Analysis of Phase Trajectories of the Third - Order Dynamic Objects

This article is dedicated to a burning problem of classification of phase portrait of a set of three simultaneous linear differential equations rendered in two-dimensional space R2. The suggested classification is a generalization of the existing researches and gives the possibility to distinctly outline dynamic tendencies of the third order linear patterns under the conditions of bifurcation inherent for automatic control systems (ACS) with changeable structure. The analysis of phase paths of the third-order linear system which is described by nondegenerated system of differential equations around single regular balance point M(0;0;0) and also phase paths with composite points of codimension 1 and codimension 2. It has been shown that the paths appearance depends upon the structure of characteristic matrix of the system of its own figures and the projections of phase portraits in phase space R3 on specific surfaces correspond to phase portraits of linear system of other order. Within this framework, phase portraits of the third-order linear system have been assigned names. The system of nondegenerated type of balance points with corresponding analytical conditions of origin and names is represented in a tabulated form. Special focus was made on the case of zero characteristic matrix which corresponds to phase portrait consisting of stationary point only. Application of the suggested classification system has been considered by the example of the automatic control system of the voltage of the DC generator. The sequence of actions and conclusion as per dynamic properties of ACS based on its phase portrait has been shown.