Home Note on a parameter switching method for nonlinear ODEs
Article
Licensed
Unlicensed Requires Authentication

Note on a parameter switching method for nonlinear ODEs

  • Marius-F. Danca EMAIL logo and Michal Fečkan
Published/Copyright: July 5, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 2

Abstract

In this paper we study analytically a parameter switching (PS) algorithm applied to a class of systems of ODE, depending on a single real parameter. The algorithm allows the numerical approximation of any solution of the underlying system by simple periodical switches of the control parameter. Near a general approach of the convergence of the PS algorithm, some dissipative properties are investigated and the dynamical behavior of solutions is investigated with the Lyapunov function method. A numerical example is presented.

MSC 2010: Primary 34K28; Secondary 34C29, 37B25, 34H10
Keywords: numerical approximation of solutions; averaging method; Lyapunov function; chaos control; anticontrol

Dedicated to Professor Anatolij Dvurečenskij on the occasion of his 65th birthday

(Communicated by Jozef Džurina)

M. Fečkan is partially supported by Grants VEGA-MS 1/0071/14 and VEGA-SAV 2/0153/16.

References

[1] ALMEIDA, J.—PERALTA-SALAS, D.—ROMERA, M.: Can two chaotic systems give rise to order?, Phys. D 200 (2005), 124–132.10.1016/j.physd.2004.10.003Search in Google Scholar

[2] DANCA, M.-F.: Convergence of a parameter switching algorithm for a class of nonlinear continuous systems and a generalization of Parrondo’s paradox, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 500–510.10.1016/j.cnsns.2012.08.019Search in Google Scholar

[3] DANCA, M.-F.: Random parameter-switching synthesis of a class of hyperbolic attractors, Chaos 18 (2008), 033111.10.1063/1.2965524Search in Google Scholar PubMed

[4] DANCA, M.-F.—FEČKAN, M.—ROMERA, M.: Chaos control of logistic map by means of a generalized Parrondo’s game, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24 (2014), 1450008, 17 pp.10.1142/S0218127414500084Search in Google Scholar

[5] DANCA, M.-F.—ROMERA, M.—PASTOR, G.—MONTOYA, F.: Finding attractors of continuous-time systems by parameter switching, Nonlinear Dynam. 67 (2012), 2317–2342.10.1007/s11071-011-0172-6Search in Google Scholar

[6] DIECI, L.—LOPEZ, L.: A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side, J. Comput. Appl. Math. 236 (2012), 3967–3991.10.1016/j.cam.2012.02.011Search in Google Scholar

[7] FExČKAN, M.: Topological Degree Approach to Bifurcation Problems (1st ed.), Springer, Berlin, 2008.10.1007/978-1-4020-8724-0Search in Google Scholar

[8] GUCKENHEIMER, J.—HOLMES, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New-York, 1983.10.1007/978-1-4612-1140-2Search in Google Scholar

[9] HAIRER, E.—LUBICH, CH.—WANNER, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.), Springer-Verlag, Berlin, 2006.Search in Google Scholar

[10] HALE, J. K.: Ordinary Differential Equations, Dover Publications, New York, 2009.Search in Google Scholar

[11] MAO, Y.—TANG, W. K. S.—DANCA, M-F.: An averaging model for chaotic system with periodic time-varying parameter, Appl. Math. Comput. 217 (2010), 355–362.10.1016/j.amc.2010.05.068Search in Google Scholar

[12] ROMERA, M.—SMALL, M.—DANCA, M.-F.: Deterministic and random synthesis of discrete chaos, Appl. Math. Comput. 192 (2007), 283–297.10.1016/j.amc.2007.02.142Search in Google Scholar

[13] ROUCHE, N.—HABETS, P.—LALOY, M.: Stability Theory by Liapunov’s Direct Method, Springer-Verlag, New York, 1977.10.1007/978-1-4684-9362-7Search in Google Scholar

[14] RUDIN, W.: Functional Analysis, McGraw-Hill, New York, 1973.Search in Google Scholar

[15] SANDERS, J. A.—VERHULST, F.—MURDOCK, J.: Averaging Methods in Nonlinear Dynamical Systems (2nd ed.), Springer-Verlag, New York, 2007.Search in Google Scholar

[16] STAMOVA, I. M.—STAMOV, G. T.: Stability analysis of differential equations with maximum, Math. Slovaca 63 (2013), 1291–1302.10.2478/s12175-013-0171-9Search in Google Scholar

[17] STUART, A. M.—HUMPHRIES, A. R.: Dynamical Systems and Numerical Analysis, Cambridge Univ. Press, Cambridge, 1998.Search in Google Scholar

[18] TANG, W. K. S.—DANCA, M.-F.: Emulating “Chaos + Chaos = Order” in Chen’s circuit of fractional order by parameter switching, Internat. J. Bifur. Chaos Appl. Sci. Engrg. (To appear).10.1142/S0218127416500966Search in Google Scholar

Received: 2013-5-27
Accepted: 2014-1-5
Published Online: 2016-7-5
Published in Print: 2016-4-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. Holistic logical arguments in quantum computation
  3. Research Article
  4. States with values in the Łukasiewicz groupoid
  5. Research Article
  6. On realization of effect algebras
  7. Research Article
  8. States on symmetric logics: extensions
  9. Research Article
  10. Extensions and measurability in quantum measure spaces
  11. Research Article
  12. Convex subalgebras of upper bounded GMV-algebras
  13. Research Article
  14. Combining boolean algebras and ℓ-groups in the variety generated by chang’s mv-algebra
  15. Research Article
  16. Large rigid sets of algebras with respect to embeddability
  17. Research Article
  18. Conformal algebras, vertex algebras, and the logic of locality
  19. Research Article
  20. Extension of measures on pseudo-D-lattices
  21. Research Article
  22. Note on a parameter switching method for nonlinear ODEs
  23. Research Article
  24. Compatibility for probabilistic theories
  25. Research Article
  26. Completeness of Gelfand-Neumark-Segal inner product space on Jordan algebras
  27. Research Article
  28. Commutativity in a synaptic algebra
  29. Research Article
  30. Discrete averaged mixing applied to the logarithmic distributions
  31. Research Article
  32. Automorphisms of decompositions
https://doi.org/10.1515/ms-2015-0148
Keywords for this article
numerical approximation of solutions; averaging method; Lyapunov function; chaos control; anticontrol

Articles in the same Issue

  1. Research Article
  2. Holistic logical arguments in quantum computation
  3. Research Article
  4. States with values in the Łukasiewicz groupoid
  5. Research Article
  6. On realization of effect algebras
  7. Research Article
  8. States on symmetric logics: extensions
  9. Research Article
  10. Extensions and measurability in quantum measure spaces
  11. Research Article
  12. Convex subalgebras of upper bounded GMV-algebras
  13. Research Article
  14. Combining boolean algebras and ℓ-groups in the variety generated by chang’s mv-algebra
  15. Research Article
  16. Large rigid sets of algebras with respect to embeddability
  17. Research Article
  18. Conformal algebras, vertex algebras, and the logic of locality
  19. Research Article
  20. Extension of measures on pseudo-D-lattices
  21. Research Article
  22. Note on a parameter switching method for nonlinear ODEs
  23. Research Article
  24. Compatibility for probabilistic theories
  25. Research Article
  26. Completeness of Gelfand-Neumark-Segal inner product space on Jordan algebras
  27. Research Article
  28. Commutativity in a synaptic algebra
  29. Research Article
  30. Discrete averaged mixing applied to the logarithmic distributions
  31. Research Article
  32. Automorphisms of decompositions
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 8.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0148/html
Scroll to top button