Some basic theorems of the geometric theory of differential equations are reviewed, without proofs, in an attempt to clarify: (a) what relationship exists between the general solution of a set of nonlinear differential equations and the solution of its linear approximation and under what conditions this relationship can be used; and (b) how the geometric theory can be used to find properties of boundedness, stability, and periodicity of the solutions of nonlinear differential systems. These theorems are illustrated by means of two-third order examples. The first is the xenon controlled reactor and the second a two-region reactor with two temperature coefficients of reactivity. It is shown without involved computations or any approximations that: (a) Xenon controlled reactor—when the reactivity controlled by xenon is smaller than the prompt xenon yield, the reactor power is always bounded but periodic oscillations may arise. When the reactivity controlled by xenon is greater than the prompt xenon yield the reactor power is unbounded; (b) Two-region reactor—this reactor does not admit periodic solutions. When the temperature coeffi.cients are of opposite sign, conditions are derived for the reactor power to be bounded.