Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
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Responses from mathematicians concerning Flash trading [closed]
Have there been any responses from the mathematics community regarding flash trading, for example from a game theory or system dynamics point of view? Please answer with personal comments or ...
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Do there exist Markov partitions with (nearly) uniform Riemannian measures?
This question complements this one; the difference is in considering Riemannian versus SRB measures.
Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an ...
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A topologically mixing subshift with multiple measures of maximal entropy
Let Σp={1,...,p}ℤ be the full shift on p symbols, and let X ⊂ Σp be a subshift -- that is, a closed σ-invariant subset, where $\sigma\colon \Sigma_p\to \Sigma_p$ is the ...
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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 ...
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Do there exist Markov partitions with (nearly) uniform SRB measures?
Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an Anosov diffeomorphism, and let $\mu$ be a Sinai-Ruelle-Bowen (probability) measure. Write $\mathcal{R} = \{ ...
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Global stability for dynamical systems in $R^n$
Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs).
Assume that:
(1) The system has an absorbing ball, that is every trajectory eventually enters this ball
and stays in ...
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Denjoy counterexamples $C^1$ close to conjugates of rotations.
It is well known and easy to prove that any Denjoy counterexample in the circle is approximated by homeomorphisms of the circle which have the same rotation number and are transitive (in particular, ...
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Automorphisms of $\pi_1$ induced by pseudo-Anosov maps
Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$.
For any $\...
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Are Markov partitions structurally stable?
Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an Anosov diffeomorphism, and let $\mathcal{R} = \{ R_1,\dots,R_n \}$ be a Markov partition. Because of ...
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Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle....
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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 ...
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Spectral curve of Elliptic Calogero-Moser systems
First, why all the coefficients in the characteristic polynomial of L are elliptic functions, since the diagonal entries of the matrix L are the momentums?
second, how to understand the ramification ...
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Is there a notion of "Morse index" for geodesics in a manifold with indefinite metric that is well-behaved under cutting and gluing?
More generally, I'm interested in the situation of Lagrangian mechanics. And actually my question is local, so you can work on $\mathbb R^n$ if you like. I will begin with some background on ...
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Uniqueness in Composition of Polynomials
The following situation came up in my research:
Suppose two functions $f$ and $g$ map $[0,\infty)$ to (a subset of) itself. The function $f$ is linear and $g$ is quadratic, but $g$ is one-to-one on ...
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Does this problem have a name? [Ducci Sequences]
Let $a_1, ... a_n$ be real numbers. Consider the operation which replaces these numbers with $|a_1 - a_2|, |a_2 - a_3|, ... |a_n - a_1|$, and iterate. Under the assumption that $a_i \in \mathbb{Z}$, ...
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How to figure out the type of the bifurcation in a dynamical system?
Suppose we have a dynamical system
$\dot{x} = f(x,r)$
in which x is a state variable and r is a bifurcation parameter. How to figure out which kind of bifurcation(s) (e.g. saddle-node, ...
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When deRham curve is bijection?
Motivation: Suppose we have deRham curve. From wikipedia:
Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M:
$d_0:\ M \...
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Partitions and Expansiveness
Why if one have an $\varepsilon$-expansive homeomorphism $T:X \rightarrow X$ ($X$ a compact metric space) and a given partition $D$ of $X$ which has diameter smaller than $\varepsilon$ the sequence ...
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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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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 ...
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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
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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 ...
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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 ...
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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 ...
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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 *?
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Is there a name for this differential operator and/or its corresponding spectrum?
Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional
$$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$
where $X_p(f)$ is the ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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