Graphical Analysis of a General Class of Third Order Non-Linear Systems

Abstract

The following method is proposed for graphical analysis of various third order non-linear systems, defined by

The technique involves treating ẋ as a parameter and introduces dummy variables, such as N = ẍ/ẋ = dẋ/dx and M = dN/dx, to obtain a single plane, to be known as the (N, x) master plane, which would contain all the data about the variation and initial conditions of all the three variables, N, ẋ and x. The variation of ẋ is incorporated into the (N, x) plane by effecting a change of x-scale, and change of the N-axis for various ẋ values. The given examples illustrate the use of this master plane and a simple procedure, by which the required (x, ẋ) and (N, x) trajectories for a third order system are obtained. It has been shown that this technique is also applicable to system defined by where f₄ is a composite non-linear function having unique values for specific ẋ and ẍ.