Multiple-scale analysis for 2D Hamiltonian?

[Random137]

I think all your questions regarding behavior close to epsilon = 0.25 can be answered with suitable numerical experiments using your current computational technique in MMa.

I would be confident that a variable step size Runge-Kutta would likely be able to investigate the dynamics for epsilon = 0.01 to 0.001 and likely even down to 0.0001.

The following contains 500 different trajectories with initial condition satisfying: x(0) = 0, y(0) = 1 with p_x^2(0) + p_y^2(0) = 1

 

[Dr. Courtney]

Very nice.

 

[Random137]

Doing the same thing with a smaller epsilon, it seems like the behaviour is quite different?

How can I extract information from these Poincare plots?

 

[Dr. Courtney]

I think you should compute your long time integrals for the squared momenta for different trajectories, overlay the values with the tori they represent in the graph and look for trends between the tori and resulting integrals.

 

[Random137]

I think you should compute your long time integrals for the squared momenta for different trajectories, overlay the values with the tori they represent in the graph and look for trends between the tori and resulting integrals.

Looking at the 500 trajectories, I calculated <p_x^2> and <p_y^2> for each of the trajectories by evolving the trajectories over a long period of time. Then I arrange the trajectories according to their <p_x^2> value (from smallest to largest).

There seems to be a pattern, similar values of <p_x^2> have similar poincare plots. As <p_x^2> increases, the poincare plots extend further.
http://gph.is/1OOtACG

 

[Dr. Courtney]

Wow! Great work. It seems like you have this one sorted out. Congratulations.

 

[Random137]

Wow! Great work. It seems like you have this one sorted out. Congratulations.

Well, I still don't really know how to find the invariant. I still don't know how to tackle the questions that I want to answer. Do you have any recommendations?

 

[Dr. Courtney]

Well, I still don't really know how to find the invariant. I still don't know how to tackle the questions that I want to answer. Do you have any recommendations?

In general, there usually is not a closed form expression for the constant of motion for the remaining tori. There may be some empirical numerical procedure for determining some expression (in terms of positions and momenta) that is approximately constant. Is this what you are trying to do?

What are the questions you want to answer?

 

[Random137]

In general, there usually is not a closed form expression for the constant of motion for the remaining tori. There may be some empirical numerical procedure for determining some expression (in terms of positions and momenta) that is approximately constant. Is this what you are trying to do?

What are the questions you want to answer?

What I wanted to answer is, for a given set of initial conditions (x(0), y(0), p_x(0), p_y(0)) can I roughly estimate the values of <p_x^2> and <p_y^2> without running the numerical integrations. Actually, I know the value of <p_x^2> + <p_y^2> (= constant * energy) for a given set of initial conditions because of Virial Theorem but I don't know how they are distributed between <p_x^2> and <p_y^2>.

 

[Dr. Courtney]

What I wanted to answer is, for a given set of initial conditions (x(0), y(0), p_x(0), p_y(0)) can I roughly estimate the values of <p_x^2> and <p_y^2> without running the numerical integrations. Actually, I know the value of <p_x^2> + <p_y^2> (= constant * energy) for a given set of initial conditions because of Virial Theorem but I don't know how they are distributed between <p_x^2> and <p_y^2>.

I think the phase space is too complicated to do what you want. All the tori close to a given tori at one point in phase space do not remain close to it. Two orbits close to each other at one point can sample very different regions of phase space over time and thus have very different <p_x^2> and <p_y^2>.

 

[Random137]

I think the phase space is too complicated to do what you want. All the tori close to a given tori at one point in phase space do not remain close to it. Two orbits close to each other at one point can sample very different regions of phase space over time and thus have very different <p_x^2> and <p_y^2>.

Does that mean the system is chaotic?

Would you say in the case where epsilon tending to zero would be easier/feasible to do?

 

[Dr. Courtney]

Does that mean the system is chaotic?

Would you say in the case where epsilon tending to zero would be easier/feasible to do?

The fact that there are so many tori show that the system is not chaotic. The fact that the tori show intricate and complicated patterns prevents you from expressing <p_x^2> and <p_y^2> as simple functions of the initial conditions.

Will the case where epsilon --> zero be easier? Maybe. It depends on how simple the structure of the tori tends to be. Do most of the tori remain next to the ones next to it, are are they a convoluted and tangled mess?

 

[Random137]

The fact that there are so many tori show that the system is not chaotic. The fact that the tori show intricate and complicated patterns prevents you from expressing <p_x^2> and <p_y^2> as simple functions of the initial conditions.

Will the case where epsilon --> zero be easier? Maybe. It depends on how simple the structure of the tori tends to be. Do most of the tori remain next to the ones next to it, are are they a convoluted and tangled mess?

Can you explain what do you mean by tori in the poincare plots?

 

[Dr. Courtney]

When all the intersections of a given trajectory trace out a closed curve, this indicates that the motion is confined to a torus - a shape with the topology of a donut winding around through phase space. Trajectories that are confined to tori give evidence of the existence of a constant of motion in addition to the energy.

 

[Random137]

When all the intersections of a given trajectory trace out a closed curve, this indicates that the motion is confined to a torus - a shape with the topology of a donut winding around through phase space. Trajectories that are confined to tori give evidence of the existence of a constant of motion in addition to the energy.

From the poincare points, it doesn't look it will form a completely closed curve, is it safe to assume it will form a completely closed curve as time tends to infinity?

 

[Dr. Courtney]

The point is more the trajectory is confined to the surface of the tori. Some orbits are periodic in that they retrace the same path and so will make a dotted line rather than closing the curve. Non-periodic orbits will eventually fill in all the gaps to infinitesimally close to each other.