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.

I have:

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?