2
votes
6answers
74 views

conjecture: a supremum property of the cosine fixed point?

in a previous question a composition of circular functions was defined for each binary string of finite length. this question will use the same terminology. if the existence of a fixed point is ...
3
votes
1answer
77 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
103 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 ...
0
votes
1answer
47 views

Rendezvous problem - Dynamical Systems

I'm learning about graph-based distributed control and there's a problem called "the rendezvous problem" that uses the Laplacian matrix as the state matrix of the system. I have a graph with 4 nodes ...
1
vote
1answer
40 views

Limiting value of iteration $x(k+1) = A x(k) + B u(k)$ for summable $u(k)$

A matrix $A$ is known to converge such that $\lim_{k\rightarrow \infty} A^k = \bar{A} \neq 0$. We have an iteration defined as $$x(k+1) = A x(k) + B u(k), \ \ k\in \mathbb{Z}_+.$$ $\{u(k), ...
1
vote
1answer
68 views

Orbit of $\frac{1}{2}$ of the dynamical system defined by $f:[0,1] \to [0,\frac{3}{4}]$ where $f(x)=3x(1-x)$ converges.

I view the orbit of $\frac{1}{2}$ of the dynamical system defined by $f:[0,1] \to [0,\frac{3}{4}]$ where $f(x)=3x(1-x)$ as the sequence $(x_{n})$. So $$x_n = \frac{1}{2}, f\bigg(\frac{1}{2}\bigg), ...
2
votes
1answer
249 views

Limsup of continuous functions between metric spaces

Let me start with a simple example: Let $f_n:[0,1]\to[-1,1],x\mapsto \sin 2\pi nx$. For each $x\in[0,1]$, consider the sequence $\lbrace f_n(x):n\ge1\rbrace$ and denote by $F(x)$ the set of points of ...
4
votes
0answers
279 views

How to prove stability of this dynamic system?

I'm trying to prove stability of the following dynamic system but I think my Mathematics knowledge is not deep enough. My dynamic system consists of a state vector $x \in \mathbb{R}^n$. The system ...
6
votes
1answer
115 views

converging to cosine by iteration

In what sense (if at all) does the iteration $x \mapsto 2x^2 - 1$ converge to $\cos 2^n x$ in the unit interval [-1,1]? One might try to plot in Mathematica: ...
2
votes
1answer
69 views

solving coupled discrete systems

Suppose you have a discrete system, whose evolution is governed by the following equations: $\mathbf{x}[k+1] = f_1(\mathbf{F}[k], \mathbf{x}[k])$ $\mathbf{F}[k+1] = f_2(\mathbf{F}[k], ...
0
votes
1answer
160 views

convergence rate of matrix product

Suppose you have a linear system like this: $$\mathbf{x}[k+1] = \mathbf{D} \mathbf{x}[k]$$ where matrix $\mathbf{D}$ is diagonal. Assume its diagonal entries are real, greater than zero and less than ...
0
votes
1answer
197 views

does the following dynamic system converge to a steady state?

This is an economics problem, but I'm pretty sure this kind of thing comes up elsewhere. I've used dynamic programming to find the optimal path of a system (law of motion), which is: ...