# 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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### Does a spiral fixed point going from stable to unstable always indicate Hopf bifurcation?

Does a spiral fixed point going from stable to unstable (with change in some parameter) always indicate Hopf bifurcation? Could it be a homoclinic bifurcation instead? (Homoclinic bifurcation keeps ...
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### Plotting the bifurcation diagram for Ikeda map

I'm trying to plot the bifurcation diagram for Ikeda map. I wrote a code in Python to get the points of this diagram, but it seems that for $u > 1$ the points diverge and my code doesn't work ...
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### center manifold and bifurcation: 2D Bifurcation system reduction

I have this system to study \left\{ \begin{aligned} \frac{dx}{dt} &= y-x - x^2 \\[5pt] \frac{dy}{dt} &= \mu x - y - y^2 \end{aligned} \right. I have derived the Jacobian around fixed ...
1 vote
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### Period doubling on Möbius strip

I have a hard time understanding a simple argument for the following. Quite often I find in papers that limit cycles before and after period-doubling bifurcations lie on the Möbius strip in the state-...
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### 2D bifurcation problem

I come across this problem which is about bifurcation. I am trying to take all the cases. I am expecting Hopf bifurcation to occur here but the last case I could not find the fixed point. Could you ...
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### Analyzing bifurcation point of a 2D dynamical system

I have the following nonlinear dynamical system: \begin{align} \frac{dX}{dt} &= Y(\gamma - 2X) \\\ \frac{dY}{dt} &= - \gamma X + X^2- Y^2 + 1 \end{align} And I'm trying to understand what type ...
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### Reference for one-dimensional "singular" implicit function theorem (i.e. the shape of a nondegenerate saddle-node bifurcation)

I think I can easily prove the seemingly very basic result below. For a paper, I would like to be able to cite a reference for it, rather than having to either write out a proof [which would be ...
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### Equation for curves of high density in bifurcation diagram

Is there an equation for the high density curves in the chaotic regions of the bifurcation diagram for the logistic map? I'm talking about the sinusoidal-looking dark curves in the following picture. ...
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### About the stability of cycles for the logistic map $x_{n+1}=rx_n(1-x_n)$

Please indicate a reference (if there exists any) proving that when a cycle appears (i.e. for the minimal value of the logistic parameter $r$ for which a $k$-cycle exists) we also have, "...
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### How to know when a transcritical bifurcation occurs (Example 3.2.1 Strogaz Nonlinear Dynamics and chaos)

Picture of Question + solution Hi, In this question After we expand $\dot{x}$ around $x^*$ = 0 , we get $\dot{x}$ = (1-ab)x + ($\frac{1}{2}$a$b^2$)$x^2$ + O($x^3$). How do we jump from this to knowing ...
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### Are 'eye' bifurcations possible?

I am currently studying bifurcations in dynamical systems and have seen saddle-node, transcritical and pitchfork bifurcations. My question is can you have a system which has an 'eye-shaped' ...
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### What is the classification of the bifurcation of a tent map?

Considering the tent map where $x_{n+1} = f(x_{n})$ and $f(x)$ is defined as  f(x)= \begin{cases} \mu x, & 0 \leq x\leq \frac{1}{2} \\ \mu - \mu x, & \frac{1}{2}\leq ...
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### Dynamical System that exhibits a fold bifurcation of Limit Tori?

Fold Bifurcations of a fixed points (i.e. saddle node bifurcations) and Fold bifurcations of limit cycles (i.e. when a stable limit and unstable limit cycle annihilate) are observed in plenty of ...
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