# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

1,593 questions

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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 $\...

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### Reading list in dynamical systems

So I’ve managed to gather from various sources, a plethora of books in dynamical systems. Now I’m wondering which of them to read, and in what order.
So far these are the books I’ve found/been ...

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### What are the applications of topological quantum field theory to continuous-time dynamical systems?

From wikipedia:
In dynamics, all continuous time dynamical systems, with and without noise, are Witten-type TQFTs and the phenomenon of the spontaneous breakdown of the corresponding topological ...

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### Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...

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### Why do unstable manifolds of two close point intersect each other in Baker map?

Let $M$ be $S^1 \times [-1,1]$, $f$ a baker map on $M$ and for $p, q \in M$ consider $W^s_p$ the stable manifold in $p$ (i.e. the set of points whose forward orbit tend to the forward orbit of $p$) ...

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### A sufficient condition for uniform convergence of ergodic averages?

Given a measure preserving system $(X, mu, T)$ where $\mu(X) = 1$, we say $T$ is uniformly weak mixing if for all measurable $A$, $B$ and every $\varepsilon > 0$, there exists some $N$ independent ...

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### Does an asymptotic component with large size in a minimal subshift always exist?

Let $(X, T)$ be a minimal subshift, i.e. $X$ is a closed $T$-invariant subset of $A^\mathbb{Z}$, where $T$ is the shift. A pair $x,y\in X$ is asymptotic if $d(T^nx, T^ny)$ goes to zero as $n\to\infty$....

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### A bisector line bundle over the total space of a principal bundle

Let $(P,X,G)$ be a principal bundle where $G$ is a Lie group which acts on $P$. We fix a principal connection on $P$ and a right invariant metric for $G$. These structures define a unique Riemannian ...

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### 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-...

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### Topological entropy of logistic map $f(x) = \mu x (1-x)$, $f:[0,1] \to [0,1]$ for $\mu \in (1,3)$

As stated in the question, I want to find the topological entropy of the logistic map on the interval $[0,1]$ for a "nice" range of the parameter $\mu$, namely $\mu \in (1,3)$. I think the fact that $...

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### Lyapunov functions for determining the stability of invariant sets?

Suppose we have a dynamical system $\dot{x}=f(x)$ with an equillibrium $x_0$. It is known that $x_0$ is Lyapunov stable in this sense if there exists $V:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $V(...

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### Can a nonlinear dynamical system be rewritten in terms of constraints?

My question is based on thoughts after reading to a specific section in the paper "On Contraction Analysis for Nonlinear Systems" by W. Lohmiller and JJ. Slotine, Section 4.2 Constrained Systems. ...

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### conditions for asymptotic comparison to hold

I have the following simple dynamical system:
\begin{align}
x_1' &= a - f(x_2)x_1\\
x_2' &= bx_1 - cx_2,
\end{align}
where all parameters and initial conditions are positive. $f(x_2)$ is a ...

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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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### Phase space of double pendulum [migrated]

Let $M$ be the phase space of a double pendulum system. What is known about the manifold $M$? Do we know it's homotopy typed? Do we know its (co)homology? Are there any references tackling these ...