# Questions tagged [fixed-points]

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. A set of fixed points is sometimes called a fixed set.

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### Sketch the phase portrait of the system in the vicinity of the fixed point using isoclines.

Given is a system of differential equations $$\dot{x} = x^2,$$ $$\dot{y} = y(2x - y).$$ Sketch the phase portrait of the system in the vicinity of the fixed point using isoclines. Attempt: I start ...
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### (Updated) Seeking proof to Dynamical Systems past exam question

I have searched and found one proof for the below which I have not been able to completely process and AI has been unable to provide sufficient reasoning. I hope you may be able to assist. We have ...
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### How to find a function with several bifurcations of specified type

Suppose I want to construct a function $h(x,r)$ such that it exhibits a bifurcation of type $a$ at the point $(x_{1},r_{1})$ and bifurcation of type $b$ at the point $(x_{2},r_{2})$. One alogirhtm to ...
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### How many permutations of set are there with 4 fixed points?

Given the set of numbers $A = \left \{ 1, 2, 3, 4, 5, 6, 7, 8 \right \}$. How many permutations leave exactly four numbers fixed? I've considered this as a solution... but then I stumbled upon a ...
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### How do I find the fixed points of one isometry, from the fixed points of another isometry?

I have an isometry group $G$ which acts on units of a ring $R$ (not neessarily the full isometry group). I know the set $P_f$ of fixed points in $R$ of one member $f$ of $G$. Can I use the group ...
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### $(M,d)$ complete metric space and $f : M \longrightarrow M$ such that $f^p$ is a contraction. Then, $lim f^n(x) = a$, for any $x \in M$.

I try to solve this problem: "Show that if $(M,d)$ is a complete metric space and $f: M \longrightarrow M$ is a map such that exists $p \in \mathbb{N}$ for which $f^p$ is a contraction, then, $f$ ...
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Let $f: [0,1] \to [0, 1]$ be a continuous function. Let $Fix(f^2)=\{x \in [0,1]: f^2(x)=x\}$ (where $f^2=f \circ f$),$\hspace{0.1cm}$ $Fix(f)=\{x \in [0,1]: f(x)=x\}$ and $Per_2(f)=\{x \in [0,1]: mín\... • 293 2 votes 1 answer 150 views ### Warsaw circle has the fixed point property I'm looking for any hint on proving that the Warsaw circle has the fixed point property. By this I mean that for every continuous function$f$from the Warsaw circle onto itself there exists a point$...
Assume I have $f : A \to B$, $g_a : A \to A$, $g_b : B \to B$ and $R$ an equivalence relation on $B$. If I know $f(a)$ is $R$-related to $g_b (f (g_a (a)))$ for all $a\in A$, is it the case that there ...