Poincare section and bifurcation diagram

[poiuy]

Homework Statement

This is the bouncing ball experiment, a circuit is used as an analog of a ball bouncing on an oscillating table. The amplitude of the tables oscillations is varied and data representing the balls position and velocity is gathered.

I have to plot a poincare section for when the ball is bouncing with a single period, another once period doubling is observed and another once it goes to chaos. Then I have to plot a bifurcation diagram.

I know from my phase-space maps that my data is good and that I have actually observed period doubling and chaos

Homework Equations

None

The Attempt at a Solution

I have read every book in my university library on the subject and have been trawling the web for hours, I can find plenty of poincare sections and bifurcation diagrams but nothing about how to extract the data which needs to be plotted.

It was explained to me that for the poincare section a sample is taken of the ball velocity and ball position for a given table position, and the two plotted against each other. Then another sample is taken exactly one period of the tables osciallation later. This means that for single periodic motion of the ball the poincare section is only one point, then after period doubling it is two points and so on. However every poincare section I can find in a book or in a paper is a whole series of points and not just one. I was also told by our lab supervisor that the bifurcation diagram is just all the poincare sections added together but I don't think that it right.

Please tell me what I need to plot for the poincare section and bifurcation diagram!

 

[anathema]

First of all, great job on doing your research and seeking out resources on the subject. it is important to thoroughly understand the concepts and methods involved in an experiment.

For the poincare section, you are correct in your understanding that it is a plot of the ball's position and velocity at a specific point in time, and then again at a later time that is one period of the table's oscillation. This means that for single periodic motion, you will have only one point on the plot, and for period doubling, you will have two points, and so on. However, as you have noticed, most poincare sections in books and papers show multiple points. This is because they are typically plotting the data from multiple periods of the table's oscillation. This is useful for identifying patterns and trends in the data.

For your experiment, you can choose to plot a single poincare section for each type of motion (single period, period doubling, chaos) or you can plot multiple poincare sections to show the progression from one type of motion to another.

As for the bifurcation diagram, it is not simply a sum of all the poincare sections. It is a plot of the parameter (in this case, the amplitude of the table's oscillation) versus the behavior of the system (in this case, the type of motion observed). This will show how the behavior of the system changes as the parameter is varied.

I hope this helps clarify things for you. Good luck with your experiment!

 

[Moetasim]

A Poincare section is a powerful tool for analyzing the dynamics of a system, particularly in cases where the system exhibits periodic or chaotic behavior. In the bouncing ball experiment, a Poincare section can provide insight into the behavior of the ball as it bounces on an oscillating table. By taking samples of the ball's position and velocity at regular intervals, and plotting them against each other, we can observe how these variables change over time and how they relate to each other. The result is a scatter plot of points, with each point representing a specific state of the system at a given time.

In the case of single periodic motion, as you have described, the Poincare section will only have one point, as the system returns to the same state after each period. However, as the system exhibits period doubling or chaotic behavior, the Poincare section will have multiple points, representing the different states of the system at different times.

A bifurcation diagram, on the other hand, is a plot of the parameter values of a system against the resulting behavior of the system. In the case of the bouncing ball experiment, the parameter would be the amplitude of the table's oscillations, and the resulting behavior would be the number of points on the Poincare section. As the parameter is varied, we can observe how the behavior of the system changes, such as the occurrence of period doubling or chaos.

In order to plot a Poincare section and a bifurcation diagram, you will need to collect data from your experiment and plot it accordingly. For the Poincare section, you will need to take samples of the ball's position and velocity at regular intervals, and plot them against each other. For the bifurcation diagram, you will need to vary the amplitude of the table's oscillations and record the resulting number of points on the Poincare section.

I would recommend consulting with your lab supervisor or a colleague who is familiar with the bouncing ball experiment for guidance on how to collect and plot the data accurately. Additionally, you can also refer to other research papers or sources for examples of Poincare sections and bifurcation diagrams in similar experiments.

Overall, the Poincare section and bifurcation diagram are important tools for analyzing and understanding the dynamics of a system, and I am confident that with proper guidance and further research, you will be able to successfully plot these diagrams for your experiment.