# Tagged Questions

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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### What is this bifurcation of a fixed point of a two-dimensional diffeomorphism with two parameters?

Suppose I have a diffeomorphism of a plane, $$\bar{x} = F(x,s,t)$$ where $x \in \mathbb{R}^{2}$ and $s \in [a,b] \subset \mathbb{R}$ and $t \in I_{2} \subset{ \mathbb{R}}$ are parameters. Suppose ...
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### Numerical analysis of bifurcation problems - notational confusion

I'm going through the following set of review notes about numerical analysis of bifurcation problems and was wondering if someone could explain to me what is going on on the page 50. First of all, I'...
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### Implicit function theorem for Banach spaces

I was wondering if someone could give a bit of broad advice regarding working with Implicit Function Theorem (IFT) and, I guess, the Catastrophe theory. This is something completely new to me. ...
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### 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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### 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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### 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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### 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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### 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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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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### 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 PoincarÃ© or are this completely different ...
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### 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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### 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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### 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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### 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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### 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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### 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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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}{... 0answers 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 ... 0answers 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 ... 0answers 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 ... 1answer 115 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 ... 1answer 93 views ### Find the values of r at which bifurcation occurs Consider the system\dot{x}= rx - \frac{x}{1 + x^2}$$where r \in \mathbb R. Find the value(s) at which bifurcations occur and where possible classify those as saddle-node, transcritical or ... 1answer 22 views ### Draw the orbits in the {\rho,\theta}-plane and describe the evolution in each cases, classfying subcritical and supercritical if appropriate. Given$$\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 ...
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 ...
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 ...