ON NUMERICAL METHODS IN LOCALIZATION PROBLEMS

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

When solving localization problem numerically, the main problem is to construct a universal cross section corresponding to a given localizing function. The paper proposes two methods for solving this problem, which use estimates of the first and second order derivatives. A comparative analysis of these methods with a method based on the use of all nodes of a regular grid was carried out. A comparative analysis shows that the proposed methods are superior both in terms of computational complexity and in the quality of the resulting approximation of the universal section.

About the authors

A. N Kanatnikov

Bauman Moscow State Technical University; V.A. Trapeznikov Institute of Control Sciences of RAS

Email: skipper@bmstu.ru
Moscow, Russia

O. S Tkacheva

Bauman Moscow State Technical University

Email: tkolga17@gmail.com
Moscow, Russia

References

  1. Krishchenko, A.P., Localization of invariant compact sets of dynamical systems, Differ. Equat., 2005, vol. 41, no. 12, pp. 1669–1676.
  2. Kanatnikov, A.N. and Krishchenko, A.P., Invariantnye kompakty dinamicheskikh sistem (Invariant Compact Sets of Dynamical Systems), Moscow: Izd. MGTU im. N.E. Baumana, 2011.
  3. Kanatnikov, A.N. and Krishchenko, A.P., Localizing sets and trajectory behavior, Dokl. Math., 2016, vol. 94, no. 2, pp. 506–509.
  4. Krishchenko, A.P., Localization of simple and complex dynamics in nonlinear systems, Differ. Equat., 2015, vol. 51, no. 11, pp. 1432–1439.
  5. Krishchenko, A.P., Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets, Dokl. Math., 2016, vol. 94, no. 1, pp. 365–368.
  6. Krishchenko, A.P., Construction of Lyapunov functions by the method of localization of invariant compact sets, Differ. Equat., 2017, vol. 53, no. 11, pp. 1413–1418.
  7. Kanatnikov, A.N. and Krishchenko, A.P., Localization of invariant compact sets of nonautonomous systems, Differ. Equat., 2009, vol. 45, no. 1, pp. 46–52.
  8. Kanatnikov, A.N., Korovin, S.K., and Krishchenko, A.P., Localization of invariant compact sets of discrete systems, Dokl. Math., 2010, vol. 81, no. 2, pp. 326–328.
  9. Kanatnikov, A.N. and Krishchenko, A.P., Localization of compact invariant sets of continuous-time systems with disturbance, Dokl. Math., 2012, vol. 86, no. 2, pp. 720–722.
  10. Kanatnikov, A.N., Localization of invariant compact sets in differential inclusions, Differ. Equat., 2015, vol. 51, no. 11, pp. 1425–1431.
  11. Krishchenko, A.P. and Podderegin, O.A., Hopf bifurcation in a predator–prey system with infection, Differ. Equat., 2023, vol. 59, no. 11, pp. 1573–1578.
  12. Coria, L.N. Bounding a domain containing all compact invariant sets of the permanent-magnet motor system / L.N. Coria, K.E. Starkov // Commun. Nonlin. Sci. Numer. Simul. — 2009. — V. 14, № 11. — P. 3879-3888.
  13. Starkov, K.E. Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems / K.E. Starkov // Phys. Lett. A. — 2011. — V. 375, № 36. — P. 3184-3187.
  14. Starkov, K.E. Eradication conditions of infected cell populations in the 7-order HIV model with viral mutations and related results / K.E. Starkov, A.N. Kanatnikov // Mathematics. — 2021. — V. 9, № 16. — Art. 1862.
  15. Starkov, K.E. On the dynamics of immune-tumor conjugates in a four-dimensional tumor model / K.E. Starkov, A.P. Krishchenko // Mathematics. — 2024. — V. 12, № 6. — Art. 843.
  16. Vorkel’, A.A. and Krishchenko, A.P., Numerical analysis of asymptotic stability of equilibrium points, Mathematics Math. Model., 2017, no. 3, pp. 44–63.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2024 Russian Academy of Sciences