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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.

6
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1answer
675 views

Moshe Rosenfeld's Salmon Problem

As an amusement at the start of this talk, Moshe Rosenfeld poses the following question. Suppose that there are n salmon which begin at distinct points on a unit circle, each facing either ...
2
votes
1answer
321 views

Switching function for Bang-Bang nagivation

I'm attempting to develop an equation to determine the "switching time" for a control system. I've managed to work out a specific solution for when starting and ending velocities are are the same, ...
8
votes
0answers
441 views

What is the “category of bifurcations”?

While reading the introduction to this paper by Curtis McMullen, I came to the following (bold added): In this paper we show that every bifurcation set contains a copy of the boundary of the ...
28
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3answers
2k views

Improving a sequence of 1s and -1s

Suppose you take a $\pm 1$ sequence and you want to "improve it" by taking pointwise limits of translates. What properties can you guarantee to get in the limit? Two examples illustrate what I think ...
5
votes
2answers
445 views

Asymptotics of iterated polynomials

Let the sequence $u_1, u_2, \ldots$ satisfy $u_{n+1} = u_n - u_n^2 + O(u_n^3)$. Then it can be shown that if $u_n \to 0$ as $n \to \infty$, then $u_n = n^{-1} + O(n^{-2} \log n)$. (See N. G. de ...
4
votes
2answers
544 views

Gaps in nx (mod 1)

It is known that if you choose n point at random on S1 = [0,1], the nearest neighbor spacings between the points are exponentially distributed with mean 1. For example, two of our n points could be ...
13
votes
2answers
773 views

Is there an analogue of the Lefschetz fixed point theorem for discrete dynamical systems?

Background/Motivation Let $(X, f)$ be a discrete dynamical system. For now, $X$ is just a set and $f$ is just a function $f : X \to X$. Suppose that $f^n$ has a finite number of fixed points for ...
13
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3answers
4k views

Mapping Class Groups of Punctured Surfaces (and maybe Billiards)

Where can I find a concrete description of mapping class group of surfaces? I know the mapping class group of the torus is $SL(2, \mathbb{Z})$. Perhaps, there is a simple description for the sphere ...
18
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0answers
796 views

Almost complex 4-manifolds with a “holomorphic” vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$? The following sub question is ...
15
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2answers
2k views

Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$. I found this in claim a ...
8
votes
1answer
543 views

A regularity property of transition matrices for the cat map

I've noticed a rather strange phenomenon (not important for my particular research, but interesting) and wouldn't be surprised if someone more versed in symbolic dynamics (i.e., just about anyone who ...
0
votes
1answer
179 views

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 ...
4
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2answers
1k views

Proper families for Anosov flows

So I've been skimming Bowen's 1972 paper "Symbolic Dynamics for Hyperbolic Flows" hoping it would give me some insight into how to build a Markov family for the cat flow (i.e., the Anosov flow ...
7
votes
3answers
1k views

Suggested reading for thermodynamic formalism

Are there any good books out there that can serve as an introduction to thermodynamical formalism in dynamical systems? I know only Zinsmeister's short "Thermodynamical formalism and holomorphic ...
128
votes
14answers
26k views

What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...
3
votes
1answer
339 views

Anosov diffeomorphisms and the chaotic hypothesis

There is a well-known "chaotic hypothesis" dating from 1995 or so in statistical physics that suggests that classical statistical-physical systems should be "effectively" Anosov. I won't get into the ...
4
votes
4answers
533 views

Does it help to learn statistical mechanics in order to learn thermodynamic formalism?

Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal ...
14
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6answers
3k views

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 ...
8
votes
0answers
229 views

Monotone invariants of braid forcing

Let $\phi$ be a diffeomorphism of the unit disk $D^2$, fixed on the boundary, and suppose that $Q$ is a finite subset of the interior permuted by $\phi$. The isotopy class of $\phi$ relative to $Q$ ...
11
votes
3answers
757 views

How much “Morse theory” can be accomplished given only a continuous transformation of a space?

If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold ...
20
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6answers
2k views

Quantitative versions of ergodic theorem

Are there any general theorems similar to Birkhoff's ergodic theorem, but giving quantitative estimates on the rate of convergence or average time of recurrence (perhaps with additional assumptions)? ...
2
votes
1answer
767 views

The Arnold cat map

How can I compute the SRB measure for the cat map? Also any pointers to references for obtaining Markov partitions and recurrence times would be lovely. Thanks
2
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1answer
280 views

ODE system question

Consider a system of the form: dx/dt = f(x,y) , dy/dt=g(x,y), with the property that the associated ODE dy/dx = g(x,y)/f(x,y) has a unique solution to IVP y(0)=0. Also, f(x,y) is smooth every except ...
5
votes
1answer
298 views

Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed. Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...
3
votes
3answers
1k views

What are some conserved quantities of Poisson brackets?

Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets. Let's say we are working on T^n x R^n (T^n ...
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5answers
2k views

Rational maps with all critical points fixed

What can be said about rational self-maps of $\mathbb P^1$ for which all critical points are also fixed points ? If all but one of the fixed points are critical, there is a characterization in http://...
8
votes
6answers
1k views

When does the sequence of iterates of a rational function converge?

Darsh asks at the 20-questions seminar: Let $f:P^1 \rightarrow P^1$ be rational function. Can you say when the sequence $\{ f^n(x)\}_n=\{ x,f(x),f(f(x)),\cdots\} $ converges? What about the sequence ...