TMLAI-9947.pdf

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Transactions on Machine Learning and Artificial Intelligence - Vol. 9, No. 3

Publication Date: June, 25, 2021

DOI:10.14738/tmlai.93.9947.

Kravets, V., Kravets, V., & Burov, O. (2021). Analytical Modeling of the Dynamic System of the Fourth Order. Transactions on

Machine Learning and Artificial Intelligence, 9(3). 14-24.

Services for Science and Education – United Kingdom

Analytical Modeling of the Dynamic System of the Fourth Order

Victor Kravets

Department of Mechanics

Dnipro University of Technology, Dnipro, Ukraine

Volodymyr Kravets

Department of Mechanics

Dnipro State Agrarian and Economic University, Dnipro, Ukraine

Olexiy Burov

Jack Baskin School of Engineering

University of California-Santa Cruz, United States

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.

Keywords: Mathematical model, symmetric polynomials, central symmetry, phase

coordinates.

INTRODUCTION

Mathematical modeling as a tool for solving problems of analysis and synthesis of deterministic

or stochastic dynamic systems is developing in a complex way, combining the improvement of

mathematical methods and computer technologies.

Analytical modeling is an important stage in the dynamic design of technical systems, which

precedes computational and full-scale experiments [1-4].

The central place in analytical modeling is occupied by a correctly composed mathematical

model that adequately describes deterministic or stochastic processes in differential form,

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Kravets, V., Kravets, V., & Burov, O. (2021). Analytical Modeling of the Dynamic System of the Fourth Order. Transactions on Machine Learning and

Artificial Intelligence, 9(3). 14-24.

URL: http://dx.doi.org/10.14738/tmlai.93.9947.

relying on fundamental laws and heuristic postulates. Various mathematical models of

dynamical systems are represented by the canonical, matrix form [5].

Analytical modeling is carried out on the basis of classical mathematical methods for solving

systems of linear differential equations, operational calculus, theory of residues [6, 7].

Analytical modeling of a second-order linear dynamical system is reduced to the algebraic

problem of determining the roots of a quadratic characteristic equation and the subsequent

analysis of differential equations [8-11].

With an increase in the order of the dynamical system, analytical modeling encounters the

problem of solvability of the characteristic equations of high degrees [5].

In this paper, using the example of a fourth-order dynamical system, we show the possibility of

overcoming this fundamental limitation by using the proposed special solution of the complete

algebraic equation of the fourth degree [12].

Approbation of the method is illustrated by considering an asymmetric Markov chain with four

states and continuous time, which is described by the Kolmogorov equations [13].

The proposed mathematical models make it possible to expand the range of problems of

analysis and synthesis of dynamical systems, simulated analytically.

SETTING UP A PROBLEM

A wide class of problems in the dynamics of both deterministic and stochastic design schemes

of objects is reduced to the consideration of mathematical models in the form of systems of

linear, homogeneous differential equations with constant coefficients for a given state of the

object at the initial moment. [1,4,9].

As an illustration, a mathematical model of the fourth order in canonical form is considered [5]:

ẋ1 = a11x1 + a12x2 + a13x3 + a14x4

ẋ2 = a21x1 + a22x2 + a23x3 + a24x4

ẋ3 = a31x1 + a32x2 + a33x3 + a34x4

ẋ4 = a41x1 + a42x2 + a43x3 + a44x4

(1)

We are looking for the possibility of analytical representation of dynamic processes in an

ordered, universal form, convenient for solving engineering problems of analysis and synthesis.

ANALYTICAL SOLUTION

The analytical solution of the original system of differential equations is reduced to the

algebraic problem of determining the roots of the characteristic equation [7]:

a11 −  a12

a21 a22 − 

a13 a14

a23 a24

a31 a32

a41 a42

a33 −  a34

a43 a44 − 

| = 0. (2)

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Transactions on Machine Learning and Artificial Intelligence (TMLAI) Vol 9, Issue 3, June - 2021

Services for Science and Education – United Kingdom

Here the characteristic determinant is transformed to a fourth degree polynomial:

∑a4−i

i=0



4−i = 0,

(3)

(i = 0, 1, 2, 3, 4).

According to Vieta's formulas, the coefficients of an algebraic equation of the fourth degree are

related to the four roots of the characteristic equation 1

, 2

, 3

, 4 by the following

symmetric polynomials [5]:

i = 4: a0

4 = (−1)

0∏j

j=1

(4)

i = 3: a1

4 = (−1)

1 ∑ jks

j,k,s=1

(j<k<s)

, (5)

i = 2: a2

4 = (−1)

2 ∑ jk

j,k=1

(j<k)

, (6)

i = 1: a3

4 = (−1)

3∑j

j=1

, (7)

i = 0: a4

4 = (−1)

. (8)

It is easy to show the similarity of the structures of the formulas for the coefficients of the

characteristic equation of the fourth degree, expressed by symmetric polynomials of the roots

and symmetric polynomials of special determinants of the fourth order, constructed from the

columns of the original characteristic determinant as follows:

i = 0: a4

4 = |

−1

|, (9)