Nonlinear differential equations is hard to find good references on-partly due to the difficulty of the subject and partly due to the highly specialized nature of most of the research problems connected with them. But a lot of these problems are really problems of numerical approximation-so I think you'll have greater luck if you begin searching the literature on THAT,math.
A very good book to start with that has a lot of great material on this is Atkinson and Kan's Theoretical Numerical Analysis. Not only is it terrifically written and comprehensive with lots of examples,it's one of the most scholarly texts I've ever seen with complete and opiniated references. I think you'll find this book's references will give you a great deal of direction for further study on nonlinear solution of ODE's.
An older book that has a lot of nice material on power series and other numerical methods for ODE's is Einar Hille's Lectures On Ordinary Differential Equations. Why most of Hille's texts-which are all wonderful-are out of print mystifies me.
That should help you get started,especially the Atkinson/Han book. Good hunting!