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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8
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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.
1
vote
1answer
169 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 ...
4
votes
3answers
243 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\} $
2
votes
2answers
172 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 ...
10
votes
1answer
312 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 ...
15
votes
3answers
626 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 ...
6
votes
2answers
188 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
3answers
229 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 ...
0
votes
0answers
193 views

Show that the set of all periodic points of the tent map are dense in $[0,1]$ [duplicate]

Let us consider the tent map: $f: [0,1] \rightarrow [0,1]$ where $f(x) = 2x$ if $0\leq x \leq \frac{1}{2}$ and $f(x) = 2(1-x)$ if $\frac{1}{2}\leq x \leq 1$. I am facing an issue with this problem: ...
6
votes
1answer
131 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
1answer
81 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 ...
3
votes
2answers
245 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 ...
7
votes
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
5answers
742 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 ...
1
vote
2answers
445 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 ...
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 ...
5
votes
2answers
186 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
65 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 ...
1
vote
2answers
125 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
381 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 $3n+3$ ...
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 ...
5
votes
0answers
325 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, ...
4
votes
0answers
122 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
157 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 ...
2
votes
2answers
118 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 ...
2
votes
1answer
73 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
1answer
791 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
1answer
218 views

Stability of nonlinear system with borderline linearization

I have the following nonlinear system: \begin{align} ...
2
votes
1answer
92 views

The system $x'=Ax$ is an attractor if and only if there is a positive quadratic form q such that $Dq(x)\cdot A(x)<0$ for all x

I need to show this result: Given the system of ODEs $x'=Ax$, the origin, $0$, is an attractor (equivalently, all the eigenvalues of the real matrix $A$ are negative) if and only if there exists a ...
2
votes
1answer
175 views

Can linear systems always be represented as differential or difference equations?

On my note, it was written that linear systems can always be represented as either differential equations or difference equations. I forgot the source of the quote. But I am not sure if it is correct. ...
1
vote
1answer
329 views

Nonlinear phase portrait and linearization

Consider the nonlinear system $x^{'} = y$, $y^{'}= -8 \sin x - 2y$ where $-2\pi$ < or = x < or = $2\pi$ Find the equilibrium points of the system. $(-2\pi,0)$$(-\pi,0)$$(0,0)$ ...
1
vote
5answers
189 views

Limit of a recursively defined bivariate function.

Let m and n be positive integers. Let $f(m,0)=m$ Let $f(m,n)= e \ln(f(m,n-1))$ $$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$$ Where $C$ is a constant. It seems ...
1
vote
2answers
279 views

Stability of discrete time nonlinear dynamical systems

In this problem we consider one dimensional discrete time dynamical system $x(k+1)=f(x(k))$ with a fixed point $u$, at which $|f'(u)|=1$. For each of the following systems, find out the stability ...
0
votes
1answer
124 views

Kepler, cartesian coordinates and ellipses

I am trying to see if I am on the right track with this. The problem: A kepler orbit (an ellipse) in Cartesian coordinates is: $$(1−\epsilon^2)x^2 + 2\alpha \epsilon x + y^2 = \alpha^2.$$ The task ...
0
votes
1answer
107 views

System, dynamic system and feedback system

Can a system be defined as a mapping from a set of mappings, called input signals, to another set of mappings, called output signals, where the two sets of mappings may or may not have the same ...
5
votes
2answers
107 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 ...
5
votes
2answers
290 views

The cosine function by iteration

Since $\cos (2x) = 2 \cos^2 (x) - 1$, I wonder about the iteration $(x,y) \mapsto (2x\mod 2\pi, 2y^2-1)$. Will it converge to the cosine graph? I tried it in mathematica and got quite a few points, ...
4
votes
0answers
45 views

Definition and some elementary properties of the “vector turn map”

This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
4
votes
1answer
212 views

Explicit form of Poincare's map for the spring-mass-damper

Problem: Write in explicit form Poincare's map for $\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$. Find the stationary points and examine their stability. An attempt at a solution: The ...
3
votes
1answer
65 views

Brouwer's fixed-point theorem and iterative convergence of a composition of circular functions

let $\psi:[0,1]\times \{0,1\} \rightarrow [0,1]$ be defined by: $$ \psi(x,\beta) = \beta \cos x + (1-\beta) \sin x $$ define $B_n$ as the set of $2^n$ binary strings $b=b_0b_1\dots b_{n-1}$ where ...
3
votes
1answer
165 views

Constructing a non-linear system with prerequisites about the nature of its critical points.

An exercise from the book I am reading is: "Construct a non-linear system that has four critical points:two saddle points, one stable focus, and one unstable focus." I have tried many systems. I ...
3
votes
1answer
91 views

Repeated Eigenvalues 2

Two problema from Differential Equations; Dynamical Systems, and an Introduction to Chaos (Morris W. Hirsch,Stephen Smale.Robert L. Devaney). Examples (pages 112-113): If $$A= \begin{pmatrix} ...
3
votes
1answer
258 views

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 ...
3
votes
1answer
86 views

inequality in a differential equation

Let $u:\mathbb{R}\to\mathbb{R}^3$ where $u(t)=(u_1(t),u_2(t), u_3(t))$ be a function that satisfies $$\frac{d}{dt}|u(t)|^2+|u|^2\le 1,\tag{1}$$where $|\cdot|$ is the Euclidean norm. According to ...
3
votes
1answer
549 views

How to obtain a possible state space representation of this 2nd order transfer function?

I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ ...
2
votes
1answer
92 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 ...
2
votes
2answers
87 views

How does the singularity of a system matrix affect the system's stability?

What can be said about system stability, given a singular system matrix below? \begin{align} A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ...
2
votes
1answer
185 views

Picard iteration (general)

This a general question about Picard iterations and is as follows. Let A be a $n\times n$ matrix. show that the Picard method for solving $X^{'}=AX$, $X(0)=X_{0}$ gives the solution $e^{tA}X_{0}$ I ...
2
votes
1answer
77 views

Surfaces without conjugate points

I'm trying to understand some aspects of the geodesics of surfaces without conjugate points and the following question came out: consider the hyperbolic space and two geodesics wich are asymptotic to ...
2
votes
3answers
185 views

How to solve simple systems of differential equations

Say we are given a system of differential equations $$ \left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix} $$ Where $A$ is a $2\times 2$ matrix. ...