Scientific and Technical Journal


ISSN Print 2221-3937
ISSN Online 2221-3805
A significant number of real systems are subject to an exponential distribution, therefore, in order to analyze the influence of innovation on a bistable system, a unified form of entropy (the Renyi entro-py) is used. For this system, using the mathematical apparatus of Markov chains, the Kolmogorov equations are generated, the solution of which makes it possible to determine the dynamics of the system entropy at the time of the phase transition. To simulate changes in system modes, the S-curve is used. The corresponding analytical expressions are obtained. It is shown that at the bifurcation points the system has a local maximum of entropy. To confirm the thesis about local maxima of entropy at bifurcation points, the entropy for the classical two-sided map is determined. Based on the similarity of the entropy curves of a system that experiences an innovative influence and the Gartner curve, it is assumed that a particular physical interpretation is possible. Interpretation is the conclusion that the Gartner curve not only adequately de-scribes the stages of the formation of technology, but also describes the information-entropy exchanges pre-sent in the system in the process of its evolution. Examples of digital technologies are shown, the dynamics of their development illustrate the adequacy of the obtained results.
1. Kurzweil, R. (1999), “The Age of Spiritual Machines: When Computers Exceed Human Intelligence”. – Viking, New York, 1999.
2. Schumpeter, J. (1982), “Theory of economy development” [“Teoriya economicheskogo razvitiya”], Progress, Moscow.
3. Kondratiev, N. (2002), “Large cycles of conjuncture and prediction theory” [“Bolshiye cycly konyuktury i teoriya predvideniya”], Economica, Moscow, 767 p.
4. Prigogine, I., Stengers, I. (1984), “ORDER OUT OF CHAOS Man's new dialogue with nature”, Heinemann, London.
5. Nikolis, J. (1989), “Dynamics of Hierarhical Systems. An Evolutionary Approach” [“Dinamika iyerarhicheskih system: evolyucionnoye predstavleniye”, per s angl., predisl. Kadomcev, B.], Mir, Moscow, 488 p.
6. Haken, G. (1985), “Synergetics: hierarchy of instabilities in self-organizing systems and devices” [“Sinergetica. Iyerarhii neustoychivostey v samoorganizuyuschihsya sistemah i ustroystvah”], Mir, Moscow, 424 p.
7. Anischenko, V., Vadivasova T. (2011), “Nonlinear dynamics. Lections” [“Lekcii po nelineynoy dinamike”, ucheb. posobiye dlya vuzov], Regulyarnaya i haoticheskaya dinamica, Izhevsk, 516 p.
8. Shennon, K. (1963), “The work on information theory and cybernetics” [“Raboty po teorii informacii i kibernetike”], Izdatelstvo innostrannoy literatury, Moscow, 830 p.
9. Klimontovich, Y. (2002), “An introduction to open systems physics” [“Vvedeniye v fiziku otkrytyh sistem”], Yanus-K, Moscow, 284 p.
10. Goodman, M. (2016), “Future crimes: inside the digital underground and the battle for our
connected world”, Anchor Books, a division of Penguin Random House, LLC, New York.
11. Yaneer, B. (2015), “Concepts: Power Law”, Complex Systems Institute, New England, 2015.
12. Bashkirov, A. (2006), “Renyi Entropy as statistic entropy for complex systems” [“Entropiya Renii kak statisticheskaya entropiya dlya slozshnih system”], TMF, Vol 149:2, pp. 299-317.
13. Rényi, A. (1960), “On measures of information and entropy”, Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, pp. 547–561.
14. Tihonov, M., Mironov, V. (1977), “Markov chains” [“Markovskiye procesy”], Sovetskoye Radio, Moscow,488p.
15. “Technology Research”, Gartner Inc., available at:
16. “Google Trends”, available at:
Last download:
2018-02-25 20:47:45

[ © KarelWintersky ] [ All articles ] [ All authors ]
[ © Odessa National Polytechnic University, 2014. Any use of information from the site is possible only under the condition that the source link! ]