Period doubling for a damped, driven, harmonic oscillator

[speed racer 5]

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

 

[obafgkmrns]

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!

 

[AlephZero]

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 $x_{n+1} = a x_n(1 - x_n)$

 

[speed racer 5]

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!

 

[blue_raver22]

Hello,

Thank you for sharing your simulation results and questions. It is great to see someone exploring the behavior of a damped, driven harmonic oscillator and observing period doubling.

To answer your first question, it is possible that you are in a "window" of chaos. This means that you have found a range of values for your driving force where the system exhibits chaotic behavior, but it is not necessarily the full range of chaotic behavior. It is also possible that you have overshot the 8, 16, 32, etc. periods, as you mentioned. This could be due to small variations in your simulation or the sensitivity of the system to initial conditions.

For your second question, there are indeed analytical ways to determine for which values of control parameters period doubling occurs. One approach is to use the bifurcation diagram, as you mentioned. This is a graphical representation of the steady-state behavior of a system as a parameter is varied. In the case of the damped, driven harmonic oscillator, the bifurcation diagram would show the relationship between the amplitude of the oscillator and the driving force. You can also use analytical techniques such as the Lyapunov exponent or the Feigenbaum constant to determine the onset of period doubling. These methods require more advanced mathematical understanding, but they can provide more precise results compared to trial and error.

In terms of finding specific values for the period doubling bifurcations, such as the r value for the 128 period of the logistic map, it is not always possible to obtain these values analytically. In most cases, these values are determined through numerical methods and simulations, as you have done. However, there are some cases where analytical solutions are possible, such as the period doubling route to chaos in the logistic map.

Overall, your simulation results and questions show a great understanding of period doubling and chaotic behavior. Keep exploring and asking questions, as there is still much to learn about this fascinating phenomenon.