1. Abstract
We have seen that at low driving forces the damped, nonlinear pendulum exhibits simple oscillatory motion, while at high drive it can be chaotic. However, when we consider this question again, the question must come out, that is, exactly how does the transition from simple to chaotic behavior take place? This is the main problem that we will deal with in the following content. Of course, we also will disscuss when the driving force is larger than 1.2, what will happen then?
I'm so sorry that the vpython didn't work on my computer. So, I have to use upperclassman's GIF.
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.
2. Background and Introduction
At last homework, we have ever taked about the phenomenon-chaos and we have disscussed a example of Poincare section-strange attractor. When the system is chaos, assume that
Quickly, it's Poincare section just like below.
But now, we will introduce more examples of Poincare section.
Bifurcation diagram is very helpful to analyze the transition to chaos. It can show us lines for \theta as a function of drive amplitude, which was constructed in the following manner. For each value of F_D we have calculated \theta as a function of time. After waiting for 300 driving periods so that the initial transients have decayed away, we plotted \theta at times that were in phase with the driving force as a function of F_D. Here we plotted points up to the 428th driving period.
After that, we can calculate the Feigenbaum \delta parameter through following formula
In theory, when n levels off to infinity, \delta=4.669.
3. Body Content and Conclusion
1. Period Doubling with different values of the drive amplitude.
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.
Here Click the Code!
Surprisingly, when F_D=1.2, the system is in chaotic state. When F_D=1.35, the system become well-aligned again, its period is the same as the drive period. When F_D=1.44, its period is the twice as the drive period. When F_D=1.465, its period is the fourth times as the drive period.
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Conclusion: If we were to increase the driven amplitude further, the period would double again as the pendulum would switch to a motion that has a period eight times that of the drive. The period-doubling cascade would continue if the drive were increased further.
2.Question 3.18
If the time, which we plot, satisfys conditions below, from above figure, there must be some points instead of lines.
Click the Code
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Analysis: As the same as 1., when F_D=1.4, there is just one point. When F_D=1.44, there are two points. When F_D=1.465, there are four points. Be careful, my each figure has two points that are used to mark conditions!
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Conclusion: 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.
Click the Code
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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.
In fact, if we didn't set the values of each axis, one point will be a line, why is it happens?
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As a matter of fact, computer can't give accurate \pi, for example, it may give 3.141592654. Thus, in a microscopic regime, it may not a point, just like what I showed you above.
3. 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.
Click the Code
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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.
In this part, the other probelm is to find the Feigenbaum \delta. It requires high accuracy. So, I spend almost an hour to get the result.
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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).
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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.
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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.