Numerical Analysis - Construction of a Poincare surface of section

[pendulum]

(I am not sure whether I'm posting in the right forum. I apologize if I do)

Does anyone have an alrorithm or a code (in any language) that constructs a Poincare surface of section?

I want to do so for a Hamiltonian model: A mass under the influense of the Henon-Heiles potential. It has to include a symplectic algorithm for the integration of the equations of motion ,but that is no problem. The general idea of the P. surface, is what I'm having problem with.

So even a hint with what it's need to be done to construct it would be fine.
Thanks

 

[lofgran]

Dear fellow scientist,

Thank you for reaching out to the forum for assistance with constructing a Poincare surface of section. This is indeed the appropriate forum for such a question.

I am not able to provide a specific algorithm or code for this task, as it may vary depending on your specific Hamiltonian model and numerical approach. However, I can offer some general guidance and resources that may be helpful to you.

First, let's briefly discuss the concept of a Poincare surface of section. This is a two-dimensional slice of a higher-dimensional phase space that allows us to visualize the behavior of a dynamical system. In the case of a Hamiltonian model, this surface of section is often used to study the periodic orbits and chaotic behavior of the system.

To construct a Poincare surface of section, you will need to integrate the equations of motion for your Hamiltonian model using a symplectic algorithm. This ensures that the numerical solution preserves the symplectic structure of the system, which is essential for accurately capturing the dynamics.

Next, you will need to choose a specific plane in phase space to serve as your surface of section. This can be done by fixing certain variables in your Hamiltonian and plotting the remaining variables as a function of time. The intersection points of the trajectory with this plane will form the points on your Poincare surface.

As for the specific steps and code for constructing the surface, I recommend consulting with a numerical methods textbook or searching for online resources that provide step-by-step instructions for constructing a Poincare surface of section for a Hamiltonian model. You may also find it helpful to look at existing codes and algorithms for similar systems and adapt them to your specific model.

I hope this helps to point you in the right direction. Best of luck with your project!

 

[ravenprp]

Hi there,

Constructing a Poincare surface of section involves plotting the intersection points of a trajectory with a particular plane in phase space. This can be achieved by first integrating the equations of motion using a symplectic algorithm, as you mentioned. Then, at regular intervals of time, you can record the position and momentum of the particle and plot it on the designated plane. This process is repeated for multiple initial conditions to get a clearer picture of the phase space.

As for the algorithm or code, it would depend on the specific language and software you are using. However, a general outline for constructing a Poincare surface of section would be as follows:

1. Define the Hamiltonian function, which in this case would be the Henon-Heiles potential.

2. Choose a plane in phase space that you want to plot the intersection points on. This can be any plane that is perpendicular to one of the coordinate axes.

3. Choose a set of initial conditions, including the position and momentum of the particle.

4. Use a symplectic algorithm, such as the Verlet or Leapfrog method, to integrate the equations of motion for a specific time interval.

5. At regular intervals of time, record the position and momentum of the particle and plot it on the designated plane.

6. Repeat this process for multiple initial conditions to get a clearer picture of the phase space.

I hope this helps guide you in constructing your Poincare surface of section. Best of luck!