Geometric Field Stability and Normal Field Curvature of Solution Sets of Ordinary Differential Equations in Two Variables

The classical linearization approach to stability theory determines whether or not a system is stable in the vicinity of its equilibrium points. This classical approach partly depends on the validity of the linear approximation. The definition of stability developed in this article takes a different approach and uses a curvature function to assess the relative locations of solutions within a field of solutions (the underlying solution set of the ODE). The present approach involves calculations that directly yield stability information, without having to enter into the often lengthy eigenvalue-eigenvector method. The present results both complement and are compatible with the classical results based on linearization near an equilibrium point.