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.
639 questions
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Iterating the map $f(n)=1+D(\sigma(n)-1)$
For integers $n$, the arithmetic derivative $D(n)$ is defined as follows:
$D(p) = 1$, for any prime $p$.
$D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule).
The Leibniz rule implies ...
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Unclear step in van Maaren's theorem proof by Schechter
I am trying to disentangle the proof of Brouwer's fixed point theorem via van Maaren's geometry-free Sperner lemma in Eric Schechter's Handbook of Analysis and its Foundations (sections 3.28-3.37). ...
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Pentation from three fixed points
When b is a real number non inclusively between 1 and $e^{e^{-1}}$, $b^x$ has two real fixed points.
If b is increased to $e^{e^{-1}}$, the two real fixed points combine into one real fixed point. If ...
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Can even periodic points of a map undergo tangent bifurcation?
My intuition for the answer is NO, here is my thought:
Let $T(x,\lambda)$ be the map depending on one parameter $\lambda$, assume at $(x_0,\lambda_0)$ a tangent bifurcation occurs for the $T^2$ map, ...
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A function $f$ such that repeated application gives you identity? i.e. $f^n(x) = x$ [duplicate]
Given a fixed $n$, I wanted a continuous function $f$ (preferably whose domain is $\mathbb{R}$) such that, applying $f^n$ (i.e. applying it $n$ times) to $x$ gives you back $x$ for any $x$ in the ...
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Types of bifurcations in $\dot{x} = \mu + 2x^2 - x^4$
Consider the 1-dimensional autonomous ODE
\begin{align}
\dot{x} = \mu + 2x^2 - x^4
\end{align}
where $\mu \in \mathbb{R}$ is a parameter. I have found the fixed points, their stability, and plotted ...
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Subgroups as fixed-point set of some subgroup of the automorphism group.
Let $G$ be a group. Amongst the subgroups of $G$ defined as the fixed-point set of some subgroup $\Xi(G)$ of $\operatorname{Aut}(G)$:
$$X(G):=\{g\in G\mid \forall \varphi \in\Xi(G),\varphi(g)=g\}$$
...
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determine the sets $A\subset Q$ for which there exists a function $f:Q\to Q$ with the property that $Fix(f)=A$ and $f(f(x))=x$ for any $x\in Q$
The problem
determine the sets $A\subset \mathbb Q$ for which there exists a function $f:\mathbb Q\to \mathbb Q$ with the property that $Fix(f)=A$ and $f(f(x))=x$ for any $x\in \mathbb Q$ (We have ...
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System of non-linear Equations with only 3 saddle points
I am puzzled by a question in my Non-Linear Dynamics lecture. Is there a System of equations of the form $a\in \mathbb{N}, a > 1 $
$$\dot x = x[(x-1)(x+1)]-y$$
$$\dot y = y(1-y^2-ax^2)$$
with only ...
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Proving Uniqueness of Nonlinear Equation Solution
Consider the following system of two nonlinear equations:
$h'_{x}( x) -C y -C_{1}=0$
$h'_{y}( y) -C x -C_{2}=0$
With $x,y\in[0,1]$
The $h$ functions both satisfy $h(0)=0$, $\lim _{t\rightarrow 1} h( t)...
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Proving that a function is not a Rakotch contraction
I want to prove that the function $f:[-1,1]\rightarrow [-1,1]$ given by
\begin{equation*}
f(x)=\frac{1}{2}x^2
\end{equation*}
is not a Rakotch contraction. To give the definition of a Rakotch ...
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Does the unique fixed point of the cosine function have a closed-form expression? [duplicate]
The real function $cos$ has a unique fixed point. But I have never seen a closed-form expression for that fixed point. I now suspect that there isn't any closed-form expression for that real number. ...
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Proving the uniqueness of a fixed point on $(0,1]^N$
I'll preface this by saying I am not very experienced with formal proofs, so that's mainly why I'm asking for help here. I found several other threads on this topic here on Math SE, but most of them ...
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What is the expected number of fixed points between an integer sequence and its random permutation, both mod k?
Let $s_n$ be a random permutation of the first $n$ nonzero positive integers. By a well-known combinatorial result of Montmort, the expected number of fixed points is 1: $E(fp(s_n)) = 1$. But suppose ...
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The iterates of a picard operator are locally uniformly convergent
Let $(X,d)$ be a metric spaces and $T$ be a self mapping of $X$ with a unique fixed point $z$ such that for any $x\in X$ the sequence $T^n(x)$ converges to $z$. Then, is at always the case that there ...
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