Tagged Questions
3
votes
1answer
140 views
Convergence of fixed-point iteration for convex function
Let $f:[0,1]\to[0,1]$ be a smooth, convex (downward) function satisfying
$$
f(0)=f(1)=1,\quad \lim_{x\to 0}f'(x)=-\infty,\quad \lim_{x\to 1}f'(x)=+\infty.
$$
I am confident to be able to argue that ...
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
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 ...
2
votes
0answers
91 views
Fixed Point of a complex dynamical spiral system
Last semester I finished my first class on complex variables and of course we had to show that $i^i$ was real. That got me wondering about quantities like $i^{i^i}$ and similar power towers.
For my ...
0
votes
1answer
102 views
Implicit Function theorem and Bifurcation points
So let us say we have a function $\dot{x} = f(x,r)$ that has some critical point at $(x_0,r_0)$ such that $f(x_0,r_0)=0$. The question now is: when is this a bifurcation point? I understand that ...
1
vote
1answer
75 views
No fixed point problem for iterations
Let $f(z)$ be an entire function that is not a polynomial of degree 1 or degree 0 , where $z$ is a complex number.
Let $f(z,1) = f(z)$ and let $f(z,n) = f(f(z,n-1))$.
Let $g(f,1)$ be the amount of ...
3
votes
1answer
140 views
Fixed point: a consequence of symmetry?
I'm studying a dynamical system with $\mathbf{D}_{3}$ symmetry (the symmetry group of an equilateral triangle), which is given by:
$\begin{align*}
d\mathbf{x}_{0}/dt &= ...
