# What is Fixed points: Definition and 49 Discussions

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 if f(c) = c. This means f(f(...f(c)...)) = f n(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.
For example, if f is defined on the real numbers by

f
(
x
)
=

x

2

3
x
+
4
,

{\displaystyle f(x)=x^{2}-3x+4,}
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, f(x) = x + 1, has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point x means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.
Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point.In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.

View More On Wikipedia.org
1. ### Calculate tilt angle of a bar lifted via two ropes on fixed points

Summary: How to calculate the equilibrium angle of a bar that is lifted on its two ends with ropes attached to fixed lifting points? Hello and good day all, First of all I would like to apologize in advance for my english, I am not a native speaker so some grammar errors may be left. My...
2. ### Understanding Fixed Points in Hamiltonian Systems

Hi, I was attempting a question about Hamiltonian systems from dynamic systems and wanted to ask a question that arose from it. Homework Question: Given the system below: \dot x_1 = x_2 \dot x_2 = x_1 - x_1 ^4 (a) Prove that the system is a Hamiltonian function and find the potential...
3. ### Equilibrium points doubt (ODE system solution)

Hi. I'm not sure about something related to the equilibrium points (or fixed points) of a non linear ode system solution. As far as I know, to check if an equilibrium point exists, I need to put the function of my ode system equal to zero. Then once the point is found, I can use it to evaluate...
4. ### A Path between fixed points in a logistic map

Hello, I have a question about period doubling and fixed points in the logistic map. Let's say I have a basic logistic map, ##f(x) = 4\lambda x(1-x)##. Let me then compare 1,2 and 4 iterations of this map on fixed points. I assume that ##\lambda## is large enough such that two period doublings...
5. M

### Distance between fixed points of contracting maps

Homework Statement Let ##V## be a Banach space. Let ##f:V\to V## and ##g:V\to V## be two ##q##-contracting maps, ##q\in(0,1)##. Assume they are uniformly close to each other. Show the distance between fixed points of ##f,g## is at most ##\epsilon/(1-q)##. Homework Equations Definitions...
6. ### A Fine structure and fixed points

I apologize for creating a new thread which has significant overlap with two other ongoing threads ("Quantization isn't fundamental", "Atiyah's arithmetic physics"). But both those threads discuss theories or paradigms of extreme breadth, whereas here I want to focus on a very specific bundle of...
7. ### Classify the fixed points of this dynamical system

Homework Statement $$\dot{x_1}=x_2-x_2^3,~~~~~~\dot{x_2}=-x_1-3x_2^2+x_1^2x_2+x_2$$ I need help in determining the type and stability of the fixed points in this system. Homework Equations The Jordan Normal Form[/B] Let A be a 2x2 matrix, then there exists a real and non singular matrix M...
8. ### Existence of point(s) within k distance of other fixed points

How do I derive an expression or algorithm that determines the existence of a point or set of points within k distance of an N number of other fixed, given points? In application, I expect to only need to determine that this region exists for three to five points. This is part of a greater...
9. ### 2D Phase portrait - Black hole?

Homework Statement Trajectories around a black hole can be described by ## \frac{d^2u}{d\theta^2} + u = \alpha \epsilon u^2 ##, where ##u = \frac{1}{r}## and ##\theta## is azimuthal angle. (a) By using ##v = \frac{du}{d\theta}##, reduce system to 2D and find fixed points and their stability...
10. ### One-dimesional system non-existence fixed points

Homework Statement First things first, this is not a HW but a coursework question. I try to understand a concept. Assume we have a one-dimensional dynamic system with: x'=f(x)=rx-x^3 Homework Equations Fixed points are simply calculated by setting f(x)=0. The Attempt at a Solution If I...
11. ### Pendulum - Stability and fixed points

Homework Statement (a): Show the lagrangian derivative in phase space (b)i: Show how the phase space evolves over time and how they converge (b)ii: Find the fixed points and stability and sketch phase diagram (c)i: Find fixed points and stability (c)ii: Show stable limit cycles exist for T>ga...
12. ### Can there be non-trivial IR fixed points in asymptotically free theories?

I understand that asymptotically free theories must be based on UV fixed points rather than IR ones, because the RG flow goes into rather than out of an IR fixed point, so an asymptotically free theory based on an IR fixed point is trivial at low energies. But at higher energies the coupling...
13. ### MHB Runge-Kutte: stability of fixed points

Show that the explicit Runge-Kutta scheme $$\frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})]$$ where $k_{1} = f(t,y_{n})$applied to the equation $y'= y(1-y)$ has two spurious fixed points if $h>2$.Briefy describe how you would investigate...
14. ### MHB Difference between distance from two fixed points is a positive constant

The set of all points $$P(x,y)$$ in a plane, such that the difference of their distance from two fixed points is a positive constant is called? ellipse hyperbola parabola circle How do I work this out? Are the two fixed points supposed to be the foci? Wouldn't this also depend on the how one...
15. ### Fixed points of map and norm

Homework Statement The Attempt at a Solution set x(t)=1+∫2cos(s(f^2(s)))ds(from 0 to t) then check x(0)=1+∫2cos(s(f^2(s)))ds(from 0 to 0)=1 then the initial condition hold, by FTC, we have dx(t)/dt=2cos(tx^(t)), then solutions can be found as fixed points of the map but for...
16. ### Fixed Points of two differential equations

Homework Statement Determine all fixed points of: dx/dt = x(β-x-ay) dy/dt = y(-1+ax-y) β and a are parameters. I get what to do when there is just one differential equation, but not two.
17. ### MHB Finding Fixed Points for F, B, A

Is there a clean may to get the fixed points for \begin{alignat*}{9} F - 2B' - cB - \frac{3}{4}AB^2 - \frac{3}{4}A^3 & = & 0 & \quad & \Rightarrow & \quad & B' & = & \frac{1}{2}F - \frac{c}{2}B - \frac{3}{8}AB^2 - \frac{3}{8}A^3\\ 2A' + cA - \frac{3}{4}A^2B - \frac{3}{4}B^3 & = & 0 & \quad &...
18. ### Regarding fixed points in finite groups of isometries

There is a theorem for finite groups of isometries in a plane which says that there is a point in the plane fixed by every element in the group (theorem 6.4.7 in Algebra - M Artin). While the proof itself is fairly simple to understand, there is an unstated belief that this is the only point...
19. ### Fixed Points of analytic functions

Hi, I am in honors track Complex Analysis, and I think I've reached my limit. We got this proof, and I don't know where to start. "We saw in class that a mobius transformation can have at most one fixed point (or else is the identity map), extend this idea to all analytic functions mapping...
20. ### Finding Fixed Points of a Mobius transformation

Homework Statement Find all the fixed points to the following Mobius transformation. Homework Equations m(z) = (2z + 5)/(3z - 1) The Attempt at a Solution Aren't all fixed points going to map to themselves? So shouldn't it be solving for m(z) = z and coming up with roots of a quadratic...
21. ### Fixed points of conjugate functions

Homework Statement suppose f and g are conjugate show that if p is an attractive fixed point of f(x), then h(p) is an attractive fixed point of g(x). Homework Equations f and g being conjugate means there exist continuous bijections h and h^-1 so that h(f(x)) = g(h(x)) a point p...
22. ### Finding fixed points non-algebraically

Homework Statement let f = \muex let 0 < \mu < 1/e Show that f has two fixed points q and p with q < p Homework Equations a fixed point p is a point such that f(p) = p The Attempt at a Solution solving f(x) = x: f(x) - x = 0 \muex - x = 0 Now I want to take logarithms...
23. ### 'Slackline' equipment load formula (weighted line between 2 fixed points)

Hello, I have a real world sports question I need help with: I am looking for an accurate formula to calculate the load of Slackline equipment (will explain shortly), based on: - The length of the line - The sag of the line - The weight on the line Slacklining is like tight rope...
24. ### Find Fixed Points of e^z Complex Equation

Homework Statement How would one go about finding the fixed points of e^z, where z is complex (i.e. all z s.t. e^z = z)? Homework Equations Nothing. The Attempt at a Solution I've considered all the relevant formulas (de Moivre's forumla, power series, z = re^i*theta, ...). For some reason...
25. ### Fixed points of Van Der Pol equation

Homework Statement "Discuss the fixed points of the Van Der Pol equation depending on the perturbation parameter µ when µ >= 0" Homework Equations x'' - \mu(1-x^2)x' + x = 0 is the Van Der Pol Equation. The Attempt at a Solution Well, this question is really easy for most values of µ...
26. ### Exploring the Tent-Map: Fixed Points & Stability

Homework Statement The "tent-map" is given by: xn+1 = g(xn) where g(x) = 2x if 0 <= x<= 1/2 and g(x) = 2-2x if 1/2 < x <= 1 a) Find the fixed points and their stability. Draw a cobweb plot of the tent map to demonstrate that your stability calculations are correct. b) Find a period-2 orbit...
27. ### Classification of fixed points of N-dimensional linear dynamical system?

I'm familiar with the classification of fixed points of linear dynamical systems in two dimensions; it's readily available in many a book, as well as good ol' Wiki (http://en.wikipedia.org/wiki/Linear_dynamical_system#Classification_in_two_dimensions). However, what happens with higher-order...
28. ### Fixed points on compact spaces

Homework Statement Let X be a compact metric space. if f:X-->X is continuous and d(f(x),f(y))<d(x,y) for all x,y in X, prove f has a fixed point. Homework Equations The Attempt at a Solution Assume f does not have a fixed point. By I problem I proved before if f is continuous with...
29. ### Find any fixed points for the following mapping:

Find any fixed points for following mapping f(z) = z2 - z + 1 Map onto the same point gives: z = z2 - z + 1 0 = z2 - 2z + 1 Therefore z = 1 and z = 1
30. ### Finding fixed points of mobius transform

Homework Statement What is the procedure to find fixed points of a mobius transform? I don't really have an example, how about: f(z)= (z-i)/(z+i) Homework Equations From what I understand, fixed points are points that when attempting to transform get mapped back to themselves. So one...
31. ### Classifying Fixed Points in a Second-Order Differential Equation

Homework Statement Write the second-order diﬀerential equation \ddot{x} + 2\epsilon \dot{x} + sin x =0,\epsilon \geq 0, as a pair of coupled ﬁrst-order equations.Find all its ﬁxedpoints, and determine how the classiﬁcation of these ﬁxed points changes with \epsilon Homework Equations...
32. ### Classify fixed points non homogeneous system of linear differential equations

Homework Statement \dot{x}=2x+5y+1, \dot{y}=-x+3y-4 Homework Equations The Attempt at a Solution Well, if system was: \dot{x}=2x+5y, \dot{y}=-x+3y we let a=2, b=5, c=-1, d=3. Then p = a + d and q =ad - bc and we investigate p^2-4q Don't know what to do when \dot{x}=2x+5y+1...
33. ### Testing Stability of Linear System Fixed Points

Homework Statement Classify the fixed points of the following linear system and state whether they are stable or unstable \dot{x}=x + y \dot{y} = x + 3y Homework Equations The Attempt at a Solution Fixed point at dy/dx = 0/0. Therefore fixed point = (0,0) How does one test...
34. ### What is the role of combinatorics in the depths of string theory?

I'm not sure I understand this correctly, but http://www-library.desy.de/preparch/desy/proc/proc02-02/Proceedings/pl.6/deboer_pr.pdf seems to say that in gauge/gravity, if the gauge theory is run from IR to UV fixed point, the bulk theory goes from supergravity to string theory. (i) Are...
35. ### Mathematica How can I find fixed points for a map with a range of r values?

Hello , i wanted to ask sth I have a map : xn+1=rxne-xn , r>0 and r<20 I want to find fixed points .My problem is that i don't have a value or r.I can't manipulate the r values. (if i had an r value i would do : r=2;) My code is: y[x_]:= r x Exp[-x]; To find the fixed points ...
36. ### MVT question + fixed points

Homework Statement Consider the function g: [0,∞) -> R defined by G(x)=x+e-2x A. Use the mean value theorem to prove that |g(x2)-g(x1)|<|x2-x1| for all x1,x2 E [0,∞) with x1≠x2. B. Find all fixed points of g on [0,∞). Homework Equations MVT: f'(c) = f(b)-f(a)/b-a. The Attempt...
37. ### Linear Maps and Fixed Points in RPn

Regard the n-dimensional real projective space RPn as the space of lines in Rn+1 through {0}, i.e. RPn = (Rn+1 − {0}) /~ with x ~ y if y = λx for λ not equal to 0 ∈ R ;with the equivalence class of x denoted by [x]. (i) Work out the necessary and sufficient condition on a linear map f...
38. ### Fixed Points of ODE: Clarifying Conditions

In a book on synchronization it is stated that given the ODE \frac{d\psi}{dt}=-\nu+\epsilon q(\psi) there is at least one pair of fixed points if \epsilon q_{min}<\nu<\epsilon q_{max} were q_{min}, q_{max} are the min and max values of q(\psi) respectively. While this could be...
39. ### Find Fixed Points for x^2+3x+1=0

ive got a question on how to get a fixed point. on the equation for. \frac{1}{8}X^2+\frac{11}{8}X+\frac{1}{2} do you find the two factors to get the fixed points. or run the equation though a quadratic formula to get the fixed points. i have an = which is x^2-3X-4=0 but i don't know how...
40. ### Fixed Points Locus with Real Parameter t | z1 and z2 Fixed Points

Homework Statement Given z1 and z2 as fixed points describe the locus of the point z1+t(z2-z1) a)t is real b)0<t<1 Homework Equations The Attempt at a Solution My problem is that I don't even understand what a locus is. b)z1 or z2 if t is 1.
41. ### Finding Fixed Points of Affine Transformations | EWW

Hello, I'm having difficulties with finding fixed points of affine transformations. I understand that given a matrix A of barycentric coefficients, I want to produce a point that is equal to the given point, i.e. Ax = y, where y = x. But all I get is a homogenous linear system whose only...
42. ### What is the expected number of fixed points in permutations?

What is the expected number of fixed points in permutations? I got 1 as answer.
43. ### How many fixed points on a circle S/Z2?

I am working on Zweibach's First Course in String Theory and question 2.4 asks: Show that there are two points on the circle that are left fixed by the Z2 action. (For those without the text, the circle is the space -1 < x <= +1, identified by x ~ x + 2. And the Z2 mod imposes the x ~ -x...
44. ### Linearization of an equation around fixed points

Homework Statement Find the linearization of the equation y' = y(-1+4y-3y^2) about each of the fixed points The Attempt at a Solution I think this is correct for finding fixed points: Set y' = 0 = y(-1+4y+3y^2), so the fixed points are y = 0, 1/3, 1 What exactly does it mean by...
45. ### Probablity of fixed points in permutations

Randomly permute (1,...,n). What is the probability that exactly i points are fixed? I think it should be \binom{n}{i}\frac{(n-i-1)!}{n!} Is it right? If so, is the expected number of fixed points (I know it's 1): \sum_{i=0}^{n}i\binom{n}{i}\frac{(n-i-1)!}{n!} But it doesn't sum to 1, I think
46. ### Convexity and fixed points

Suppose that K is a nonempty compact convex set in R^n. If f:K->K is not continuous, then f will not have any fixed point. I believe this statement is false, but I cannot think of a function(not continuous) that maps a compact convex set to another compact convex set. any tips would be...
47. ### Fixed Points and Critical Points in Continuous Scalar Functions

Hi, I was thinking about the following and would like some clarification. Suppose that we have a continuous scalar function f:R^n \to R with a critical point at say x_0, where the dimension of x_0 depends on the value of n. Consider as an example f(x) = x (n = 1). The point x = 0 is a...
48. ### Tension in string attached to two fixed points

I'm having trouble with this physics problem: A piece of steel wire (diameter 2mm) is connected between two fixed points. The tension in the wire is 120N at 0 degrees Celcius. At what temperature is the tension 0? I assume that I first have to calculate how much "too short" the string is...
49. ### Fixed Points of Quantum Gravity

A very nice short paper, http://arxiv.org/abs/hep-th/0312114 giving an analytical (Wilson renormalization group) development of the fixed points of the quantum gravity models as previously suggested by numerical studies. Conclusions, 1) At very short distances, the coupling runs and...