| Stability properties of the zero solution of a scalar ordinary differential equation are studied using functions, called fences, which act as bounds to the solutions of the differential equation. It is shown, under simple restrictions, that these fences are bounds for the solutions of the perturbed differential equations as well. Hence one finds that total stability of the zero solution of a scalar ordinary differential equation is equivalent to the existence of these functions. In several cases the isoclines, which arise from the right hand side of the differential equation can be used as the fences for the solutions of the differential equation. Hence the isoclines can also be used to show that the zero solution is uniformly totally stable.;Many properties have been studied using numerical approximations. It is shown, for scalar autonomous differential equations, that uniform total stability is preserved under numerical one step methods. The extension to the nonautonomous; case can not, in general, be done using the one step methods. This extension will be considered at a later date.;Bifurcation can be studied using radii of stability and instability. A number of the properties of the radii of stability and instability are considered in this thesis. |
