The basic principles of stability analysis are set forth, including the development of a mathematical model, steady state analysis, integration of the dynamic system equations, and predictions of transient responses. The equations for the mathematical model are developed for a continuous stirred tank reactor with a cooling jacket, containing an autocatalytic reaction, in unsteady state operation. These equations are numerically integrated to obtain a good picture of reactor dynamics about the steady state. State variable methods, including the classical linearization theorems and Lyapunov's Second Method, are utilized to predict bounds of stable reactor behavior. A discussion of the shortcomings of each method is presented. Finally the results are compared and recommendations are set forth.