Stability of the Vallis System

Abstract

In this paper, a variational method is proposed for obtaining the necessary (and sufficient) stability conditions for perturbed solutions of the system of Vallis equations. This method allows us to establish the necessary conditions for Lyapunov stability. It is effective even in cases when the application of the classical Lyapunov method causes difficulties associated with the construction of the Lyapunov function or with inaccuracies of Taylor linearization, which is typical for dynamical systems of large dimension.
In some cases, this method can be used to find regions of phase variables in which the necessary stability conditions coincide with sufficient stability conditions (asymptotic stability) according to Lyapunov.
In these cases, the metric function itself, as shown in this paper, can play the role of the Lyapunov function to obtain sufficient stability conditions.

Author Biographies

Vladimir Vadimovich Nefedov, Lomonosov Moscow State University

Associate Professor of the Chair of Automation for Scientific Research, Faculty of Computational Mathematics and Cybernetics, Cand.Sci. (Phys.-Math.), Associate Professor

Vasiliy Vasilevich Tikhomirov, Lomonosov Moscow State University

Associate Professor of the Chair of General Mathematics, Faculty of Computational Mathematics and Cybernetics, Cand.Sci. (Phys.-Math.), Associate Professor

Yana Dmitrievna Maximova, Lomonosov Moscow State University

student at the Faculty of Computational Mathematics and Cybernetics

References

1. Vallis G.K. Conceptual models of El Niño and the Southern Oscillation. Journal of Geophysical Research: Oceans. 1988; 93(C11):13979-13991. (In Eng.) doi: https://doi.org/10.1029/JC093iC11p13979
2. Smol'yakov E.R. Jeffektivnyj metod ustojchivosti sushhestvenno nelinejnyh dinamicheskih sistem [An efficient method of stability analysis for highly nonlinear dynamic systems]. Kibernetika i sistemnyj analiz = Cybernetics and Systems Analysis. 2019; 55(4):15-23. (In Russ.)
3. Vallis G.K. El Niño: A Chaotic Dynamical System? Science. 1986; 232(4747):243-245. (In Eng.) doi: https://doi.org/10.1126/science.232.4747.243
4. Tikhomirov V.V., Isaev R.R. Application of the variational method for studying stability of the 3D Lotka – Volterra system. Proceedings of the International Conference on Actual Problems of Applied Mathematics, Informatics and Mechanics. VSU, Voronezh; 2021. p. 17-21. Available at: https://www.elibrary.ru/item.asp?id=46371385 (accessed 10.01.2022). (In Eng.)
5. Tikhomirov V.V., Isaev R.R., Maltseva A.V., Nefedov V.V. Stability of the Lorentz System. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2021; 17(2):241-249. (In Russ., abstract in Eng.) doi: https://doi.org/10.25559/SITITO.17.202102.241-249
6. Kalantarov V.K., Yilmaz Y. Decay and growth estimates for solutions of second-order and third-order differential-operator equations. Nonlinear Analysis: Theory, Methods & Applications. 2013; 89:1-7. (In Eng.) doi: https://doi.org/10.1016/j.na.2013.04.016
7. Lorenz E.N. Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences. 1963; 20:130-141. Available at: https://cdanfort.w3.uvm.edu/research/lorenz-1963.pdf (accessed 10.01.2022). (In Eng.)
8. Arnold V.I. Teorija katastrof [Catastrophe Theory]. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Vol. 5. VINITI, Moscow; 1985. p. 5-218. (In Russ.)
9. Zhu C., Liu Y., Guo Y. Theoretic and Numerical Study of a New Chaotic System. Intelligent Information Management. 2010; 2(2):104-109. (In Eng.) doi: https://doi.org/10.4236/iim.2010.22013
10. Wang C., Liu J.G., Johnston H. Analysis of a fourth order finite difference method for the incompressible Boussinesq equations. Numerische Mathematik. 2004; 97(3):555-594. (In Eng.) doi: https://doi.org/10.1007/s00211-003-0508-3
11. Magnitskii N.A., Sidorov S.V. Transition to chaos in nonlinear dynamical systems described by ordinary differential equations. Computational Mathematics and Modeling. 2007; 18(2):128-147. (In Eng.) doi: https://doi.org/10.1007/s10598-007-0014-z
12. Sidorov S.V. Structure of solutions and dynamic chaos in nonlinear differential equations. Vestnik Rossijskogo universiteta druzhby narodov. Serija: Matematika, informatika, fizika = RUDN Journal of Mathematics, Information Sciences and Physics. 2013; (2):45-63. Available at: https://www.elibrary.ru/item.asp?id=18932923 (accessed 10.01.2022). (In Russ., abstract in Eng.)
13. Magnitskii N.A. Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations. Communications in Nonlinear Science and Numerical Simulation. 2008; 13(2):416-433. (In Eng.) doi: https://doi.org/10.1016/j.cnsns.2006.05.006
14. Evstigneev N.M., Magnitskii N.A., Sidorov, S.V. On the nature of turbulence in Rayleigh-Benard convection. Differential Equations. 2009; 45(6):909-912. (In Eng.) doi: https://doi.org/10.1134/S0012266109060135
15. Evstigneev N.M., Magnitskii N.A. On possible scenarios of the transition to turbulence in Rayleigh-Bénard convection. Doklady Mathematics. 2010; 82(1):659-662. (In Eng.) doi: https://doi.org/10.1134/S106456241004040X
16. Kaloshin D.A., Magnitskii N.A. A Complete Bifurcation Diagram of Nonlocal Bifurcations of Singular Points in the Lorenz System. Computational Mathematics and Modeling. 2011; 22(4):444-453. (In Eng.) doi: https://doi.org/10.1007/s10598-011-9112-z
17. Kaloshin D.A. On the construction of a bifurcation surface of existence of heteroclinic saddle-focus contours in the Lorenz system. Differential Equations. 2004; 40(12):1790-1793. (In Eng.) doi: https://doi.org/10.1007/s10625-005-0112-7
18. Chen X. Lorenz Equations Part I: Existence and Nonexistence of Homoclinic Orbits. SIAM Journal on Mathematical Analysis. 1996; 27(4):1057-1069. (In Eng.) doi: https://doi.org/10.1137/S0036141094264414
19. Evstigneev N.M., Magnitskii N.A., Sidorov S.V. Nonlinear dynamics of laminar-turbulent transition in three dimensional Rayleigh-Benard convection. Communications in Nonlinear Science and Numerical Simulation. 2010; 15(10):2851-2859. (In Eng.) doi: https://doi.org/10.1016/j.cnsns.2009.10.022
20. Hirata Y., Judd K. Constructing dynamical systems with specified symbolic dynamics. Chaos. 2005; 15(3):033102. (In Eng.) doi: https://doi.org/10.1063/1.1944467
21. Guckenheimer J., Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences. Vol. 42. Springer, New York, NY; 1983. 462 p. (In Eng.) doi: https://doi.org/10.1007/978-1-4612-1140-2
22. Shil'nikov A., Shil'nikov L., Turaev D. Normal forms and Lorenz attractors. International Journal of Bifurcation and Chaos. 1993; 03(05):1123-1139. (In Eng.) doi: https://doi.org/10.1142/S0218127493000933
23. Sparrou C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Applied Mathematical Sciences. Vol. 41. Springer, New York, NY; 1982. 270 p. (In Eng.) doi: https://doi.org/10.1007/978-1-4612-5767-7
24. Hirsch M.W., Smale S., Devaney R.L. Differential Equations, Dynamical Systems, and an Introduction to Chaos. 3rd ed. Academic Press, Elsevier; 2013. 432 p. (In Eng.) doi: https://doi.org/10.1016/C2009-0-61160-0
25. Firsov A.N., Inovenkov I.N., Tikhomirov V.V., Nefedov V.V. Numerical Study of the Effect of Stochastic Disturbances on the Behavior of Solutions of Some Differential Equations. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2021; 17(1):37-43. (In Eng.) doi: https://doi.org/10.25559/SITITO.17.202101.730
Published
2022-03-31
How to Cite
NEFEDOV, Vladimir Vadimovich; TIKHOMIROV, Vasiliy Vasilevich; MAXIMOVA, Yana Dmitrievna. Stability of the Vallis System. Modern Information Technologies and IT-Education, [S.l.], v. 18, n. 1, p. 13-19, mar. 2022. ISSN 2411-1473. Available at: <http://sitito.cs.msu.ru/index.php/SITITO/article/view/838>. Date accessed: 09 oct. 2025. doi: https://doi.org/10.25559/SITITO.18.202201.13-19.
Section
Theoretical Questions of Computer Science, Computer Mathematics