Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. (Def: http://en.m.wikipedia.org/wiki/Bifurcation_theory)

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How can I show that the lag-1 autocorrelation must increase?

Assume that we have a dynamical system $$ x' = f(x) $$ and that there is a repeated disturbance in the system after each time step $\Delta T$. Between disturbances, there is a return to equilibrium $...
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27 views

Bifurcations in two dimensional systems

Given the following autonomous differential equations, which illustrates an elliptic limit cycle, $r' = \alpha -\left(\frac{r}{a}\right)^2 \cos^2\theta - \left(\frac{r}{b}\right)^2 \sin^2\theta \\ \...
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14 views

Function of x and r with n number of bifurcation

I'm looking for a function $f(x, r)$ that bifurcates to give $n$ number of solutions suddenly at $r > 0$, and $0$ solutions if $n$ is even or $1$ solution if $n$ is odd for $r < 0$. For example: ...
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1answer
29 views

Fixed point analysis in the Wilson-Cowan model

i guess this is a rather simple question, but given my non-mathematical background, i'm a bit stuck. i'm trying to find the jacobian matrix for the follwing dynamical system (wilson-cowan model). the ...
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9 views

Eigenvalues occur in quadrupels, Turing bifurcation

First of all, sorry, if the following is much to vague. In which sense might the special situation that eigenvalues occur in quadruples $c,\bar{c},-c,-\bar{c}\in\mathbb{C}$ be connected with the ...
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23 views

1D Flows, Local bifurcation, method.

I am working through a problem sheet which consists of questions such as "Find the type of bifurcation which occurs in the 1D system defined by $\dot{x}= f(r,x):= rx - \sinh{x}$, and state the ...
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47 views

Poincare first return map, stability and bifurcations

Let $X= \mathbb R^3$ and consider the autonomous dynamical system $$\dot{x_1} = -x_2 + x_1 (1 - (x_1^2 +x_2^2)^2), \qquad{} \dot{x_2} = x_1 + x_2(1-(x_1^2 +x_2^2)^2), \qquad{} \dot{x_3}= \epsilon x_3$...
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1answer
26 views

How to Show that Lorenz equations are invariant?

I am struggling a little bit with this question. I know that that the Lorenz equations are: \begin{align} \dot{x} &= \sigma(y-x)\\ \dot{y} &= rx - y- xz\\ \dot{z} &= xy - bz \end{align} ...
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34 views

A simple question about bifurcation theory

it seems that I need some elements of bifurcation theory for my research, and I'm a bit puzzled at the moment by some basic stuff. I'm reading the beginning of the book 'Singularities and groups in ...
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2answers
26 views

Programming Bifurcation analysis

Next year I'm going to do my final project for graduation. I've been assigned to search and study some biomathematical models with bifurcation theory and numerical bifurcation analysis. I'm thinking ...
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1answer
36 views

Finding bifurcation of trigonometric system

I'm really struggling to find the bifurcation(s) of the system $x'=x^2 + \cos(x+ \mu)$, $\mu \in [0,2\pi)$. I've tried substituting $y=\mu+x$, taylor expanding, and just about everything else I ...
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85 views

Periodic solution to DDE: $\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0$

Consider differential equation with delay: $$\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0.$$ Let's use $t=(1+c)\tau$ substitution to normalize time t: $$\frac{d}{(1+c)d\left(\...
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1answer
47 views

Bifurcation Problem

I am trying to classify the type of bifurcation for the dynamical system given by: $\dot x = x^2+y^2-2my$ $\dot y= mx-y$ with m as a varying parameter The fixed points are at (0,0) and ($2m^2 \...
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1answer
19 views

system of differential equation finding pitchfork biforcation

Image of the question \begin{align} \frac{\text{d} x}{\text{d} t} = f(x,y) &= - x+\gamma(b x+y)-\epsilon(b x+y)^3\\ \frac{\text{d} y}{\text{d} t} = g(x,y) &= - r y-\alpha b x \end{align}...
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1answer
53 views

2D Bifurcation Classification

Given the system with m as a varying parameter: $\dot x = mx^2-y$ and $\dot y = m+y - x$ Determine any bifurcations that occur Attempt: x nullcline $y=mx^2$ y nullcline $y=x-m$ Fixed ...
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1answer
313 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ...
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1answer
20 views

Name of degenerate parameter dependent ODE

I am looking for literature on specific type of degeneracy for odes. Consider the phase space $M= \{ (x_1,x_2)\in \mathbb{R}^2 \}$ and the ode of the general form: \begin{align} \alpha \frac{d x_1}{...
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32 views

Transform $dx/dt = r + x/2 - x/(x+1)$ into normal form $dX/dt = R +- X^2$ (saddle node bifurcation)

I am trying to show that $dx/dt = r + x/2 - x/(x+1)$ has a saddle node bifurcation by putting it into normal form. I've found that the bifurcation point must be at $$x=-1+-\sqrt(2)$$ and that from ...
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54 views

Transform $dx/dt = 1 + r + x^2$ into normal form $dX/dt = R +- X^2$ (saddle node bifurcation)

I am trying to find a substitution $X(x,r)=?$ and $R(x,r)=?$ to allow me to transform $dx/dt = 1 + r + x^2$ into $dX/dt = R +- X^2$ (normal form for saddle node bifurcation) I'm sure it's a ...
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1answer
58 views

Analyzing a trajectory

I have the discrete dynamical system $f(x) = 13xe^{-x}$ and want to know its stable period 4 orbit, its fixed points, and period 2 orbit. Of course also check stability of these things as well. I've ...
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49 views

Finding the stable period $4$ orbit of a trajectory

I am told to find the stable period $4$ orbit of $f(x) = 13xe^{-x}$ for my discrete dynamical system through direct numerical iteration. However, I am a bit confused on what is meant by direct ...
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86 views

Asymptotics for solutions of a version of Lienard's differential equation

Consider the second order differential equation $ x'' + f(x)x' + g(x) = 0 $ with $$ f(x) = -\lambda + x^2, \quad g(x) = (-1 + x^2)x \, . $$ with $\lambda > 0$. Note: The original post had a ...
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52 views

Signs of a Neimark-Sacker bifurcation?

This bifurcation diagram looks to me like it could just be a lot of little pitchforks. Is there something visual/graphical about it that gives it away as a Neimark-Sacker bifurcation?
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1answer
35 views

bifurcation values

Given differential equation $\frac{dy}{dt} = f(y, \alpha),$ we solve $f(y, \alpha) = 0$ to find equilibrium solutions. By definition, $\overline y$ is a bifurcation point and $\overline \alpha$ is ...
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1answer
41 views

Confusion over Bifurcation Diagram [closed]

I have the function $f(x,a) = \frac{2}{x} + 0.75x + a$ and want to create a bifurcation diagram of this function as $a$ varies. An $\textit{equilibrium}$ point (I thought) of the trajectory made by ...
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1answer
229 views

Continued fraction analog to zeta function - how to properly define it and find its properties?

I do not mean the continued fraction representation of zeta function; I mean the function which has the form: $$f(s)=\cfrac{1}{1^s+\cfrac{1}{2^s+\cfrac{1}{3^s+\cfrac{1}{4^s+\cdots}}}}$$ For some ...
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1answer
45 views

Why do we need to find the multiple roots? (bifurcation curve)

Consider the system $$ \dot{x}=x+ay-y^3,\quad \dot{y}=b-2y+x. $$ The task is to give the bifurcation curve for the equilibria. First of all, equilibria are determined by $$ x+ay-y^3=0\...
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50 views

Planar system: Analyse the existence of equilibria and determine their bifurcations

Consider the system $$ \dot{x}=x+ay-y^3,\quad \dot{y}=b-2y+x. $$ Analyse the existence of equilibria and determine their bidurcation. The equilibria can be determined by setting $$ 0=x+ay-...
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1answer
25 views

Not sure how to get this normal form

Consider $$ \dot{x}=f(x,\alpha), x\in\mathbb{R},\alpha\in\mathbb{R} $$ with smooth $f$ and equilibrium $x=0$ at $\alpha=0$, $\lambda=f_{x}(0,0)=0$ and, moreover, $f_{xx}(0,0)\neq 0, f_{\alpha}(...
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1answer
28 views

Normal form of subcritical pitchfork bifurcation.

I'm working the a dynamical system $\dot{x} = r x - \frac{x}{1+x^2}$. I have already worked out that it is a subcritical pitchfork bifurcation. At least, that what my bifurcation diagram shows. ...
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51 views

Name of bifurcation that causes eigenvalues to switch sign in a saddle?

What is the name for a bifurcation where the signs of the eigenvalues switch? E.g. Given a 4-dimensional saddle (two positive, two negative real eigenvalues), as I bifurcate a parameter two ...
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42 views

Does the normal form of the Fold bifurcation has something to do with the Dulac-Poincaré normal form?

Maybe this is a silly question but does the normal form $\dot{x}=\mu\pm x^2$ of the fold bifurcation has something to do with the normal form by Dulac and Poincaré or are this completely different ...
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18 views

Examples of state-space models that show strong homeostasis but also substantial change after critical threshold?

The question is, can can anyone provide examples of systems or math models that exhibit patterns of homeostasis but which can be exhibit substantial transitions or bifurcations after some critical ...
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75 views

Bifurcation diagram and bifurcation value

Determine the bifurcation values of $\dot{x} = x(x-r^2)$, and sketch the bifurcation diagram. My attempt: First, we see that if $f(x_0, r_0) = Df(x_0, r_0) = 0$, then $x_0$ is a non-hyperbolic ...
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38 views

Bifurcation diagram

Consider the logistic map $x_{n+1}=rx_n(1-x_n)$, whose bifurcation diagram is shown below for $2.4 < r < 4.0$: I need to find a particular value of $r$ so that "attracting $2^k$ periodic points ...
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27 views

Find all bifurcation values of a function

I need to find all bifurcation values of the function $x' = u + cos(x) + cos(2x)$. How do I find all bifurcation values of $u$? I know the solution is $u < -2, u = -2, -2 < u < 0$, and that ...
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1answer
58 views

Bifurcation - first-order ODEs

Construct a first-order ODE with one critical point if $\left\lvert \mu \right\rvert \ge 1$ and three critical points if $\left\lvert \mu \right\rvert \lt 1$ and draw a bifurcation diagram. Having ...
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2answers
105 views

What is the correct approach for studying bifurcations?

Probably a trivial question. Let's say I have the following system of equations: \begin{cases} f\left(x,y,p\right)=0\\ \\ y=g\left(x\right) \end{cases} where $p$ is a parameter, and I want to study ...
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57 views

Finding bifurcation values of one-parameter families of first-order differential equations

Consider the following one-parameter families of first-order differential equations defined on the reals: $$ \dot x = \mu - x - e^{-x} $$ $$ \dot x = x(\mu + e^x) $$ $$ \dot x = x - \frac{\mu x}{...
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45 views

How to draw bifurcation diagram:$\dot{x}=x^3-C*sin(\frac{\pi x}{2})$

I want to draw the bifurcation diagram but since I can't solve this equation by hand it is difficult. I can graph it by having $f1=x^3$ and $f2 = C*sin(\frac{\pi x}{2})$ and the intersection points ...
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19 views

Find the bifurcation value given the differential equations

Given $\dot{x} =-2x+y$ $\dot{y} =x^2-y+r$ Where r is a real number. Find the bifurcation values. The answer is $r=1$ but I don't see how that can be obtained so if someone can help me further I ...
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29 views

Explanation for the following bifurcation diagrams.

I am asked to plot the bifurcation diagrams of $x'(t)=ax+3$, $x'(t)=x^3-x+a$, and $x'(t)=x^2-ax$ respectively. The solutions (from an instructor) are as follows. Can anyone explain how to do this ...
2
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1answer
113 views

Find and classify the bifurcations that occur as $\mu$ varies for the system

Find and classify the bifurcations that occur as $\mu$ varies for the system \begin{align}\frac{dx}{dt}&= y-2x \\ \frac{dy}{dt}&=\mu +x^2 -y\end{align} What I have so far: The equilibrium ...
2
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1answer
89 views

Find the values of $r$ at which bifurcation occurs

Consider the system $$\dot{x}= rx - \frac{x}{1 + x^2}$$ where $r \in \mathbb R$. Find the value(s) at which bifurcations occur and where possible classify those as saddle-node, transcritical or ...
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1answer
22 views

Draw the orbits in the {$\rho,\theta$}-plane and describe the evolution in each cases, classfying subcritical and supercritical if appropriate.

Given $$\frac{dA}{d\tau}=\sigma A-\beta A|A|^2, $$ where $\sigma=\sigma_r+i\sigma_i$, $\beta$ is real and $A(\tau)=\rho(\tau)\exp(i\theta(\tau))$. Draw the orbits in the {$\rho,\theta$}-plane and ...
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44 views

What type of Hopf bifurcation takes place here?

Consider the system: $\dot{x} = \mu x-y-xy^2-x^3$ $\dot{y} = x+\mu y - x^2y-y^3$ I have shown that a Hopf bifurcaiton takes place at the origin $(0,0)$ as a stable spiral becomes an unstable spiral ...
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44 views

How is this an untable limit cycle?

I am investigating the Lorenz equations and in MATLAB I have plotted a case with $\sigma = 10, b = 8/3, r = 21$ and I have this phase portrait: However I am not exactly sure how this is an unstable ...
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58 views

Is this a spontaneous symmetry-breaking?

I have a system of equations: \begin{cases} f\left(x_{1}\right)+f\left(x_{2}\right)+P=0\\ \\ g\left(x_{1}\right)+g\left(x_{2}\right)=0 \end{cases} where $f,g$ are some functions, $\boldsymbol{x}=\...
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2answers
117 views

Need a textbook for math course

The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and ...