Period doubling for a damped, driven, harmonic oscillator

  • #1
I'm not sure I'm in the right forum but I'll try and ask anyways.

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
 

Answers and Replies

  • #2
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1) Almost certainly you overshot. Try varying the forcing function much more slowly -- the doublings become very closely spaced as you approach chaos.

2) If you find a way, be sure to publish it!
 
  • #3
AlephZero
Science Advisor
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Google for "Feigenbaum constant".

But be warned that the errors in your numerical integration will make it hard to "see" more than a small number of period doublings. If you want to study this numerically you might do better with the fundamental "logistic map" equation [itex]x_{n+1} = a x_n(1 - x_n)[/itex]
 
  • #4
Yes absolutely i realize now that the step size has to be miniscule! This is the paper I'm trying to simulate http://prl.aps.org/abstract/PRL/v47/i19/p1349_1

I wish I had been given a "simpler" model but alas, as all professors demand the impossible, I was given this ;p

Well I see what you both mean...i'll have to go smaller and smaller!
 

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