Thus far, we have presented methods of solution primarily for linear dynamical systems. Here, we address nonlinear ones, albeit for the limited case in which the nonlinear terms are suitably smaller than the linear ones. Approximate solutions in this case may be obtained by using asymptotic expansions (specifically, singular perturbation methods). Two of them (multiple–scale and the normal–form methods) are addressed in detail. Much emphasis is given to the so–called Hopf bifurcations and limit cycles.