ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2018. Vol. 23

Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions

N. A. Sidorov, D. N. Sidorov, Li Yong

The dynamical model consisting of the differential equation with a non- linear operator acting in Banach spaces and a nonlinear operator equation with respect to two elements from different Banach spaces is considered. It is assumed that the system has stationary solutions (rest points). The Cauchy problem with the initial condition with respect to one of the unknown functions is formulated. The second function playing the role of controlling the corresponding nonlinear dynamic process, the initial conditions are not set. Sufficient conditions are obtained for which the problem has the global classical solution stabilizing at infinity to the rest point. Under suitable sufficient conditions it is shown that a solution can be constructed by the method of successive approximations. If the conditions of the main theorem are not satisfied, then several solutions can exists. Some of them can blow-up in a finite time, while others stabilize to a rest point. Examples are given to illustrate the constructed theory.

About the Authors

Nikolai A. Sidorov, Dr. Sci. (Phys.–Math.), Prof., Institute of Mathematics, Economics and Informatics, Irkutsk State University, 1, K. Marx st., Irkutsk, 664003, Russian Federation, e-mail: sidorovisu@gmail.com

Denis N. Sidorov, Dr. Sci. (Phys.–Math.), Lead Research Fellow, Melentiev Energy Systems Institute SB RAS, 130, Lermontov st., Irkutsk, 664033, Russian Federation Institute of Solar-Terrestrial Physics SB RAS, 126a, Lermontov st., Irkutsk, 664033, Russian Federation, e-mail: dsidorov@isem.irk.ru

Yong Li, PhD (Power Engineering), College of Electrical and Information Engineering, Hunan University, Changsha 410082, People’s Republic of China, e-mail: yongli@hnu.edu.cn

For citation:
Sidorov N.A., Sidorov D.N., Li Yong. Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 23, pp. 46-63. (In Russian). https://doi.org/10.26516/1997-7670.2018.23.46
dynamical models, rest point, stability, blow-up, branching, Cauchy problem, bifurcation

1. Barbashin E.A. Introduction to stability theory. M., Libercom, 2014. 230 p. (in Russian)

2. Vainberg M.M., Trenogin V.A. A Theory of branching of solutions of non-linearequations. Leyden, 1974.

3. Voropai N.I., Kurbatsky V.G. et al. Complex of intelligent tools for preventingmajor accidents in electric power systems. Novsibirsk, Nauka, 2016. 332 p. (in Russian)

4. Daleckii Ju.L., Krein M.G. Stability of solutions of differential equations in Banachspace. Ser. “Translations of Mathematical Monographs”, vol. 43. Rhode Island, AMS Publ., 2002. 386 p.

5. Demidovich B.P. Lectures on mathematical stability theory. Moscow, Nauka Publ.,1967. 471 p. (in Russian)

6. Erugin N.P. The Book for Reading on General Course of Differential Equations. Minsk, Nauka i Tekhnika Publ., 1972. 668 p. (in Russian)

7. Matrosov V.M. The comparison principle with a vector-valued Ljapunov function. III. Differ. Uravn., 1969, vol. 5, no 7, pp. 1171-1185.

8. Matrosov V.M. Differential equations and inequalities with discontinuous rightmember. I. Differ. Uravn., 1969, vol. 3, no 3, pp. 395–409.

9. Sidorov N.A., Trenogin V.A. Bifurcation points of nonlinear equation. In the book “Nonlinear analysis and nonlinear differential equations”. Edts V.A. Trenogin and A.F. Filippov. Moscow, Fizmatlit Publ., 2013, pp. 5–50. (in Russian)

10. Sidorov D.N. Methods of analysis of integral dynamical models. Theory and applications. Irkutsk, ISU Publ., 2013. 293 p. (in Russian)

11. Sidorov D.N. Existence and blow-up of Kantorovich principal continuous solutionsof nonlinear integral equations. Differential Equations, 2014, vol. 50, issue 9, pp. 1217-1224. https://doi.org/10.1134/S0012266114090080

12. Sidorov N.A. General issues of regularization in branching problems. Irkutsk, ISU Publ., 1982. 312 p. (in Russian)

13. Trenogin V.A. Functional analysis. Moscow, Fizmatlit Publ., 2002. 488 p. (in Russian)

14. Khalil H.K. Nonlinear systems. Prentice hall, 1991.

15. Ayasun S., Nwankpa C.O., Kwatny H.G. Computation of singular and singularity induced bifurcation points of differential-algebraic power system model. IEEET ransactions on Circuits and Systems - I: Fundamental Theory and Applications, 2004, vol. 51, no 8, pp. 1525–1538. https://doi.org/10.1109/TCSI.2004.832741

16. Buffoni B., Toland J. Analytic Theory of Global Bifurcation: An Introduction. Princeton series in applied mathematics. Princeton University Press, 2003. 169 p. https://doi.org/10.1515/9781400884339

17. Machowski J., Bialek J.W., Bumby J.R. Power system dynamics. Stability andcontrol. Oxford, John Wiley, 2008. 658 p.

18. Milano F. Power system modelling and scripting, Berlin, Springer, 2010. 578 p. https://doi.org/10.1007/978-3-642-13669-6

19. Sidorov D., Sidorov N. Convex majorants method in the theory of nonlinear Volterra equations. Banach J. of Mathematical Analysis, 2014, vol. 6, no 1, pp. 1-10. https://doi.org/10.15352/bjma/1337014661

20. Sjoberg J., Fujimoto K., Glad T. Model reduction of nonlinear differential algebraic equations. IFAC Proceedings Volumes, vol. 40, issue 12, 2007, pp. 176-181. https://doi.org/10.3182/20070822-3-ZA-2920.00030

21. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. Lyapunov-Schmidt methodsin nonlinear analysis and applications. Springer Series: Mathematics and Its Applications, 2013, vol. 550. 568 p. https://doi.org/10.1007/978-94-017-2122-6

22. Sidorov D. Integral Dynamical Models: Singularities, Signals and Control Ed. by L. O. Chua, Singapore, London: World Scientific Publ., 2015,vol. 87 of World Scientific Series on Nonlinear Science, Series A. 258 p. https://doi.org/10.1142/9789814619196_bmatter

23. Sidorov D.N., Sidorov N.A. Solution of irregular systems of partial differential equations using skeleton decomposition of linear operators. Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2017, vol. 10, no 2, pp. 63-73. https://doi.org/10.14529/mmp170205

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