Generalized Stability Analysis of a Non-Linear Dynamic Model

Abstract

In the study of dynamic systems, modellers have usually enquired after the existence, uniqueness, and stability of equilibrium solutions (Casti 1985). Unless a systems model hails from a non-equilibrium tradition, establishing the existence of an equilibrium solution (usually long-term) has been deemed important for finding states in which system components are in some sense well-aligned, but also for ensuring that the model is properly constructed and ‘closed’ from a theoretical perspective. Determining whether or not an equilibrium solution is unique has been considered important for delimiting the scope of solution space that needs to be investigated to understand system behaviour. In non-linear systems, however, unique solutions are unlikely. Moreover, there may be good theoretical reasons for why one should expect to observe multiple equilibria in some complex systems (see Farmer 1993). The stability of a solution and the nature of this stability are analysed to determine whether the solution represents a state the system will persist in or diverge from if shocked and whether it is a state to which all or only some paths will converge. Stability analysis also concerns whether a model is structurally stable over some relevant range of parameter values such as the confidence intervals of the parameter estimates (Wymer 1987; Gandolfo 1992).