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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 causes bifurcation?

Suppose let's say we have a recursive function, $$x_{n+1}=rx_{n}(1-x_{n})$$ From what I understand, the $x_{n+1}$ VS r graph starts to split (bifurcate) after a particular value of r (apparently it's around 3). Does that mean after they split up, we…
basilisk
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Logistic Map Bifurcation diagram

So in the bifurcation diagram of the logistic map, there is period doubling from about $r=3$ to about $r=3.54409$. There are two fluctuation points between $r=3$ and $r=1+\sqrt{6}$. My question is, how would one obtain the $r$ value of $1+\sqrt{6}$?
AMorales93
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Bifurcation in PDE

How do we characterize bifurcation in nonlinear PDE instead of ODE i.e. ht=f(x,h,hx,hxx,hxxx,...)? For example, study the temporal evolution of a regular pattern into a chaotic one. Can someone please point out the branch of mathematics I should…
a non-normal user
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Show that the family $E_\mu$ undergoes tangential bifurcation

Let $E_\mu(x)=\mu e^x$. Show that the family $E_\mu$ undergoes tangential bifurcation at $\mu=1/e$. In particular follow out the following steps: (a) Plot out the diagonal and the graph of $E_\mu (x)$ for $\mu < 1/e$, $\mu=1/e$, and $\mu>1/e$.…
jerry2144
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Bifurcated limit cycles from real eigenvalues

I have never heard of limit cycles bifurcated from real eigenvalues crossing zero. For the below system I believe that is the case. Is this bifurcation known and has a specific name? For $d = b + i c \in \mathbb{C}$, $z(t) = x(t) + i y(t) \in…
pitonist
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Bifurcation values for logistic map

To find the bifurcation values for $$x_{i+1}=f(x_i) = rx_i(1-x_i)$$first I set $rx(1-x) = 0$ and found the x values and then used the x values to find $r = 0$ and $r = 1$. Do you think what I did here is correct? If not, can you help me find the…
user136422
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How to read two-dim bifurcation diagrams?

Suppose we have some 2-dimensional bifurcation diagram, say, the following which I found when using google; this is just meant as a general question about how to read such diagrams, I am not dealing with the concrete equations here. Now my question…
mathfemi
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Bifurcation Diagram(f '(x)=r-cosh(x))

I am trying to write a bifurcation diagram for $f'(x)=r-cosh(x)$ and I am having a little trouble. I also have to show that a saddle-node bifurcation occurs at a critical value of $r$. So, I know $f(x^*,r_c)=r_c-cosh(x^*)$ and…
Brian
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What is symmetry in bifurcation analysis?

I did a quick google but I couldn't find much. Could someone please explain when a system has symmetries or link me to some good resources? For example, the system $x'=\mu x-y+x^3$, $y'=bx-y$ has certain symmetries which are not present if the…
user153253
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bifurcation in differential equations

I have an equation $$ y' = (2+y)(k-y^2) $$ and I am asked to find the equilibrium solutions and bifurcation values for all values of $k$. My approach was that... there exists $2$ equlibrium solutions for $k>0$, and $1$ equilibrium soltuion for $k =…
Topstar
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Taylor expansion of center manifold with respect to a central variable and control parameter

I am being asked to find the Taylor expansion up to order two with respect to $(w, \mu)$ of the center manifold for the following ODE: $\frac{dx^2}{dt^2}+\frac{dx}{dt}-\mu x + x^2=0$, where $\mu \in \mathbb{R}$ is a small parameter. We usually…
anna
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Infinite periods in bifurcation diagram

A slightly more theoretical question for you all. Recently I was looking at the logistic map and the resulting bifurcation diagram (shown). Wikipedia says that prior to roughly r = 3.56995, there is a period-doubling cascade that goes from 2 to 4 to…
Alex Mac
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Bifurcation Diagram- The Logistic Map

I recently got to know about the logistic map, given by $f(x) = rx(1-x)$, and it’s bifurcation diagram when mapped as $r$ along the x-axis. At $r=3.56995 (approx.)$ we enter the chaotic part with some islands of stability. All this continues till…
Aryan
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Divergent limit cycle frequency along a hopf boundary

I am studying the linear stability of continuous system as a function of two parameters (a and b) and I observe that a hopf bifurcation with frequency w happens along the line described by f(a,b). Now, it turns out that there is a critical point…
MatemTest Testi
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Is there a bifurcation node in the activation energy of chemical reactions?

I was thinking about the activation energy of chemical reactions (obviously), and I was wondering if there exists a bifurcation node somewhere in the transition state. Let me give you a link to an image:…
Mlagma
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