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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507 views

If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of ...
30
votes
2answers
734 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 ...
28
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1answer
3k 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
3answers
803 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 ...
22
votes
2answers
963 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
votes
2answers
4k 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). ...
21
votes
0answers
558 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
18
votes
4answers
637 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 ...
17
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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 $\mathbb{C}$?
17
votes
1answer
782 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 ...
16
votes
4answers
619 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 ...
15
votes
3answers
868 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
votes
1answer
557 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, ...
15
votes
2answers
318 views

Do any of these sequences have infinitely-many distinct iterates under run-length substitution?

Let $$S = \{x \in \{1,2\}^\mathbb{N}: \ \text{every run in }x\text{ has finite length}\}$$ and define $$T: S\to \mathbb{N}^\mathbb{N} $$ such that for any $x\in S$, ${T}x$ is the sequence of ...
14
votes
1answer
540 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
12
votes
3answers
3k 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 ...
12
votes
1answer
245 views

Equality by iteratively applying $(a,b)\rightarrow [(a+1,2b)\text{ or }(2a,b+1)]$?

I play a game starting with $2$ positive integers $a$ and $b$. At each step of the game I can double one of the integers and add $1$ to the other integer. Is there always a procedure for any ...
12
votes
2answers
4k 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 ...
12
votes
4answers
436 views

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
12
votes
2answers
270 views

How did Ulam and Neumann find this solution?

In the book "Chaos, Fractals and Noise - Stochastic Aspects of Dynamics" from Lasota and Mackey the operator $P: L^1[0,1] \to L^1[0,1]$ $$ (Pf)(x) = \frac{1}{4\sqrt{1-x}} \left[ ...
12
votes
0answers
209 views

Notions of stability for differential equations

Consider a system of differential equations $$\dot{x} = f(x,u)$$ $$y = h(x,u)$$ where $x(t), u(t)$ are vectors in some $\mathbb{R}^n$. We define the infinity norm of a function in more-or-less in the ...
11
votes
1answer
163 views

limit cycle of an ODE system

Consider a planar ODE system $z'=f(z)$ with $z=(x,y)$, $$ f(x,y)=(xy^2-x-y,y^3+x-y). $$ Using polar coordinates, one can get $$ r'=r(r^2\sin^2\theta-1),\quad \theta'=1. $$ With Mathematica one can ...
11
votes
2answers
345 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 ...
11
votes
2answers
152 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 ...
10
votes
2answers
675 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 ...
10
votes
5answers
331 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 ...
10
votes
1answer
254 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 ...
9
votes
1answer
384 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.
9
votes
1answer
295 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 ...
9
votes
1answer
131 views

Bifurcations in the Duffing oscillator

I'm trying to describe all the bifurcations in the two parameter Duffing oscillator: $$\ddot{x} + ax + bx^3 = 0$$ In phase space with $y = \dot{x}$ I've found the origin to be a centre for $a>0$ ...
9
votes
3answers
444 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
3answers
572 views

ODE with singular coefficients

I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - q(x) y(x) = \lambda y(x).$ Then I got a more sophisticated differential equation ( second one) and is given by $$-(1-x^2)y''(x) +x ...
9
votes
1answer
83 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
458 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
1answer
474 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 ...
9
votes
1answer
568 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 Lyapunov) I don't know ...
9
votes
0answers
234 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, ...
8
votes
5answers
1k 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 ...
8
votes
3answers
255 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
8
votes
2answers
168 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
4answers
347 views

Concentric Equilateral Triangles

I'm currently researching a particular dynamical system that is very geometric in nature. As part of this, I need to prove the following results (the second obviously implies the first). They are ...
8
votes
1answer
194 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
239 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
180 views

Fixed, attracting points are Fatou points

Let $f$ be a holomorphic function on an open, connected set $\Omega\subset \mathbb{C}$ with $z_0\in \Omega$ a fixed point, and $\{f^n\}_{n\in \mathbb{N}}$ the sequence of iterates. I want to prove ...
8
votes
1answer
141 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
5k 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
3answers
390 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 ...
7
votes
6answers
211 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
2answers
368 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem. Let $f(x):\Omega\longrightarrow \mathbb R^n$ be a $C^2$ function where $\Omega$ is an open subset of ...
7
votes
2answers
84 views

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?