Poincare Return Map

[Nusc]

How do I write u''+u*e^x = 0 as a planar system?

[Nusc]

Okay its:

Let z=u', then z'=u''=-exp(x)*u.

So the planar system is {u'=z,z'=-exp(x)*u}, or in matrix form, U'=A*U, where U is the column vector (u,z) and A is the 2x2 matrix (0,1,-exp(x),0).

What can be said about the Poincare return map on the transversal u>0?