Fixed points Definition and 49 Threads

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.

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 &...

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...

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

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...

What is the expected number of fixed points in permutations? I got 1 as answer.

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...