The first problem on this quiz was primarily just about vocabulary
In the logistic map, the parameter is R, and the state variable is x
And changes in that parameter can most certainly induce bifurcations in the dynamics
The answers to the first two problems are r is the parameter and that changes in that parameter value can certainly induce a bifurcation in the dynamics, which changes the topology of the attractor
The third problem was about zeroing in on the bifurcation point
For that, you needed to do a bunch of experiments with the app
Heres r = 2.9, so I would definitely say that this is still fixed-point dynamics
Heres r = 3.1. I would definitely say thats two-cycle dynamics
So someplace between 2.9 and 3.1, theres a bifurcation
Heres 3.0. When I only plot 50 iterates, its not clear whether or not this is converging, so let me plot some more
Thats a hundred iterates; thats three hundred iterates.
It looks to me like this is still converging down to the fixed point
Lets try slightly higher than 3
Ah. So theres definitely a fixed point below three, and theres definitely a two-cycle above three
So I would say that the bifurcation occurred right at 3, which would suggest that the second answer is the right one
Question 4 is about basins of attraction and attractors
When you start from 0.1, with r = 3.2, you reach a two-cycle
Heres x = 0.2, x = 0.3, x = 0.4: these all appear to be going to the same two-cycle
x = 0.5, x = 0.7, x = 0.9
That was the range that the problem suggested that you explore, and it sure looked like all orbits limited to the same two-cycle, so the answer is yes
And if an orbit limits to some subset of the state space, that subset of the state space is an attractor, so indeed that two-cycle is an attractor
As we raise r slowly from 3.4 to 3.41, to 3.42, to 3.43, to 3.44, to 3.45, youre starting to see another bifurcation
At 3.46 its pretty clear; lets plot a few more points
Looks like a 4-cycle to me, so the answer is 4
Lets explore one more range, from 3.5 to 3.55 to 3.551
The 4-cycle has turned into an 8-cycle, so the answer to the seventh problem is 8
In the thought question here, we went from 1 to 2 to 4 to 8
So what youre seeing is whats called a period-doubling bifurcation sequence