Abstract

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    We have seen that at low driving forces the damped, nonlinear pendulum exhibits simple oscillatory motion, while at high drive it can be chaotic.This raise an obvious question, Exactly how does the transition from simple to chaotic behavior take place?It turns out that the penduum exhibits transitions to chaotic behavior a several values of the driving force.

Background

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Question 3.18

    Calculate Poincare section for the pendulum as it undergoes the period-doubling route to chaos. Plot \omega versus \theta, with one point plotted for each drive cycle, as in Figure 3.9. Do this for F_D =1.4, 1.44, 1.465, using the other parameters as given in connection with Figure 3.10. You should find that after removing the points corresponding to the initial transient the attractor in the period-1 regime will contain only a single point. Likewise, if the behavior is period n, the attractor will contain n discrete points.

Question 3.20

Calculate the bifurcation diagrams for the pendulum in the vicinity of F_D=1.35 to 1.5. Make a magnified plot of the diagram (as compared to Figure 3.11) and obtain an estimate of the Feigenbaum \delta parameter.

Main Body

This part deals with the question: how does the transition from simple to chaotic behacior take place?First, we draw the results for θ as a function of time with driving force a little bit different  here is the       code 

.In the last time, we just gave the results when F_D is smaller than 1.2, now,  I'll give results for \theta as a fuction of time for our pendulum for several different values of the drive amplitude.

If the time, which we plot, satisfys conditions below, from above figure, there must be some points instead of lines.

from figure, it quite easy find that after removing the points corresponding to the initial transient the attractor in the period-1 regime will contain only a single point. Likewise, if the behavior is period n, the attractor will contain n discrete points.

Here is the code2 

In order to understand it more directly, I just give a figure that we plot just in the drive period. And above four diagrams were ploted when F_D=1.2, F_D=1.4, F_D=1.44 and F_D=1.465.

Question 3.20

Bifurcation diagram is a quite good method to tell us the transition to chaos. I spend a flood of time on operating the program, it need a long time to calculate. If you find that it seems that my code can't work, don't worried or surprised about, just wait for the results. It is about several minutes to several hours. Of course, it depend on your accuracy.

Here is the   code3 

Conclusion:

For each value of F_D we have some values  of \theta, just like question 1. and queation 2., it is obvious that when F_D=1.4, there just is one value of \theta, when F_D=1.44, there are two branchs. There may are more branchs when F_D is larger.

Make some magnifield plots of the diagram, in this way, we can find the points that from period 2^(n-1) to period 2^(n).

Just like figures above, we can get some some branch-points, such as, F_D1=1.4228, F_D2=1.45841, F_D=1.47505, F_D=1.47616, F_D=1.476425, F_D=1.476481, F_D=1.476493.

Calculate them, we can obtain 2.14, 14.991, 4.189, 4.732, 4.667. If we continue this process, we will get the Feigenbaum \delta approximate to 4.669.

Reference and Acknowledgement

Thanks to Tan Shan and Yuqiao Wu. Their work is admirable.

Here is the      code      for Vpython image.