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
544 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.
0
votes
1answer
430 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 ...
4
votes
3answers
274 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\} $
3
votes
1answer
52 views

Elementary properties of gradient systems

Consider $x_0\in\mathbb{R}^n$ and a $C^{1,1}$ function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ (that is, a differentiable function whose gradient is Lipschitz function). Consider the system $$ ...
1
vote
1answer
191 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
172 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 ...
3
votes
2answers
237 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 ...
1
vote
1answer
65 views

Find the flow for the following dynamical system

I have the following dynamical system: $\dot{x_1}= -x_2 + (x_1(1-(x_1^2+x_2^2)^2))$ , $ \dot{x_2}= x_1 + (x_2(1-(x_1^2+x_2^2)^2))$, $\dot{x_3}= \epsilon x_3$ . I am required to work out the flow ...
0
votes
0answers
37 views

Poincare first return map, stability and bifurcations

Let $X= \mathbb R^3$ and consider the autonomous dynamical system $$\dot{x_1} = -x_2 + x_1 (1 - (x_1^2 +x_2^2)^2), \qquad{} \dot{x_2} = x_1 + x_2(1-(x_1^2 +x_2^2)^2), \qquad{} \dot{x_3}= \epsilon ...
0
votes
2answers
174 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 ...
33
votes
1answer
1k 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 ...
21
votes
2answers
421 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
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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
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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
345 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
205 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 ...
8
votes
1answer
512 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, ...
4
votes
1answer
7k 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 ...
4
votes
1answer
272 views

Does $\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(…\cos x))}}$ exist? [duplicate]

Possible Duplicate: Explaining $\cos^\infty$ Does the following limit exist? $$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$ If yes, find the limit. If no, please ...
10
votes
1answer
578 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 ...
3
votes
1answer
854 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. ...
3
votes
1answer
1k 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
2answers
314 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
5answers
408 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
0answers
109 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
262 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
674 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
1answer
70 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
508 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
86 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, ...
7
votes
1answer
356 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 ...
6
votes
1answer
198 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 ...
5
votes
2answers
248 views

“Constrained” numerical solutions of ODEs with conservation laws?

I 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 ...
3
votes
3answers
265 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
53 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
2answers
70 views

$2$-dim dynamical system IVP

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
1
vote
0answers
120 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
41 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) ...
0
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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
46 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? ...
40
votes
5answers
692 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 ...
21
votes
1answer
1k 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 ...
13
votes
4answers
487 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
637 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
2answers
208 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 ...
6
votes
1answer
143 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. ...
10
votes
1answer
186 views

Is there any relationship between the Riemann z function and strange attractors?

I have this question in mind since the first time I saw a graphical representation of the zeta function (like in the sample below). Just by looking to them I wondered if there is any relationship ...
9
votes
1answer
594 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
5answers
2k 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 ...
5
votes
2answers
322 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...