Analytical Modeling of the Dynamic System of the Fourth Order

Abstract

A canonical mathematical model of a fourth-order dynamical system in the form of A.M. Letov. The analytical modeling methods are based on the algebraic concept and the principle of symmetry. The symmetry principle is realized on the set of four indices of the roots of the characteristic equation and the set of four indices of the phase coordinates of the dynamic system.

The problem of the quality of dynamic processes in time is reduced to the algebraic problem of distribution of four roots in the complex plane. An analogy is established in the procedure for transforming the characteristic determinant to a polynomial and elementary symmetric polynomials of four roots. On the basis of the theory of residues, a new form of analytical representation of data in time is obtained in the form of ordered determinants with respect to the indices of four roots and indices of four coordinates.

General provisions are illustrated by a stochastic dynamical system in the form of an asymmetric Markov chain with four states and continuous time, which is described by the fourth-order Kolmogorov equations.