# 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)

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### Why does ${x}^{x^{x^{x^{\,.^{\,.^{\,.}}}}}}$ bifurcate below $\sim0.065$?

When you calculate what ${x}^{x^{x^{x\cdots }}}$ converges to between $0$ and $1$, before approximately $0.065$ the graph bifurcates. Why does this happen and is there a reason for it happens at that ...
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### What are near-identity transformations?

I've started a module on Dynamical systems and we're currently looking at Hopf bifurcations. Below is an exercise in our notes revolving around near identity transformations: The notes seem to assume ...
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### Translated systems and bifurcation value

Consider the system $$x'=y$$ $$y'=x+x^2+\mu x-xy$$ we know that $\operatorname{det}(Df(0,0))=-1$, then there is a saddle at the origin. In addition det $(Df (-1,0)) = 1> 0$, so there is a focus or ...
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### How do I find the bifurcation points of the quadratic map?

I am exploring the quadratic map $x_{n+1} = x_n^2 + c$. If we plot the value of $x$ after many iterations on the vertical axis against the input $c$ value on the horizontal axis we get a classic ...
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### Classifying a Bifurcation?

For the system of differential equations $$\dot{x}=y-x, \dot{y} = \mu x - y$$ with Jacobian matrix $$J = \begin{bmatrix}-1 & 1 \\ \mu & -1\end{bmatrix},$$ the point $(x,y) = (0,0)$ is a stable ...
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### Equations for Mandelbrot bifurcation diagram

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded within a circle of radius 2 when iterated from $z_0 = 0$. Looking at only the ...
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### How to calculate fixed points and plot bifurcation diagram for non-linear ODE system

I am trying to understand how to analyse a system of coupled, non-linear ODEs taken from this paper. I want to perform a fixed point analysis and plot a bifurcation diagram to show how fixed points ...
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### Mandelbrot set and logistic map connection

I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
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### What kind of local bifurcation occurs here?

I have encountered a bifurcation diagram for my five-dimensional non-linear ode system? I am really confused to identify the kind of bifurcation that occurs in the knowns forms of local bifurcations ...
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### Identify bifurcation and sketch diagram

I am given the equation $$\dot x = rx + \frac {x}{1+x^2}$$ I believe I've found fixed points at $x=0$ and $x= \sqrt \frac{-1-r}{r}$ and I think its stable for $-1<r<0$ but I'm not sure how to ...
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### Are bifurcation diagrams meaningful without paramaters?

I am given a system of equations: $\dot{x} = -x^3 - y^2$ $\dot{y} = xy - y^3$ .. and asked to draw a bifurcation diagram. I dont think the ask is a valid one because bifurcation diagrams are only ...
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### Find nullclines for nonlinear system [closed]

I have a nonlinear system and need find and plot nullclines: $$\dot{x}=0.1(-x-1.8*10^{-3}Q(y)+1.3*10^{-3})\\ \dot{y}=0.1(-y-2.1*10^{-3}Q(x)+D)$$ here $Q(x)=\frac{100}{1+e^{(0.01-x)/0.003}}$ and D ...
1answer
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### Find a critical point

I found a method to find a critical control parameter in one article and it worked for other similiar system with saddle-node bifurcation too. In general, we have a 2 equations nonlinear system: \...
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### Find a system point

Here is given nonlinear ODEs system: where $I$ is a control parameter. When $I=I_c$ saddle-node bifurcation is happening. Need to find point $I_c$. By phase plain method via Matlab I ploted ...