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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9
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1answer
370 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.
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3answers
263 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
140 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
179 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
149 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
205 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
50 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 ...
15
votes
3answers
856 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 ...
9
votes
1answer
440 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 ...
7
votes
2answers
259 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 ...
4
votes
0answers
89 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} ...
4
votes
3answers
255 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
419 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 ...
2
votes
2answers
270 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
49 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, ...
6
votes
1answer
157 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 ...
3
votes
3answers
167 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 ...
3
votes
1answer
141 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 ...
1
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0answers
105 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 ...
1
vote
1answer
39 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) ...
37
votes
5answers
501 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 ...
16
votes
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 $\mathbb{C}$?
12
votes
4answers
427 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} ...
9
votes
1answer
554 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 ...
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 ...
7
votes
2answers
183 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 ...
1
vote
2answers
509 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 ...
14
votes
1answer
526 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 ...
9
votes
1answer
467 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 ...
7
votes
2answers
364 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 ...
6
votes
2answers
288 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 \| ...
1
vote
2answers
86 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 ...
6
votes
2answers
266 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 ...
3
votes
1answer
52 views

How to prove that $F(x) = \lim_{n \to \infty} F^{n}( f ' (0) \cdot F^{-n}(x)) $?

Let $F(x)$ be a real-analytic function near $0$ ,with $0$ as one of its fixpoints and $f ' (0) > 1$. $$F(x) = F \circ F \circ F^{-1} = \lim_{n \to \infty} F^{n} \circ F \circ F^{-n} = \lim_{n \to ...
3
votes
1answer
78 views

Compact space, continuous dynamical system, stationary point

I'm having trouble proving that if $X$ is a compact metric space and every continuous function $f : X \rightarrow X$ has a fixed point, then every continuous dynamical system $ \varphi $ on $X$ has a ...
3
votes
1answer
3k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
2
votes
2answers
140 views

Stability analysis, or, Can we prove this limit to be zero?

Let's think about this ODE $$ \dot{y}(t) = \gamma \left(g(t) - y(t)\right),\quad \gamma > 0, $$ where $g(t)$ is a Lipschitz continuous function. It can be seen that the value of $y(\cdot)$ goes ...
1
vote
1answer
40 views

Show there is only one trajectory passing through each point

I have to show the following: Let $\varphi$ be a flow on the manifold M and suppose that that the orbits {$\varphi_t (x_0)$} & {$\varphi_t (x_1)$} intersect. Prove that the orbits coincide.
1
vote
1answer
443 views

What is known about Collatz like 3n + k?

I wonder what is known about variations of Collatz where $3n+1$ is replaced by $3n + 2k + 1$ where k is a fixed positive integer. In the OP ' about Collatz $3n+3$ ' it was confirmed that $3n+3$ ...
7
votes
3answers
381 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
2answers
77 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?
6
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0answers
398 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
6
votes
3answers
234 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 ...
5
votes
1answer
3k views

nondimensionalization of predator-prey model

Given a set of differential equations for a predator-prey model (Lotka-Volterra): \begin{align*} \frac{dH}{dt} &= bH - sHP \\ \frac{dP}{dt} &= -dP + esHP \\ \end{align*} Where $P$ is the ...
4
votes
0answers
207 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
3
votes
1answer
110 views

Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
3
votes
1answer
178 views

Small question about ODE

i have this question : Given three parameters $L,a$ et $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ > t\geq0$$ 1) Show that the ...
3
votes
1answer
630 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 ...
2
votes
1answer
85 views

Bifurcation and homoclinic orbits.

In two dimensions, if we have a dynamical system: $$\dot{x}=f_k(x,y)$$ $$\dot{y}=g_k(x,y)$$ with $f$ and $g$ smooth functions and $k$ is a paremeter. If $k=k^*$ is a bifurcation at which two ...
2
votes
2answers
147 views

M-set interior point probability on the real axis

For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are ...