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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10
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1answer
521 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.
4
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3answers
272 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
0
votes
1answer
299 views

Alpha and Omega limit sets (dynamical systems)

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
1
vote
1answer
189 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
3
votes
3answers
169 views

How do we solve $c_1^d+\ldots+c_n^d=1$ for $d$?

The question is motivated by the definition of self-similarity dimension for self-similar sets: Let $M \subset \mathbb R^d$ be self-similar. That is, there are $T_1, \ldots, T_m \subsetneqq M$ and ...
2
votes
2answers
226 views

Periodic orbits

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$ My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with ...
0
votes
2answers
151 views

System of equations, limit points

This is a worked out example in my book, but I am having a little trouble understanding it: Consider the system of equations: $$x'=y+x(1-x^2-y^2)$$ $$y'=-x+y(1-x^2-y^2)$$ The orbits and limit sets ...
21
votes
2answers
357 views

Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ ...
18
votes
11answers
3k 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
3answers
1k 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 ...
5
votes
2answers
333 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Let $x(t)$ be a solution of the IVP $$ \dot x=A(t)x+h(t), $$ where $A(t), h(t)$ continuous on $1\le t<\infty$. Further assume that $$ \int_1^\infty \| ...
4
votes
1answer
166 views

Formal proof of Lyapunov stability

I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system $$\dot{x}=-4y+x^2\\\dot{y}=4x+y^2$$ The streamplot ...
10
votes
1answer
544 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 ...
8
votes
1answer
502 views

How to determine $\Omega(T)$?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and let $T\colon X\to X$ describe the following dynamics: 1 becomes a 2, 2 becomes a 0 and 0 becomes a 1 if at least one of its two neighbours is 1, ...
3
votes
1answer
655 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
6
votes
5answers
385 views

Is Collatz' conjecture the only stable solution of its type?

The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...
4
votes
3answers
258 views

Iterates of $f_b(x) = x - \log_b(x) $ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
4
votes
2answers
584 views

How can a Markov chain be written as a measure-preserving dynamic system

From http://masi.cscs.lsa.umich.edu/~crshalizi/notabene/ergodic-theory.html irreducible Markov chains with finite state spaces are ergodic processes, since they have a unique invariant ...
3
votes
1answer
968 views

Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle.

I think the title of the question says it all. I unfortunately did not seem to conclude anything. Some ideas I had: It is easy to show that (given $T$ is the rotation) $\{T^n(\theta)\}$ is a set of ...
7
votes
1answer
299 views

convergence of the iterated cosine

it can be demonstrated by elementary means that the curves $y=\cos x$ and $y=x$ meet exactly once, at a value $x=\alpha$ satisfying: $$\cos \alpha = \alpha$$ it is also evident (empirically) that ...
7
votes
2answers
304 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
votes
1answer
189 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 ...
4
votes
0answers
104 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
3
votes
3answers
223 views

Why is the Koch curve homeomorphic to $[0,1]$?

Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources ...
2
votes
1answer
65 views

Stability for higher dimensional dynamical systems

I remember learning that in order for a steady state to be locally stable in a system of two equations, it is sufficient for the Jacobian evaluated at a steady state to have: $$Tr(J)<0$$ ...
2
votes
2answers
447 views

Expressing an oscillator as a series of ODEs

Consider an oscillator satisfying the initial value problem $u''+w^2u=0$, where $u(0)=u_0$, $u'(0)=v_0$. Let $x_1 = u$, $x_2=u'$, and transform the equations given into the form $x' = Ax, x(0)$. Then ...
1
vote
1answer
76 views

Fixed Points and Graphical Analysis

For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$. I have no problem finding the fixed points, ...
1
vote
1answer
40 views

Finding normal coordinates of a system

We have coupled oscillators with equations of motion: $$\ddot{x} = -10x+18y$$ $$\ddot{y}=-3x+5y$$ At $t= 0$ we have $x=a$ and $\dot{x}=\dot{y}=y=0$. I found the solution to be $$\begin{pmatrix} x(t) ...
5
votes
2answers
207 views

“Constrained” numerical solutions of ODEs with conservation laws?

Hi know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
2
votes
1answer
52 views

Repairing solutions in ODE

Recently I encounter something interesting that I hope to hear from your opinions: Suppose we are given a ODE $\frac{dy}{dx}=y$, with no initial condition. Naively, we divide both sides by $y$ and ...
1
vote
0answers
115 views

(Still open) Convergence of a certain matrix product representing a class of piecewise linear dynamical systems.

Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$. $x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline ...
0
votes
0answers
45 views

Calculating Equilibria for a 3 ODE system with 11 unknown parameters i.e. Looking For the intersection points of 3 surfaces.

I have the following system of 3 ODEs of 3 variables $(\ell,M,h)$ and 11 parameters $(\sigma_{\ell},\mu_{\ell},d_{\ell},\sigma_M,\alpha_1,\beta,\alpha_2,\nu_M,\sigma_h,\nu_h,d_h)$: ...
0
votes
1answer
42 views

Recurrent points and rotation number

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that? ...
28
votes
1answer
965 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 ...
38
votes
5answers
640 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 ...
13
votes
4answers
480 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} ...
11
votes
1answer
631 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
1answer
557 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
2answers
204 views

eventually constant maps

Let $f:[0,1]\to [0,1]$ be a continuous function with a unique fixed point $x_{0}$ Assume that $\forall x\in [0,1], \exists n\in \mathbb{N}$ such that $f^{n}(x)=x_{0}$. Does this implies ...
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 ...
6
votes
1answer
136 views

Properties of the space of $ T $-invariant probability measures over a compact topological space.

Let $T: X \to X$ be a continuos map defined on the compact space (Maybe Haussdorff) $X$. Denote by $M_T$ the space of $T$-invariant probability measures defined over the Sigma Algebra of Borel sets. ...
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vote
2answers
575 views

About Collatz 3n+3?

While trying to prove the Collatz conjecture I came across the following Lemma : Lemma $1$ : All variables are positive integers. Define $Collatz(n)$ as the result of the (repeated) collatz ...
7
votes
2answers
401 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 ...
4
votes
1answer
133 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
2
votes
1answer
74 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
1
vote
2answers
168 views

Suggestion for a Lyapunov function

Consider the differential system $$ x'=x+y $$ $$y'=x-y+xy$$ What would be a Lyapunov function for this system at $(0,0)$? I have considered functions $V(x,y)=ax^{2n}+by^{2m}$ but none of ...
14
votes
1answer
576 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 ...
7
votes
2answers
288 views

Could this odd insight help explain part of the difficulty in proving the Collatz Conjecture?

Background: Here's a crash course on the Collatz Conjecture. Basically, you take a number and if it is even you divide it by two. If a number is odd, you multiply it by three and then add one. You ...
6
votes
2answers
308 views

How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
4
votes
1answer
100 views

A rational orbit that's provably dense in the reals?

Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational ...