In dynamical systems, the motion of a particle in some geometric space, governed by some time dependent rules, is studied. The process can be discrete (where the particle jumps from point to point) or continuous (where the particle follows a trajectory). Dynamical systems is used in mathematical ...

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687 views

Fekete's conjecture on repeated applications of the tangent function

A high-school student named Erna Fekete made a conjecture to me via email three years ago, which I could not answer. I've since lost touch with her. I repeat her interesting conjecture here, in case ...
23
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1answer
2k views

Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of ...
22
votes
2answers
690 views

Why is the topological pressure called pressure?

Let us consider a compact topological space $X$, and a continuous function $f$ acting on $X$. One of the most important quantities related to such a topological dynamical system is the entropy. For ...
21
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3answers
569 views

Self Study in Dynamical Systems

I'm trying to get into the field of dynamical systems by (self) studying one-dimensional dynamics and circle homeomorphisms; for my guidance, I'm trying to assemble materials in this field that obey ...
18
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2answers
3k views

Perfectly centered break of a perfectly aligned pool ball rack

This question is asked on Physics SE and MathOverflow by somebody else. I don't think it belongs there, but rather here (for reasons given there in my comments there; edit: now self-removed). ...
16
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4answers
354 views

A Category Theoretical view of the Kalman Filter

Some basic background The Kalman filter is a (linear) state estimation algorithm that presumes that there is some sort of uncertainty (optimally Gaussian) in the state observations of the dynamical ...
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11answers
2k views

Mandelbrot-like sets for functions other than $f(z)=z^2+c$?

Are there any well-studied analogs to the Mandelbrot set using functions other than $f(z)= z^2+c$ in $\mathbb{C}$?
15
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3answers
621 views

Are the fractional parts of $\log \log n!$ equidistributed or dense in $[0,1]$?

Are there any results relevant to the distribution of the sequence $\{\log \log n!\}$ for integers $n$, where $\{x\}$ denotes the fractional part of $x$? For instance, it is known that for irrational ...
15
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1answer
364 views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
14
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4answers
343 views

What are the important theorems in the theory of dynamical systems?

I recently stumbled over the section about dynamical systems in my physics textbook. I noticed that, although most of the rest of the book was very rigorous, this part contained nearly no firm ...
14
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1answer
421 views

Kakutani skyscraper is infinite

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
10
votes
1answer
204 views

Non-ergodic measure

Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic? I know that ergodic measures are ...
10
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2answers
281 views

An equivalent condition for strong-mixing

For a measure-preserving (finite) system $(X,\mathcal{B},\mu,T)$, is it correct that the following are equivalent? For every $A,B\in\mathcal{B}$ , $\displaystyle\lim_{n\rightarrow\infty}\mu(A\cap ...
10
votes
1answer
309 views

Is a Markov process a random dynamic system?

A random dynamic system is defined in Wikipedia. Its definition, which is not included in this post for the sake of clarity, reminds me how similar a Markov process is to a random dynamic system just ...
9
votes
3answers
296 views

Quadratic Julia sets and periodic cycles

Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting. Infinity is always one ...
9
votes
2answers
2k views

How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background?

I'm turning to you as I find myself in need of help as to how to best go about studying something related to chaos theory/dynamical systems/fluid dynamics in postgraduate school. I'm currently in my ...
9
votes
2answers
118 views

What are Green's almost primes?

In a general-audience talk, Ben Green explains his famous proof with Terence Tao as an application of Szemerédi's theorem, but placing the primes within a smaller set of almost-primes in which they ...
9
votes
1answer
71 views

Set of points dense in subset of four-dimensional space

We may assume the following theorem: Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$. Consider the following ...
9
votes
1answer
351 views

Number of limit points of a continued exponential

Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
8
votes
1answer
314 views

Starting digits of 2^n

Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits.
8
votes
1answer
247 views

The Mandelbrot Set Membership

To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions: $$ \begin{cases} z_{n+1} &= &z_n^2 + c &\text{ for }n\geq ...
8
votes
5answers
242 views

High-School Level Introduction to Dynamical Systems

In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory. I'm having trouble coming up with a topic compelling enough ...
8
votes
2answers
155 views

Is it possible to capture a light ray in a solar panel?

A while ago I was wondering how we could use mathematics to increase the efficiency of solar panels. The kind of mathematics I was thinking about in particular was Dynamical Billiards. Though I think ...
8
votes
2answers
1k views

Recommendation for a book and other material on dynamical systems

I currently have the book Dynamical Systems with Applications Using Mathematica by Stephen Lynch. I used it in an undergrad introductory course for dynamical systems, but it's extremely terse. As an ...
8
votes
1answer
131 views

What kind of polygonal surface has an interior angle > 360°?

Consider this polygon as the setting for a dynamical billiard: When it's drawn in the plane, the polygon intersects itself; it is non-simple. However, I don't want to embed the polygon in the ...
8
votes
1answer
184 views

An example for a dynamical system.

is there an example for a dynamical system $(X,T)$ where $X$ is a compact space and $T:X \to X$ is a continous map, s.t $\Pi(T) = \emptyset$, where $$\Pi(T) = \bigcup_{n=1}^{\infty} \Pi_n(T), \Pi_n(T) ...
8
votes
1answer
107 views

An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
7
votes
5answers
737 views

Period of 3 implies chaos

Let $f(x)$ be a continuous function from $\mathbb{R}\rightarrow\mathbb{R}$. Let's denote $k$-times repeated application of the function, $f(f(f(...f(x)...)))$ as $f^{(k)}(x)$. Let's call any $x$ a ...
7
votes
5answers
2k views

What is the difference between max and sup?

I am studying KS (Kolmogorov-Sinai) entropy of order q and it can be defined as $$ h_q = \sup_P \left(\lim_{m\to\infty}\left(\frac 1 m H_q(m,ε)\right)\right) $$ Why is it defined as supremum over ...
7
votes
6answers
206 views

Modeling the Decay of a Pack of Cannibalistic Hyenas

A population of $p_0$ hyenas has run out of food in their ecosystem, and so sadly they have resorted to eating each other. Hyenas need to consume one meal a day, and so exactly once per day, any ...
7
votes
1answer
155 views

Vectors fields structurally stable

(This was a question on my doctoral qualifying exam.) Let be $X$ a vector field defined in $\mathbb{R}^2$ such that $X$ is structurally stable in every compact set of $\mathbb{R}^2$. Is $X$ ...
7
votes
1answer
438 views

Bifurcations in a two-parameter dynamical system

I have the following dynamical system $$ \dot{x} = \mu - mx - xy^2 $$ $$ \dot{y} = -\mu y + xy^2 $$ and I need to "analyse" any bifurcations that occur as $m$ and $\mu$ are varied. I've worked out ...
7
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0answers
388 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Liyapunov) I don't ...
7
votes
0answers
207 views

Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, ...
6
votes
2answers
515 views

Interesting but elementary properties of the Mandelbrot Set

I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know ...
6
votes
3answers
175 views

Mathematical Limitations of Computer Experiments

One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
6
votes
2answers
223 views

A recursive sequence generated by a polynomial

Let $P(x)$ be a polynomial in one variable with integer coefficients, and define the sequence $a_0, a_1, a_2, a_3,\cdots$ $$a_0 = 0, \ a_n = P(a_{n-1})$$ If there exists a $m$ natural number such ...
6
votes
3answers
302 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
6
votes
1answer
130 views

Trajectories on the $k$-dimensional torus

Let $r_1,\dots,r_k$ be irrational and linearly independent over $\mathbb Q$. My intuition clearly tells me that the set $$\{(nr_1,\dots,nr_k)+\mathbb Z^k:n\in\mathbb N\}$$ is dense in $\mathbb ...
6
votes
2answers
88 views

Sharkovskii-type results in other topological spaces?

I recently came across Sharkovskii's Theorem which asserts that if $f:\mathbb{R} \to \mathbb{R}$ is continuous and has a cycle of length $m$, then $f$ has a cycle of length $n$ for any $n$ which comes ...
6
votes
2answers
235 views

Transition time in a Lotka-Volterra system

I am working with a set of real-valued ordinary differential equations based on the Lotka-Volterra competition equations: $$\begin{align} \dot{a_1} & = a_1 \left( 1 - a_1 - 2 a_2 \right) \\ ...
6
votes
1answer
245 views

is the geodesic flow on Hyperbolic Plane completely integrable?

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and ...
6
votes
2answers
301 views

Chaos and ergodicity in hamiltonian systems

EDIT : I formerly claimed something incorrect in my question. The Liouville measure needs NOT be ergodic on hypersurfaces of constant energy. Also, I found out that NO hamiltonian system can be ...
6
votes
1answer
110 views

Quadratic iterative system

A general linear iteative system can be represented as a matrix: $$(x,y)\mapsto(ax+by,cx+dy)$$ is essentially the same as $$\left[\begin{array}{cc} a&b\\ c&d\\ \end{array}\right] ...
6
votes
2answers
187 views

Prove that Anosov Automorphisms are chaotic

Let me add some detail first. An Anosov automorphism on $R^2$ is a mapping from the unit square $S$ onto $S$ of the form $\begin{bmatrix}x \\y\end{bmatrix}\rightarrow\begin{bmatrix}a && b \\c ...
6
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1answer
111 views

converging to cosine by iteration

In what sense (if at all) does the iteration $x \mapsto 2x^2 - 1$ converge to $\cos 2^n x$ in the unit interval [-1,1]? One might try to plot in Mathematica: ...
6
votes
1answer
117 views

What's the name of this chaotic system? (Cool pics included.)

I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a ...
6
votes
2answers
356 views

Schwarzian Derivative and One-Dimensional Dynamics - how are they connected?

During the summer, I did an REU where we focused primarily on one-dimensional dynamics and more specifically kneading theory. One thing that I was always confused about is why the Schwarzian ...
6
votes
1answer
304 views

How to prove Mandelbrot set is simply connected?

In this lecture note of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is ...
6
votes
2answers
126 views

Recurrent point of continuous transformation in a compact metric space

Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be recurrent iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, ...