# Tagged Questions

48 views

### What is the method for solving this class of problem?

$\frac{dx}{dt} = y$ $\frac{dy}{dt} = -\partial y - \mu x - x^2$ Find the fixed points and discuss stability. I'd at least like to know what I should be googling, any help is appreciated.
53 views

### Fixed point in a metric space with distance at most 1

The question is: Suppose that $X$ is a complete metric space such that the distance function is at most 1, and $f:X\rightarrow X$ is such that $d(f(x),f(y))\le d(x,y)−1/2(d(f(x),f(y)))^2$. Prove that ...
40 views

### Is there a method for finding the fixed point of logarithmic functions?

I am faced with this function (warning, I am not good at math) $x(t+1)=0.5 \ln x(t)+1$ initial condition = 1 . I know the fixed point is 1 because $0.5 \ln (1)+1=1$ but I wanted to know the ...
47 views

### Fixed points of multivariable calculus

I have discrete case. $z=1-x-y$; $x=a_1(x^{2}+2yz)+a_2(z^{2}+2xy)+a_3(y^{2}+2xz)$; $y=b_1(x^{2}+2yz)+b_2(z^{2}+2xy)+b_3(y^{2}+2xz)$; $z=c_1(x^{2}+2yz)+c_2(z^{2}+2xy)+c_3(y^{2}+2xz)$; where ...
25 views

### How to find fixed point for Hamilton equations

I am learning the dynamical system. I start with Taylor-Greene-Chirikov map as follow \begin{eqnarray*} && I_{n+1} = I_n + K\sin(\theta_n), \\ && \theta_{n+1} = \theta_n + I_{n+1} ...
23 views

### How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
106 views

### A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
175 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 ...
227 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 ...
255 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 ...
65 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), ...
165 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 ...
160 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 ...
152 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 ...
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 ...
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 &= ...