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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
Show that the explicit Runge-Kutta scheme
\begin{equation} \frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})]
\end{equation}
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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 ...
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...
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...
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...
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...
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.
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...
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...
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...
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
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...
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...
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...
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...