RE: Research diaries #4: The beauty of chaos
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Do you mind detailing more the mechanism behind the splitting of one points into two? How does that work here?
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You are viewing a single comment's thread:
Do you mind detailing more the mechanism behind the splitting of one points into two? How does that work here?
With splitting you mean what happens directly after mu \approx 0.75?
I am plotting the orbit. So denote the logistic map by f_\mu. And what I plot is the set { f^n(x_0) : n \in N, n> big number } for x_0=1/2. So after \mu \approx 0.75 the orbit is an attracting period-2 orbit.
I am not sure if that answers your question :3
I am not sure too but I think that I got it (as you insisted on the orbit). Thanks!
I think you want to know how to determine the bifurcation at mu=0.75?
So f_mu has two fixed points. One at 0 and one at (-1+4mu)/(4mu). The single line up to mu=0.75 follows (-1+4mu)/(4mu) because it is a stable fixed point. Then at mu=0.75 we have that f_\mu'(-1+4mu)/(4mu)=-1 and for mu>0.75 we have that |f_\mu'(-1+4mu)/(4mu)|>1 so it becomes unstable.
If you investigate the map f_mu \circ f_mu you see that two fixed points are being "created" as we pass mu=0.75. For the original map f_mu the fixed points of f_mu \circ f_mu correspond to a period 2-orbit.
This transition is called a transcritical bifurcation because the stability of the fixed point at (-1+4mu)/(4mu) changes.