Ferdinand Verhulst 
Nonlinear Differential Equations and Dynamical Systems [PDF ebook] 

On the subject of differential equations a great many elementary books have been written. This book bridges the gap between elementary courses and the research literature. The basic concepts necessary to study differential equations – critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds – are discussed. Stability theory is developed starting with linearisation methods going back to Lyapunov and Poincare. The global direct method is then discussed. To obtain more quantitative information the Poincare-Lindstedt method is introduced to approximate periodic solutions while at the same time proving existence by the implicit function theorem. The method of averaging is introduced as a general approximation-normalisation method. The last four chapters introduce the reader to relaxation oscillations, bifurcation theory, centre manifolds, chaos in mappings and differential equations, Hamiltonian systems (recurrence, invariant tori, periodic solutions). The book presents the subject material from both the qualitative and the quantitative point of view. There are many examples to illustrate the theory and the reader should be able to start doing research after studying this book.