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Questions tagged [ds.dynamical-systems]

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

Dynamics of flows and maps (continuous and discrete time), including topological dynamics, low-dimensional dynamics, complex dynamics, hyperbolic dynamics, including nonuniform hyperbolicity, ergodic theory, Hamiltonian dynamics, bifurcation theory, normal forms, symbolic dynamics, algebraic dynamics, dimension theory, thermodynamic formalism, multifractal analysis, classical mechanics, including infinite-dimensional dynamics (functional equations, delay differential equations).

2394 questions
42
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2 answers

Can we trap light in a polygonal room?

Suppose we have a polygonal path $P$ on the plane resulting from removal of an one of a convex polygon's edges and a ray of light "coming from infinity" (that is, if we were to trace the path backwards in time, we would end up with an infinite…
Wojowu
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25
votes
5 answers

Is there a dynamical system such that every orbit is either periodic or dense?

Let $(T,X)$ be a discrete dynamical system. By this I mean that $X$ is a compact Hausdorff space and $T: X \to X$ a homeomorphism. For example, take $X$ to be the sequence space $2^{\mathbb{Z}}$ and $T$ the Bernoulli shift. Then there is a dense set…
Akhil Mathew
  • 25,291
22
votes
9 answers

Why do dynamicists worry so much about differentiability hypotheses in smooth dynamics?

I have been learning a bit about stable and unstable manifold theory for a non-uniformly hyperbolic diffeomorphism $f: M \to M$ on a smooth manifold. It seems that there are two completely separate cases, each having its own universe of literature:…
Paul Siegel
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18
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3 answers

How to construct a topological conjugacy?

Background: One says that continuous maps $f: X \to X, g: Y \to Y$ are topologically conjugate if there exists a homeomorphism $h: X \to Y$ such that $h \circ f = g \circ h$. There are many ways one can see that two maps are not topologically…
Akhil Mathew
  • 25,291
16
votes
3 answers

Local linearization of ODE at singular point

I would like the simplest example of the failure of an ODE to be locally diffeomorphic to its linearization, despite being locally homeomorphic to it. More precisely, consider x' = f(x) with f(0) = 0 in R^n. Let A = f'(0) so that the local…
Robert Bruner
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14
votes
6 answers

Definition of a strange attractor.

May be it's not the right place for this, but I don't know the right definition of a strange attractor. Wikipedia states that "An attractor is informally described as strange if it has non-integer dimension or if the dynamics on it are chaotic." In…
Nurdin Takenov
  • 1,565
13
votes
4 answers

Dynamical systems with multidimensional, complex and other exotic kinds of time spaces

As one may know, a dynamical system can be defined with a monoid or a group action on a set, usually a manifold or similar kind of space with extra structure, which is called the phase space or state space of the dynamical system. The monoid or…
The_Sympathizer
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11
votes
2 answers

Invariant subsets of $z \mapsto z^2$

Where can I find an explicit construction of closed invariant subsets of the map $z \mapsto z^2$ on the unit circle? Furstenberg mentions that there are continuum of such disjoint minimal sets but does not provide specific details.
Arkady Kitover
  • 111
10
votes
1 answer

Topological conjugacy between homeomorphisms and diffeomorphisms

Consider a compact differentiable manifold $M$. We say that $f:M\to M$ and $g: M \to M$ are topologically conjugated if there exists $h:M\to M$ a homeomorphism such that $f\circ h= h \circ g$. The conjugacy class of a homeomorphism $f$ is the set of…
rpotrie
  • 3,878
8
votes
1 answer

In a non-compact metric space, topological transitivity need not imply onto

I had asked this question on Mathematics Stack Exchange yesterday but it got no response so I'm asking here. Let $X$ be a compact metric space and $f:X \to X$ be continuous. If $f$ is topologically transitive. Then $f$ is onto. I'm trying to show…
Mark
  • 343
8
votes
1 answer

$\omega$-limits of $1$-dimensional dynamical systems

The question that I have in mind is the following: Which kind of closed sets can arise as the $\omega$-limit of a point for a $1$-dimensional dynamical system? It is probably somewhat naive, but nowhere I looked did I find a hint of what kind of…
Selim G
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8
votes
0 answers

Volume preserving conjugacy in Hartman-Grobman theorem

According to Hartman-Grobman theorem, a $C^1$ germ of diffeomorphism $f$ on $\mathbb{R}^n$ at a fixed point $x$ whose differential $Df(x)$ is hyperbolic is always $C^0$-conjugated to its differential, that is there exists a germ of homeomorphism $h$…
Vincent H
  • 383
7
votes
0 answers

A model of self-organizing behavior

I'd just like to know if the following model has received any attention: A state at discrete time $t$ consists of a function $S_t:{\Bbb Z}^2\rightarrow S^1$. So view each cell $c$ (element of ${\Bbb Z}^2$) as having eight neighbors in the usual…
David Feldman
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7
votes
0 answers

Unbounded energy growth in a Hamiltonian system

Does there exist an orbit with unbounded velocity in the system $\ddot x = (-1)^{[t]+[x]}$, where $[*]$ denotes the integer part of *?
Vadim
  • 71
6
votes
0 answers

Dynamical phenomena in $\mathbb{R}^n$ first arising for n > 3?

For differentiable dynamical systems defined on, say, an open ball in $\mathbb{R}^n$, when $n=2$ Poincaré-Bendixson tells us a lot about what can happen. In particular, P-B precludes chaos and strange attractors. When $n=3$ these things appear, as…
Joshua Grochow
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