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So I simulated a damped, driven pendulum in Java with the goal of showing period doubling/chaotic behavior.

But then, as I was increasing the driving force, i saw the double period born. Then the 4-period...but then suddenly I noticed I was getting 5 peaks! I let it run for a while and they were stable.

So my questions are:

1) does this mean I'm in a "window" of the chaos? Did i overshoot the 8,16,32..etc periods?

2) is there an analytic way to determine for which values of control parameters period doubling occurs? I know you can look at the bifurcation diagram (if you have one!) and see..But for any example I look at, I can only make out the 16-period doubling bifurcation.

it seems like most books just plot and the values are obtained from trial/error. For example, what is the r value for the 128 period of the logistic map? I can't find it.

Anyways I'd appreciate any insight! Thank you