Tagged Questions
0
votes
1answer
56 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
$$x_n = \frac{1}{2}, f\bigg(\frac{1}{2}\bigg), ...
2
votes
1answer
187 views
Limsup of continuous functions between metric spaces
Let me start with a simple example:
Let $f_n:[0,1]\to[-1,1],x\mapsto \sin 2\pi nx$. For each $x\in[0,1]$, consider the sequence $\lbrace f_n(x):n\ge1\rbrace$ and denote by $F(x)$ the set of points of ...
4
votes
0answers
190 views
How to prove stability of this dynamic system?
I'm trying to prove stability of the following dynamic system but I think my Mathematics knowledge is not deep enough.
My dynamic system consists of a state vector $x \in \mathbb{R}^n$. The system ...
6
votes
1answer
97 views
converging to cosine by iteration
In what sense (if at all) does the iteration $x \mapsto 2x^2 - 1$ converge to $\cos 2^n x$ in the unit interval [-1,1]?
One might try to plot in Mathematica:
...
2
votes
1answer
54 views
solving coupled discrete systems
Suppose you have a discrete system, whose evolution is governed by the following equations:
$\mathbf{x}[k+1] = f_1(\mathbf{F}[k], \mathbf{x}[k])$
$\mathbf{F}[k+1] = f_2(\mathbf{F}[k], ...
0
votes
1answer
143 views
convergence rate of matrix product
Suppose you have a linear system like this:
$$\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
131 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:
...
