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Questions tagged [bifurcation]

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)

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. Reference: Wikipedia.

Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour.

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What does it mean to find a normal form of a saddle node bifurcation?

So I have just started taking a course in bifurcations and finding some parts of the course rather tricky when I don't think they should be. I was wondering if anyone could explain this question. Find a transformation putting each of the f(r, x) in…
user415903
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bifurcations,bifurcation points and bifurcation diagram

For a real number $c$, define the one-parameter family $f_a(x)=(x-a)(2x-3a)+x+c$. For what values of $c$ is there a bifurcation in this family? Describe the bifurcations and list the bifurcation points $(a, x)$, and sketch the bifurcation diagram.
Nora1987
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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: 2 branches(saddle node): $f(x, r) = r - x^2$ 3…
Ameer saadat-yazdi
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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 describe the evolution in each cases($\sigma_r$ and…
DDaren
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Non-hyperbolicity implying bifurcation

I have come to know that a bifurcation point must be non-hyperbolic. But I am not sure whether the converse is also true. As I considered $f(x)=\mu-x^2$, I got $x^2=\mu$ are bifurcation points. i) For $\mu<0$, there is no bifurcation point. ii) For…
Manjoy Das
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Bifurcation Diagram and Logistic Map

I want to create a Bifurcation Diagram with the Logistic Map and I have open questions about the correct algorithm. Here is how I understood it so far: logistic map $x_{i+1} = rx_{i}(1-x_{i})$ take a start population, e.g. $x_{0} = 0.5$ generate…
zirkelc
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Have I correctly found the Hopf Bifurcation on this system?

Given the equations: \begin{align} \frac{dx}{dt}&=1-(b-1)x+ax^{2}y\\ \frac{dy}{dt}&= bx-ax^{2}y \end{align} a) Fix $a$ and vary $b$. Show that a Hopf Bifurcation occurs at $b=a+1$. So I got two fixed points $(0,0)$ and $(1,\frac{b}{a})$ I computed…
Robbie Meaney
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How many cases are there for this bifurcation?

Here is the bifurcation: $$\frac{dx}{dt} = x(1-x)-h \frac{x}{1+x}$$ fixed points $$\ x = 0, -\sqrt {1-h}, + \sqrt {1-h} $$ How many cases are there? My guess: $$\ 0 1 $$ $$\ h < 0 $$ What are the cases?
Bob Dylan
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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 bifurcation theory' from Golubitsky and Schaeffer.…
Multiverse
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