The European Physical Journal Special Topics

, Volume 222, Issue 7, pp 1675–1684 | Cite as

Nonlinear dynamics of complex hysteretic systems: Oscillator in a magnetic field

  • G. RadonsEmail author
  • A. Zienert
Regular Article


Complex hysteresis is a well-known phenomenon in many branches of science. The most prominent examples come from materials with a complex microscopic structure such as magnetic materials, shape-memory alloys, or, porous materials. Their hysteretic behavior is characterized by the existence of multiple internal system states for a given external parameter and by a non-local memory. The input-output behavior of such systems is well studied and in a standard phenomenological approach described by the so-called Preisach operator. What is not well understood, are situations, where such a hysteretic system is dynamically coupled to its environment. Since the hysteretic sub-system provides a complicated form of nonlinearity, one expects non-trivial, possibly chaotic behavior of the combined dynamical system. We study such a combined dynamical system with hysteretic nonlinearity. In this original contribution a simple differential-operator equation with hysteretic damping, which describes a magnetic pendulum is considered. We find, for instance, a fractal dependence of the asymptotic behavior as function of the starting values. The sensitivity of the system to perturbations is investigated by several methods, such as the 0–1 test for chaos and sub-Lyapunov exponents. The power spectral density is also calculated and compared with analytical results for simple input-output scenarios.


Radon Lyapunov Exponent European Physical Journal Special Topic Chaotic Behavior Energy Harvest 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Bertotti, I.D. Mayergoyz (eds.), The Science of Hysteresis, Vols. 1–3 (Academic Press, 2006)Google Scholar
  2. 2.
    G. Bertotti, Hysteresis in Magnetism (Academic, New York, 1998)Google Scholar
  3. 3.
    E. Della Torre, Magnetic Hysteresis (IEEE, New York, 1999)Google Scholar
  4. 4.
    J. Ortin, J. Appl. Phys. 71, 1454 (1992)ADSCrossRefGoogle Scholar
  5. 5.
    C. Song, J.A. Brandon, C.A. Featherston, J. Mech. Eng. Sci. 215, 673 (2001)CrossRefGoogle Scholar
  6. 6.
    P. Ge, M. Jouaneh, Precis. Eng. 17, 211 (1995)CrossRefGoogle Scholar
  7. 7.
    I.D. Mayergoyz, T.A. Keim, J. Appl. Phys. 67, 5466 (1990)ADSCrossRefGoogle Scholar
  8. 8.
    M. Sjöström, D. Djukic, B. Dutoit, IEEE Trans. Appl. Supercond. 10, 1585 (2000)CrossRefGoogle Scholar
  9. 9.
    D. Flynn, H. McNamara, P. O’Kane, A. Pokrovskii, Application of the Preisach Model to Soil-Moisture Hysteresis, in Ref. [1], Vol. 3, p. 689Google Scholar
  10. 10.
    M.P. Lilly, P.T. Finley, R.B. Hallock, Phys. Rev. Lett. 71, 4186 (1993)ADSCrossRefGoogle Scholar
  11. 11.
    R.A. Guyer, K.R. McCall, G.N. Boitnott, Phys. Rev. Lett. 74, 3491 (1995)ADSCrossRefGoogle Scholar
  12. 12.
    M. Göcke, J. Econ. Surv. 16, 167 (2002)CrossRefGoogle Scholar
  13. 13.
    H.A. Mc Namara, A.V. Pokrovskii, Physica B 372, 202 (2006)ADSCrossRefGoogle Scholar
  14. 14.
    P. Weiss, J. de Freudenreich, Arch. Sci. Phys. Nat. 42, 449 (1916)Google Scholar
  15. 15.
    F. Preisach, Z. Phys. 94, 277 (1935)ADSCrossRefGoogle Scholar
  16. 16.
    I.D. Mayergoyz, Phys. Rev. Lett. 56, 1518 (1986)ADSCrossRefGoogle Scholar
  17. 17.
    I.D. Mayergoyz, Mathematical Models of Hysteresis and their Applications (Elsevier, New York, 2003)Google Scholar
  18. 18.
    M.A. Krasnoselskii, A.V. Pokrovskii, Systems with Hysteresis (Springer, Berlin, 1989)Google Scholar
  19. 19.
    A. Visintin, Differential Models of Hysteresis (Springer, Berlin, 1994)Google Scholar
  20. 20.
    M. Brokate, J. Sprekels, Hysteresis and Phase Transitions (Springer, New York, 1996)Google Scholar
  21. 21.
    G. Litak, M. I. Friswell, S. Adhikari, Appl. Phys. Lett. 96, 214103 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    H. Kantz, T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2003)Google Scholar
  23. 23.
    S. Kodba, M. Perc, M. Marhl, Eur. J. Phys. 26, 205 (2005)CrossRefGoogle Scholar
  24. 24.
    A. Pokrovskii, O. Rasskazov, Numerical integration of ODEs with Preisach nonlinearity, Preprints of BCRI, UCC, Ireland, 2004Google Scholar
  25. 25.
    M. Brokate, A. Pokrovskii, D. Rachinskii, O. Rasskazov, Differential equations with hysteresis via a canonical example, in Ref. [1], Vol. 1, p. 125Google Scholar
  26. 26.
    Systems with Hysteresis,
  27. 27.
    G.A. Gottwald, I. Melbourne, Proc. R. Soc. Lond. A 460, 603 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    G.A. Gottwald, I. Melbourne, Physica D 212, 100 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    K. Kaneko, Progr. Theor. Phys. Suppl. 99, 263 (1989)ADSCrossRefGoogle Scholar
  30. 30.
    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Part I: Theory, Meccanica 15, 9 (1980)ADSzbMATHGoogle Scholar
  31. 31.
    G. Radons, Phys. Rev. Lett. 100, 240602 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    G. Radons, Phys. Rev. E 77, 061133 (2008)ADSCrossRefMathSciNetGoogle Scholar
  33. 33.
    G. Radons, Phys. Rev. E 77, 061134 (2008)ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    G. Radons, F. Heße, R. Lange, S. Schubert, On the dynamics of nonlinear hysteretic systems, in P.J. Plath and E.-Ch. Haß (eds.), Vernetzte Wissenschaften, Crosslinks in Natural and Social Sciences (Logos, Berlin, 2008), p. 271Google Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Institute of Physics, Complex Systems and Nonlinear Dynamics, Chemnitz University of TechnologyChemnitzGermany
  2. 2.Institute for MechatronicsChemnitzGermany
  3. 3.Center for Microtechnologies, Chemnitz University of TechnologyChemnitzGermany

Personalised recommendations