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 ...
7
votes
1answer
268 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.
2
votes
2answers
131 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
145 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 ...
6
votes
2answers
135 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 ...
15
votes
3answers
485 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 ...
0
votes
0answers
53 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
0answers
187 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 ...
2
votes
2answers
122 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 ...
0
votes
0answers
91 views
Phase portrait of $x-x^3$ [closed]
It seems irritating when I found that there is some $A > 1(1.324717957)$ s.t. for $1 < x < A, \, f(x) \in (-1,0)$, and $f(A) = -1$, and $f^n(x)\rightarrow 0$ as $n \rightarrow \infty$, and ...
1
vote
2answers
360 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
193 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
5answers
500 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
105 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
331 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 $3n+3$ ...
1
vote
5answers
146 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 ...
0
votes
2answers
501 views
What to do with Collatz proof? [closed]
I have discovered a proof solving the Collatz problem, and I have no idea what to do with it. Given the nature of the topic, all the experts I've found that are capable of reviewing the paper ...
2
votes
1answer
138 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
0answers
102 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, ...
1
vote
1answer
76 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
2answers
200 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
80 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
196 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, ...
3
votes
0answers
28 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 ...
3
votes
1answer
80 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 ...
2
votes
1answer
78 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
45 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
134 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.
...
2
votes
2answers
89 views
Not empty omega limit set
Dynamical system is generated by:
$x'=-x+f(x,y)$
$y'=-y+g(x,y)$
$f,g \in C^1$ and $f,g$ are bounded.
Prove that the omega limit set of p: $\omega(p) \neq \emptyset$ for all $p \in \mathbb{R}^2$.
...
2
votes
1answer
138 views
Symplectic submanifolds and first integrals
I was working with symplectic submanifolds when I posed the following question:
Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard ...
2
votes
1answer
151 views
Why this vector field $f$ belongs to $C^1({\bf R}^2\times {\bf R})$?
The following system is an example in a book of dynamical system(in the section about Hopf Bifurcation).
$$
\begin{align}
\dot{x}=\mu x- y-x\sqrt{x^2+y^2} \\
\dot{y}=x + \mu y-y\sqrt{x^2+y^2}
...
2
votes
2answers
196 views
A very simple discrete dynamical system with pebbles
Let us suppose we have slots $n$ slots $1, \ldots, n$ and $k$ pebbles, each of which is initially placed in some slot. Now the pebbles want to space themselves out as evenly as possible, and so they ...
2
votes
1answer
126 views
Minimizing the cost of a path in a dynamic system
So suppose I want a path from 0 to $c>0$ on the real line, and I am going to use the function $S(t)$ to get there in (discrete) time $T$. That is, my position at time 0 is 0, my position at time $T$ ...
1
vote
1answer
46 views
Explicit form of Poincare's map for the following system of Ode’s
Problem: Write in explicit form Poincare's map for
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their stability.
An attempt at a solution: Let ...
1
vote
3answers
69 views
conditions under which real-matrix exponential are equivalent
Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$?
Thanks!
1
vote
1answer
62 views
Use Euler's method with step size 10^-n to estimate x(1), where f(x) is the solution of the initial-value problem below. f(x)=-x x(0)=1
Use Euler's method with step size $10^{-n}$ for $n=1,2,3,4.$ to estimate $x(1)$, where $f(x)$ is the solution of the initial-value problem below.
$x'=f(x)=-x$
$x(0)=1$
EDIT / UPDATE:
x_n+1=x_n + ...
1
vote
1answer
284 views
How to obtain a possible state space representation of this 2nd order transfer function?
I have this 2nd order transfer function:
$$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$$
...
0
votes
1answer
50 views
How have they calculated this maximum point of the Verhulst Model
The simplest guess is using the Vehulst model: $r = r_0( 1 - N/K)$. Then the dynamic equation is $\frac{dN}{dt} = r_0N(1 - \frac{N}{K})$. From here, we can calculate the maximum point to be at ...
0
votes
0answers
39 views
Construction of dynamical systems from ODEs with initial values
From Wikipedia
Given an initial value problem $$
\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x}) $$ $$
\boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0 $$
The solution is an ...
0
votes
1answer
41 views
Show that a set is not invariant
From my textbook (Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields). Consider the following system
\begin{equation}
\left(
\begin{array}{c}
\dot{u} \\
\dot{v} \\
...
0
votes
1answer
77 views
Reducing degrees of freedom with first integrals
Given is a Hamiltonian function with $k$ first integrals. Suppose these $k$ first integrals are closed under the Poisson bracket, is it then possible to reduce the number of independent variables by ...
0
votes
1answer
163 views
How to obtain the state matrix of this trajectory?
Continuous-time LTI case.
I have a problem getting the state matrix of this trajectory.
One element of the state matrix is known.
$$ A = \begin{pmatrix} a & 4 \\c & d \end{pmatrix}
$$
I ...
