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Mathematica Slovaca

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Volume 66, Issue 2


Note on a parameter switching method for nonlinear ODEs

Marius-F. Danca
  • Department of Mathematics and Computer Science Emanuel University of Oradea 410597 Oradea ROMANIA Romanian Institute for Science and Technology Str. Cireilor 29 400487 Cluj-Napoca ROMANIA
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/ Michal Fečkan
  • Department of Mathematical Analysis and Numerical Mathematics Comenius University Mlynská dolina 842 48 Bratislava SLOVAKIA Mathematical Institute Slovak Academy of Sciences Štefánikova 49 814 73 Bratislava SLOVAKIA
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Published Online: 2016-07-05 | DOI: https://doi.org/10.1515/ms-2015-0148


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.


  • [1]

    ALMEIDA, J.—PERALTA-SALAS, D.—ROMERA, M.: Can two chaotic systems give rise to order?, Phys. D 200 (2005), 124–132.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.Web of ScienceGoogle Scholar

  • [3]

    DANCA, M.-F.: Random parameter-switching synthesis of a class of hyperbolic attractors, Chaos 18 (2008), 033111.Web of ScienceGoogle Scholar

  • [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.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.Web of ScienceGoogle 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.Google Scholar

  • [7]

    FExČKAN, M.: Topological Degree Approach to Bifurcation Problems (1st ed.), Springer, Berlin, 2008.Google Scholar

  • [8]

    GUCKENHEIMER, J.—HOLMES, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New-York, 1983.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.Google Scholar

  • [10]

    HALE, J. K.: Ordinary Differential Equations, Dover Publications, New York, 2009.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.Web of ScienceGoogle Scholar

  • [12]

    ROMERA, M.—SMALL, M.—DANCA, M.-F.: Deterministic and random synthesis of discrete chaos, Appl. Math. Comput. 192 (2007), 283–297.Google Scholar

  • [13]

    ROUCHE, N.—HABETS, P.—LALOY, M.: Stability Theory by Liapunov’s Direct Method, Springer-Verlag, New York, 1977.Google Scholar

  • [14]

    RUDIN, W.: Functional Analysis, McGraw-Hill, New York, 1973.Google Scholar

  • [15]

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

  • [16]

    STAMOVA, I. M.—STAMOV, G. T.: Stability analysis of differential equations with maximum, Math. Slovaca 63 (2013), 1291–1302.Web of ScienceGoogle Scholar

  • [17]

    STUART, A. M.—HUMPHRIES, A. R.: Dynamical Systems and Numerical Analysis, Cambridge Univ. Press, Cambridge, 1998.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).Google Scholar

About the article

Received: 2013-05-27

Accepted: 2014-01-05

Published Online: 2016-07-05

Published in Print: 2016-04-01

Citation Information: Mathematica Slovaca, Volume 66, Issue 2, Pages 439–448, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0148.

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