004.942 Non-autonomous system as a model of the production process of technical innovation

Markova Y. N. (АУ «Технопарк–Мордовия»), Gavryushin S. S. (Bauman Moscow State Technical University), Belov V. F. (МГУ им. Н.П. Огарева/АУ «Технопарк–Мордовия»)


doi: 10.18698/2309-3684-2021-1-110131

Development of an analytical method for economic characteristics variation in an innovation process is a challenging task, starting from invention or entrepreneur idea and finishing with market implementation. The practical purpose here is both to minimize the risks and shorten design and implementation time. Theoretical and experimental results presented in the paper show the possibility to solve the mentioned task via applying a non-autonomous differential equation system along with the Lyapunov's First Method for stability analysis. A mathematical model of flow of funds related to the manufacturing and market implementation of the engineering innovation has been studied. It is presented in the form of the system of differential balance equations with unit impulse function in the right-hand side. An algorithm has been developed to analyze the stability of the equilibrium states of the manufacturing process, considering the influence of the external environment “from the right” (preparation of manufacturing) as well as “from the left” (market state). The requirements are determined regarding both the discrete model of manufacturing preparation to formulate initial conditions for the non-autonomous system and the discrete model of the market to calculate the market state dependent coefficients for the system of differential equations. The results of analysis of the economic characteristics variation in the stage of product manufacturing have been presented as 3D phase images.

Oslo Manual 2018: Guidelines for Collecting, Reporting and Using Data on Innovation, 4th Edition. Paris/Eurostat, Luxembourg, OECD Publ., 2018, 258 p.
Silkina G.Yu. Natural science categories in modelling the diffusion of innovations. Bulletin of South Ural State University. Series. Economics and Management, 2013, vol.7, no.2, pp.95–103.
Makarov V.L. Overview of mathematical models of economy with innovation. Economics and Mathematical Methods, 2009, vol.45, no.1, pp.3–14.
Bilal Nawaf Elian Suleiman. Matematicheskoe modelirovanie innovacionnyh processov na osnove avtonomnyh dinamicheskih sistem: dissertaciya na soiskanie uchenoj stepeni kandidata fiziko-matematicheskih nauk [Mathematical modeling of innovative processes based on autonomous dynamic systems: dissertation for the degree of Candidate of Physical and Mathematical Sciences]. Belgorod State University. Belgorod, 2012, 181 p.
Belov V.F., Gavryushin S.S., Markova Y.N. A mathematical model of distributed prototype design in mechanical engineering. BMSTU Journal of Mechanical Engineering, 2019, no.9, pp.7–19.
Kalitin B.S. Ustojchivost' neavtonomnyh differencial'nyh uravnenij [Stability of non-autonomous differential equations]. Minsk, BSU Publ., 2013, 227 p.
Kalitin B. S., Chabour R. Ob ustojchivosti diskretnyh neavtonomnyh sistem [On the stability of discrete non-autonomous systems]. XII vserossijskoe soveshchanie po problemam upravleniya VSPU–2014. Institut problem upravleniya im. V.A. Trapeznikova RAN [The XII All-Russian meeting on the problems of VSPU management–2014. V.A. Trapeznikov Institute of Management Problems of the Russian Academy of Sciences], Moscow, 2014, pp.868–881.
Demidovich B.P., Modenov V.P. Differencial'nye uravneniya [Differential equations]. St. Petersburg, Lan' Publ., 2008, 288 p.
Il'ichev V.G. Local and global properties of nonautonomous dynamical systems and their application to competition models. Siberian Mathematical Journal, vol.44, no.3, pp.490–499.
Lasunsky A.V. Methods of investigating the stability of equilibrium positions in the nonautonomous systems, and some applications theirof. Transactions of Karelian Research Centre of Russian Academy of Science. Mathematical Modeling and Information Technologies, 2011, no.5, pp.38–44.
Aleksandrov A.Yu., Kosov A.A. Stability analysis of equilibrium positions of nonlinear mechanical systems by means of decomposition. Vestnik of Saint Petersburg University. Applied mathematics. Computer science. Control processes, 2009, no.1, pp.143–154.
Aleksandrov A.Y., Kosov A.A. Stability and stabilization of equilibrium positions of nonlinear nonautonomous mechanical systems. Journal of Computer and Systems Sciences International, 2009, vol.48, no.4, pp.511–520.
Ratnadip Adhikari. A Treatise on stability of autonomous and non-autonomous systems: theory and illustrative practical applications paperback. Saarbruecken, LAP LAMBERT Academic Publ., 2013, 84 p.
Zhabko A.P., Kotina E.D., Chizhova O.N. Differencial'nye uravneniya i ustojchivost' [Differential equations and stability]. St. Petersburg, Lan' Publ., 2015, 320 p.
Kurakin L.G., Ostrovskaya I.V. Elementy teorii ustojchivosti [Elements of the stability theory]. Rostov-on-Don, Southern Federal University Publ., 2016, 60 p.

Белов В.Ф., Гаврюшин С.С., Маркова Ю.Н. Неавтономная система как модель процесса производства технической инновации. Математическое моделирование и численные методы, 2021, № 1, с. 110–131.

Download article

Количество скачиваний: 123