<div>
	METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS</div>

Roman O. Omorov

doi:10.17586/2226-1494-2020-20-2-257-262

2020 , VOLUME 20, NUMBER 2 ( march-april )

ISSN 2226-1494 (print), ISSN 2500-0373 (online)

Publications

Editor-in-Chief

Nikiforov
Vladimir O.
D.Sc., Prof.

Partners

doi: 10.17586/2226-1494-2020-20-2-257-262

METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS

R. O. Omorov

Read the full article

Article in Russian

For citation:

Omorov R.O. Method of topological roughness of dynamic systems: applications to synergetic systems. Scientiﬁc and Technical Journal of Information Technologies, Mechanics and Optics, 2020, vol. 20, no. 2, pp. 257–262 (in Russian). doi: 10.17586/2226-1494-2020-20-2-257-262

Abstract

The paper presents a method of dynamic system roughness research, based on Andronov-Pontryagin concept of roughness (method of topological roughness). Andronov-Pontryagin concept of roughness has been formulated. Reachability conditions of dynamic system required roughness are deﬁned. Concept deﬁnition for maximum roughness and minimum non-roughness of dynamic systems is given. Theorems on necessary and sufﬁcient conditions of eachability of maximum roughness and minimum non-roughness and occurrence of bifurcations of dynamic system topological structures are formulated. It is claimed that the sets of rough and non-rough systems are continuous in terms of the set roughness. The condition number of the matrix of bringing to the diagonal (quasi-diagonal) view of the Jacobi matrix at special points of the system phase space is used as an indicator of roughness. The method gives the possibility to control the roughness of control systems based on a theorem formulated using Sylvester’s matrix equation. The basic concepts on synergetics and synergetic systems are presented. The method can be used for studies of roughness and bifurcations of dynamic systems, as well as synergetic systems and chaos of various physical nature. The method is tested on the examples of many synergetic systems: Lorenz and Rössler, Belousov-Zhabotinsky, Chua, “predator-prey”, Henon, and Hopf bifurcation. The main provisions of the topological roughness method are given. The possibilities of the method are illustrated by examples of Belousov-Zhabotinsky and Chua synergetic systems.

Keywords: dynamic system, topological roughness, synergetic system, Andronov-Pontryagin roughness, bifurcation, maximum roughness and minimum non-roughness of systems, hyperbolic and non-hyperbolic special points

References

Andronov A.A., Pontriagin L.S. Structurally stable systems. Doklady AN SSSR, 1937, vol. 14, no. 5, pp. 247–250. (inRussian)
Anosov D.V. Structurally stable systems. Proceedings of the Steklov Institute of Mathematics, 1986, vol. 169, pp. 61–95.
Poliak B.T., Tcypkin Ya.Z. Robust stability of linear systems. Itogi nauki i tehniki. Tehnicheskaja kibernetika, 1991, vol. 32, pp. 3–31. (inRussian)
Omorov R.O., Ushakov A.V. Robustness estimates in control and observation tasks. Russian Electromechanics, 1991, no. 1, pp. 78–85. (in Russian)
Peixoto M.M. On structural stability. Annals of Mathematics, 1959, vol. 69, no. 1, pp. 199–222. doi: 10.2307/1970100
Omorov R.O. Maximal robustness of dynamical systems. Automation and Remote Control, 1991, vol. 52, no. 8, pp. 1061–1068.
Omorov R.O. Topological roughness of synergetic systems. Journal of Automation and Information Sciences, 2012, vol. 44, no. 4, pp. 61–70. doi: 10.1615/JAutomatInfScien.v44.i4.70
Omorov R.O. Theory of Topological Roughness of Systems. Bishkek, Ilim Publ., 2019, 288 p. (in Russian)
Poincaré H. Mémoire sur les courbes définies par une équation différentielle. Journal de Mathématiques Pures et Appliquées. 1881-1886. (in French)
Himmelblau D.M. Applied Nonlinear Programming. McGraw-Hill, 1972, 498 p.
Omorov R.O. Dynamical system quantitative robustness measures and their applications to control systems. Dissertation for the degree of candidate of technical sciences, St. Petersburg, Saint Petersburg Institute of Fine Mechanics and Optics, 1992, 188 p. (in Russian)
Haken H. Advanced Synergetics. Berlin, Springer, 1983.
Nicolis G., Prigogine I. Exploring Complexity: An Introduction. W.H. Freeman, 1989, 313 p.
Strange attractors. Moscow, Mir Publ., 1981, 253 p. (in Russian)
Kondepudi D.K., Prigogine I. Modern Thermodynamics: From Heat Engines to Dissipative Structures. John Wiley, 1998, 486 p.
Zhang W.-B. Synergetic Economics: Time and Change in Nonlinear Economics. Springer-Verlag, 1991, 246 p.
Kapitca S.P., Kurdiumov S.P., Malinetckii G.G. Synergetics and Future Forecasts. Moscow, URSS Publ., 2001, 288 p. (in Russian)
Leonov G.A., Kuznetsov N.V., Kudryashova E.V. Tunisia 2011-2014. Bifurcation, revolution, and controlled stabilization. Vestnik Sankt-Peterburgskogo universiteta seriya 10 Prikladnaya matematika informatika protsessy upravleniya, 2016, no. 4, pp. 92–103. (in Russian). doi: 10.21638/11701/spbu10.2016.409
Andrievskii B.R., Fradkov A.L. Control of chaos: methods and applications. I. Methods. Automation and Remote Control, 2003, vol. 64, no. 5, pp. 673–713. doi: 10.1023/A:1023684619933
Kolesnikov A.A. Synergetic Control Methods for Complex Systems: System Synthesis Theory. Moscow, LIBROKOM Publ., 2012, 240 p. (in Russian)
Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A. Algorithms for finding hidden oscillations in nonlinear systems. the Aizerman and Kalman conjectures and Chua's circuits. Journal of Computer and Systems Sciences International, 2011, vol. 50, no. 4, pp. 511–543. doi: 10.1134/S106423071104006X

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License