The role of self-similarity in ...
- 177
2
votes
0
answers
55
views
Search for period N logistic map
The logistic map is a period doubling bifurcation system.
Are there known dynamical maps, which oscillate between $N$ points where $N$ is a prime number, like 2, 3, or 5, or 7... , where each ...
- 155
2
votes
0
answers
137
views
Entropy of a sequence
I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following,
It would be interesting to rule out the possibility of $\lambda_f(n)$ ...
- 589
2
votes
0
answers
141
views
Choosing the derivative of a flow
I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
- 121
2
votes
0
answers
94
views
Lyapunov function utility in stochastic optimal control
The article Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching by (X.Mu, Q.Zhang) studies necessary and sufficient conditions on near-optimal controls.
In both ...
- 141
2
votes
0
answers
106
views
Stochastic dynamics: how do the random matrix $J_{ij}$ and coupling strengh $g$ affect the variance of the local field $h_i$?
Context: Q3 in How to understand the largest Lyapunov exponent?
We know $g$ is proportional to (square root of) the variance of $J$'s every entry ($J_{ij}\sim \mathcal{N}(0,g^2/N)$).
Why is it also ...
- 139
2
votes
0
answers
149
views
Polynomial entropy of topological dynamical systems
For a continuous mapping $\varphi: X \to X$ on a compact Hausdorff space $X$ we define the entropy as follows:
Given open covering $\mathcal{U}$, $\mathcal{V}$ of $X$, we call $\mathcal{V}$ a ...
- 467
2
votes
0
answers
74
views
Equicontinuity, Beltrami coefficients and Sequence of top/bottom semi-annuli
In the Beltrami equation literature, one approach to showing equicontinuity for pairs $(f_{n},\mu_{n})$ (where $\partial_{\bar{z}}f_{n}=\mu_{n}(z)\partial_{z}f_{n}$) is via the relations of moduli and ...
- 5,474
2
votes
0
answers
167
views
Why were these constants picked in this Lyapunov function and how did the author arrive at the final form of the Lyapunov function?
Consider the following paper:
"A note on global stability for a tuberculosis model" by Gao and Huang: https://doi.org/10.1016/j.aml.2017.05.004
The methodology is understood in this paper ...
- 185
2
votes
0
answers
192
views
Almost periodicity and approximation in tracial von Neumann algebra
Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. For a countable group $G$, let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action on $N$ which preserves the tracial ...
- 73
2
votes
0
answers
76
views
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 $Q$, ...
- 854
2
votes
0
answers
94
views
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, ...
- 121
2
votes
0
answers
72
views
Maximal Lyapunov exponent of Schrödinger-Newton equation
I am trying to determine the sign of the maximal Lyapunov exponent of the Schrödinger-Newton equation
$$
\partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|...
- 121
2
votes
0
answers
35
views
Properties of the orbit $Abx_0$ when $b$ is upper or lower triangular but not diagonal
Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $A$ denote the subgroup of $G$ ...
- 1,565
2
votes
0
answers
170
views
Symplectic structure on moduli space of holomorphic Abelian differentials
I've heard a "symplectic structure" referred to on the moduli space of holomorphic Abelian differentials by numerous people / sources. I do not know how to interpret this - I am looking for ...
- 31
2
votes
0
answers
95
views
Local decomposition of semisimple Lie groups at the identity into the product of centralizer, unstable and stable horospherical subgroups
First let us look at an example. Let $G=\text{SL}(d,\mathbb R)$ and $A$ be its subgroup consisting of diagonal matrices with positive entries.
Let $a\in A$ be an element. We define the stable ...
- 1,565
2
votes
0
answers
54
views
A.e. global existence of solution to 'encased' n-body problem already somewhere in the literature?
In the study of the Newtonian n-body problem, it seems that Von Zeipel's theorem and Saari's theorem concerning the improbability of collision singularities ought to lend themselves to a nice ...
- 1,461
2
votes
0
answers
71
views
Domain of definition of a certain mapping
Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold.
I am studying the mapping
$$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...
2
votes
0
answers
70
views
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
2
votes
0
answers
153
views
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
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
2
votes
0
answers
71
views
What are desirable properties that data should satisfy to reasonably use the dynamic mode decomposition?
In the dynamic mode decomposition, we consider a sequence of data vectors $\{z_0, \dots, z_m\}$ where $z_k \in \mathbb{R}^n$ for all $n$. We assume that the data satisfies the linear relationship $z_{...
- 263
2
votes
0
answers
43
views
Self-maps (dynamical systems) in several variables induced by functions $X^{n+1}\to X$
Self-maps $F:X\to X$ can be viewed as dynamic systems.
A function $f:X^{n+1}\to X$ induces a self-map $F:X^{n+1}\to X^{n+1}$,
$$ F(x_0\dots x_n):=
(x_1, \ldots, x_n, f(x)) $$
for every $x:=(x_0, \...
- 4,786
2
votes
0
answers
143
views
inverse of moment-generating function in terms of moments
Let $\{h_i\}$ be decreasing sequence of $n$ positive reals. Define distribution $p(X=h_i)\propto h_i$ and let $g(s)=E_X[e^{sX}]$ be the moment generating function. For instance, for $h=\{1,\frac{1}{4},...
- 2,242
2
votes
0
answers
116
views
Birth of chaos due to nonautonomous perturbation
Let $\sigma, b>0$. I want to study the dynamics of the map
$$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that
$$T_{\sigma,b}(n,\theta,y) = (\...
2
votes
0
answers
95
views
Persistence of homoclinic points in the non-compact case
It is well known that a transverse homoclinic point of a hyperbolic periodic point of a $C^1$-diffeomorphism of a compact manifold $M$ persists under small $C^1$ perturbations. This follows easily ...
- 111