2
votes
0answers
75 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.
1
vote
2answers
46 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 ...
1
vote
1answer
53 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 ...
1
vote
0answers
26 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} ...
1
vote
0answers
24 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 ...
4
votes
2answers
109 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 ...
5
votes
2answers
186 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 ...
4
votes
3answers
229 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 ...
3
votes
1answer
282 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 ...
1
vote
1answer
66 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
172 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 ...
4
votes
1answer
173 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 ...
1
vote
1answer
164 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
94 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
151 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 &= ...