I've got a rather long question, and i've done a lot of it, but i've hit a small hurdle and i'm not really sure what it wants me to work out.
and i'm considering it's behaviour on the plane .
The question is this:
I decided to set so I would just be working in the plane.On each line , the dynamics can first be reduced to two equations in two variables (since is identically equal to 0), and then by using , to one equation in one variable.
Write down the equation and solve it.
Find the Poincare map for a time step of t=1.
Describe the dynamics on such a line.
Substitutiing into the equations gives:
Then, using we get:
This shows that for all values of and (since it start at 0 and stays where it is).
We can solve and to get:
and where A and B are constants.
So my solution is:
<----- Is this right?
So now I move on to try and find the Poincare map. I know that:
Now I have to make it periodic so I get:
From here I would just integrate the LHS, but I run into problems with . I tried choosing but I still couldn't integrate that.
Am I approaching the Poincare map in the correct way?