Graphical analysis of nonlinear systems: a modification of the δ-method

Graphical solution methods are particularly valuable for the insight given the student being introduced to the subject of nonlinear systems analysis. The author has found the δ-method to be one of the most attractive methods to his students, because of its conceptual simplicity. The δ-method, however, appears to be somewhat less than ideal, in the following sense. The solution to the system equation is obtained as a sequence of circular arcs, but the approximate solution only matches the slope of the actual solution at a number of points. The graphical information which the curvature of the arcs can convey is not being utilized. If circular arcs are to be used, then it is exceedingly reasonable to seek a modification to the δ-method which provides an approximate solution that matches both the slope and the curvature of the actual solution at a number of points. The following note provides a development of the δ-method which is amenable to modification and a presentation of the modification which adds curvature matching to the solution process.