Journal of Dynamics and Differential Equations

, Volume 24, Issue 4, pp 713–759 | Cite as

Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis

  • Z. Balanov
  • W. Krawcewicz
  • D. Rachinskii
  • A. Zhezherun


The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S 4-equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.


Symmetric Hopf bifurcation Equivariant system Differential-operator equation Preisach hysteresis memory operator Twisted equivariant degree 


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  1. 1.
    Antonyan, S.A.: Equivariant generalization of Dugundji’s theorem. Mat. Zametki 38:608–616 (in Russian) (1985). English translation in 1985, Math. Notes 38, 844–848Google Scholar
  2. 2.
    Appelbe B., Rachinskii D., Zhezherun A.: Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis. Phys. B 403, 301–304 (2008)CrossRefGoogle Scholar
  3. 3.
    Appelbe B., Flynn D., McNamara H., O’Kane P., Pimenov A., Pokrovskii A., Rachinskii D., Zhezherun A.: Rate-independent hysteresis in terrestrial hydrology. IEEE Control Syst. Mag. 29, 44–69 (2009)CrossRefGoogle Scholar
  4. 4.
    Ashwin P., Podvigina O.: Hopf bifurcation with rotational symmetry of the cube and instability of ABC flow. Proc. R. Soc. A 459, 1801–1827 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aubry S.: Exact models with a complete Devil’s staircase. J. Phys. C: Solid State Phys. 16, 2497–2508 (1983)CrossRefGoogle Scholar
  6. 6.
    Balanov Z., Krawcewicz W.: Symmetric Hopf bifurcation: twisted degree approach. In: Battelli, F., Feckan, M. (eds) Handbook of Differential Equations, Ordinary Differential Equations, vol. 4, pp. 1–131. Elsevier, Amsterdam (2008)Google Scholar
  7. 7.
    Balanov Z., Krawcewicz W., Ruan H.: Applied equivariant degree, part I: an axiomatic approach to primary degree. Discr. Continuous Dyn. Syst. A 15, 983–1016 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Balanov Z., Krawcewicz W., Steinlein H.: Applied Equivariant Degree, AIMS Series on Differential Equations and Dynamical Systems, vol. 1. AIMS, Springfield (2006)Google Scholar
  9. 9.
    Balanov Z., Krawcewicz W., Rybicki S., Steinlein H.: A short treatise on the equivariant degree theory and its applications, a short treatise on the equivariant degree theory and its applications (on the occasion of S. Smale’s 80-th birthday. J. Fixed Point Theory Appl. 8, 1–74 (2010)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Barut A.O., Ra¸czka R.: Theory of Group Representations and Applications. World Scientific Publishing Co., Singapore (1986)MATHGoogle Scholar
  11. 11.
    Bredon G.E.: Introduction to Compact Transformation Groups. Academic Press, New York (1972)MATHGoogle Scholar
  12. 12.
    Brokate M., Collings I., Pokrovskii A., Stagnitti F.: Asymptotically stable oscillations in systems with hysteresis nonlinearities. Z. Anal. Anw. 19, 469–487 (2000)MathSciNetMATHGoogle Scholar
  13. 13.
    Brokate M., Sprekels J.: Hysteresis and Phase Transitions. Springer, New York (1996)MATHCrossRefGoogle Scholar
  14. 14.
    Brokate M., Pokrovskii A., Rachinskii D., Rasskazov O.: Differential equations with hysteresis via a canonical example. In: Bertotti, G., Mayergoyz , I. (eds) The Science of Hysteresis, vol. 1, pp. 127–291. Academic Press, New York (2005)Google Scholar
  15. 15.
    Brokate M., Pokrovskii A.V., Rachinskii D.I.: Asymptotic stability of continual sets of periodic solutions to systems with hysteresis. J. Math. Anal. Appl. 319, 94–109 (2006)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Colombo A., di Bernardo M., Hogan S.J., Kowalczyk P.: Complex dynamics in a hysteretic relay feedback system with delay. J. Nonlinear Sci. 17, 85–108 (2007)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Cross R., Grinfeld M., Lamba H., Seaman T.: Stylized facts from a threshold-based heterogeneous agent model. The Eur. Phys. J. B 57, 213–218 (2007)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Dahmen K.: Nonlinear dynamics: universal clues in noisy skews. Nat. Phys. 1, 13–14 (2005)CrossRefGoogle Scholar
  19. 19.
    Dahmen K., Ben-Zion Y.: The physics of jerky motion in slowly driven magnetic and earthquake fault systems. In: Marchetti, C., Meyers, R.A. (eds) Enciclopedia of Complexity and Systems Science, Springer, New York (2009)Google Scholar
  20. 20.
    Dahmen K.A., Ben-Zion Y., Uhl J.T.: A micromechanical model for deformation in solids and with universal predictions for stress strain curves and slip avalances. Phys. Rev. Lett. 102, 175501 (2009)CrossRefGoogle Scholar
  21. 21.
    Dancer E.N.: A new degree for S 1-invariant gradient mappings and applications. Ann. Inst. H. Poincaré Anal. Non Lineaire 2, 1–18 (1985)MathSciNetGoogle Scholar
  22. 22.
    Dancer E.N., Toland J.F.: The index change and global bifurcation for flows with first integrals. Proc. Lond. Math. Soc. 66, 539–567 (1993)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Davino D., Giustiniani A., Visone C.: Compensation and control of two-inputs systems with hysteresis. J. Phys. Conf. Ser. 268, 012005 (2011)CrossRefGoogle Scholar
  24. 24.
    Diamond P., Rachinskii D.I., Yumagulov M.G.: Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity. Nonlinear Anal. 42, 1017–1031 (2000)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Diamond P., Kuznetsov N.A., Rachinskii D.I.: On the Hopf bifurcation in control systems with asymptotically homogeneous at infinity bounded nonlinearities. J. Differ. Equ. 175, 1–26 (2001)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Dias A.P.S., Rodrigues A.: Hopf bifurcation with S N-symmetry. Nonlinearity 27, 627–666 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Eleuteri M., Kopfova J., Krejči P.: On a model with hysteresis arising in magnetohydrodynamics. Physica B 403, 448–450 (2008)CrossRefGoogle Scholar
  28. 28.
    Fiedler B.: Global Bifurcation of Periodic Solutions with Symmetry, Lecture Notes in Mathematics, vol. 1309. Springer, New York (1988)Google Scholar
  29. 29.
    Field M.J.: Dynamics and Symmetry, ICP Advanced Texts in Mathematics, vol. 3. Imperial College Press, London (2007)Google Scholar
  30. 30.
    Field M.J., Swift J.W.: Hopf bifurcation and Hopf fibration. Nonlinearity 7, 385–402 (1994)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Fuller F.B.: An index of fixed point type for periodic orbits. Am. J. Math. 89, 133–148 (1967)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Gleeson J.P.: Bond percolation on a class of clustered random networks. Phys. Rev. E 80, 036107 (2009)CrossRefGoogle Scholar
  33. 33.
    Godsil, C.: (2004). Association Schemes http://quoll/
  34. 34.
    Goicoechea J., Ortin J.: Hysteresis and return-point memory in deterministic cellular automata. Phys. Rev. Lett. 72, 2203 (1994)CrossRefGoogle Scholar
  35. 35.
    Golubitsky M., Stewart I.N.: The Symmetry Perspective. Berlin, Birkhäuser (2002)MATHCrossRefGoogle Scholar
  36. 36.
    Golubitsky M., Schaeffer D.G., Stewart I.N.: Singularities and Groups in Bifurcation Theory, vol. 2. Springer, New York (1988)CrossRefGoogle Scholar
  37. 37.
    Guo S., Lamb J.S.W.: Equivariiant Hopf bifurcation for neutral functional differential equations. Proc. Am. Math. Soc. 136, 2031–2041 (2008)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Guyer R.A., McCall K.R.: Capillary condensation, invasion percolation, hysteresis, and discrete memory. Phys. Rev. B. 54, 18–21 (1996)CrossRefGoogle Scholar
  39. 39.
    Iudovĭck V.I.: The onset of auto-oscillations in a fluid. Prikl. Mat. Mek. 35, 638–655 (1971)Google Scholar
  40. 40.
    Iyer, R., Tan, X.: (eds.) Hysteresis. IEEE Control Systems Magazine 1 (2009)Google Scholar
  41. 41.
    Ize, J., Vignoli, A.: Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, vol. 8, W. de Gruyter (2003)Google Scholar
  42. 42.
    Kawakubo K.: The Theory of Transformation Groups. The Clarendon Press, New York (1991)MATHGoogle Scholar
  43. 43.
    Kirillov A.A.: Elements of the Theory of Representations, Grundlehren der Mathematischen Wissenschaften, vol. 220. Springer, Berlin (1976)Google Scholar
  44. 44.
    Krasnosel’skii M.A.: Positive Solutions of Operator Equations. P. Noordhoff Ltd., Groningen (1964)Google Scholar
  45. 45.
    Krasnosel’skii M., Pokrovskii A.: Systems with Hysteresis. Springer, New York (1989)MATHCrossRefGoogle Scholar
  46. 46.
    Krasnosel’skii A.M., Rachinskii D.I.: On a bifurcation governed by hysteresis nonlinearity. Nonlinear Differ. Equ. Appl. 9, 93–115 (2002)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Krasnosel’skii A.M., Rachinskii D.I.: On continuous branches of cycles in systems with non-linearizable nonlinearities. Doklady Math. 67, 153–158 (2003)MATHGoogle Scholar
  48. 48.
    Krasnosel’skii A.M., Kuznetsov N.A., Rachinskii D.I.: Nonlinear Hopf bifurcations. Doklady Math. 61, 389–392 (2000)Google Scholar
  49. 49.
    Krasnosel’skii A.M., Rachinskii D.I.: On continua of cycles in systems with hysteresis. Doklady Math. 63, 339–344 (2001)Google Scholar
  50. 50.
    Krasnosel’skii A.M., Rachinskii D.I.: On existence of cycles in autonomous systems. Doklady Math. 65, 344–349 (2002)MATHGoogle Scholar
  51. 51.
    Krasnosel’skii A.M., Kuznetsov N.A., Rachinskii D.I.: On resonant differential equations with unbounded nonlinearities. Z. Anal. Anwendungen 21, 639–668 (2002)MathSciNetGoogle Scholar
  52. 52.
    Krawcewicz W., Wu J.: Theory of Degrees with Applications to Bifurcations and Differential Equations, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)Google Scholar
  53. 53.
    Krejci P.: On Maxwell equations with the Preisach hysteresis operator: the one-dimensional time-periodic case. Appl. Math. 34, 364–374 (1989)MathSciNetMATHGoogle Scholar
  54. 54.
    Krejči P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo (1996)MATHGoogle Scholar
  55. 55.
    Krejci P.: Resonance in Preisach systems. Appl. Math. 45, 439–468 (2000)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Krejci P., Sprekels J., Zheng S.: Asymptotic behaviour for a phase-field system with hysteresis. J. Differ. Equ. 175, 88–107 (2001)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Krejci P., O’Kane P., Pokrovskii A., Rachinskii D.: Stability results for a soil model with singular hysteretic hydrology. J. Phys. Conf. Ser. 268, 012016 (2010)CrossRefGoogle Scholar
  58. 58.
    Kuhnen K.: Compensation of parameter-dependent complex hysteretic actuator nonlinearities in smart material systems. J. Intel. Mater. Syst. Struct. 19, 1411–1424 (2008)CrossRefGoogle Scholar
  59. 59.
    Kushkuley A., Balanov Z.: Geometric Methods in Degree Theory for Equivariant Maps, Lecture Notes in Mathematics, vol. 1632. Springer, Berlin-Heidelberg (1996)Google Scholar
  60. 60.
    Kuznetsov Yu. A.: Elements of Applied Bifurcation Theory. Springer, Berlin (1995)MATHGoogle Scholar
  61. 61.
    Kuznetsov N.A., Rachinskii D., Zhezherun A.: Hopf bifurcation in systems with Preisach operator. Doklady Math. 78, 705–709 (2008)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Lamba H., Seaman T.: Market statistics of a psychology-based heterogeneous agent model. Int. J. Theor. Appl. Finance 11, 717–737 (2008)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Logemann H., Ryan E.P., Shvartsman I.: A class of differential-delay systems with hysteresis: asymptotic behaviour of solutions. Nonlinear Anal. Theory Methods Appl. 69, 363–391 (2008)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Mayergoyz I.D.: Mathematical Models of Hysteresis. Springer, New York (1991)MATHCrossRefGoogle Scholar
  65. 65.
    Mayergoyz, I.D., Bertotti, G. (eds): The Science of Hysteresis, 3-volume set. Elsevier, Academic Press (2005)Google Scholar
  66. 66.
    Mehta A., Barker G.C.: Bistability and hysteresis in tilted sandpiles. Europhys. Lett. 56, 626–632 (2001)CrossRefGoogle Scholar
  67. 67.
    Pimenov A., Rachinskii D.: Linear stability analysis of systems with Preisach memory. Discr. Continuous Dyn. Syst. B 11, 997–1018 (2009)MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Pokrovskii A., Power T., Rachinskii D., Zhezherun A.: Stability by linear approximation of ODEs with Preisach operator. J. Phys. Conf. Ser. 55, 171–190 (2006)CrossRefGoogle Scholar
  69. 69.
    Rachinskii D.I.: Asymptotic stability of large-amplitude oscillations in systems with hysteresis. Nonlinear Differ. Equ. Appl. 6, 267–288 (1999)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Rachinskii D.I., Schneider K.R.: Dynamic Hopf bifurcations generated by nonlinear terms. J. Differ. Equ. 210, 65–86 (2005)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Rezaei-Zare A., Sanaye-Pasand M., Mohseni H., Farhangi Sh., Iravani R.: Analysis of ferroresonance modes in power transformers using Preisach-type hysteretic magnetizing inductance. IEEE Trans. Power Deliv. 22, 919–929 (2007)CrossRefGoogle Scholar
  72. 72.
    Sethna J.P., Dahmen K., Kartha S., Krumhansl J.A., Robetrs B.W., Shore J.D.: Hysteresis and hierarchies: dynamics of disorder-driven first-order phase transitions. Phys. Rev Lett. 70, 3347 (1993)CrossRefGoogle Scholar
  73. 73.
    Sethna J.P., Dahmen K., Myers C.R.: Crackling noise. Nature 410, 242–250 (2001)CrossRefGoogle Scholar
  74. 74.
    Sethna J.P., Dahmen K.A., Perkovic O.: Random-field Ising models of hysteresis. In: Bertotti, G., Mayergoyz, I. (eds) The Science of Hysteresis, vol 2, pp. 107–168. Elsevier, Amsterdam (2005)Google Scholar
  75. 75.
    tom Dieck T.: Transformation Groups. W. de Gruyter, Berlin (1987)MATHCrossRefGoogle Scholar
  76. 76.
    Turing A.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. B. 237, 37–72 (1952)CrossRefGoogle Scholar
  77. 77.
    Visintin A.: Differential Models of Hysteresis. Springer, Berlin (1994)MATHGoogle Scholar
  78. 78.
    Visone C.: Hysteresis modelling and compensation for smart sensors and actuators. J. Phys. Conf. Ser. 138, 012028 (2008)CrossRefGoogle Scholar
  79. 79.
    Watts D.J.: A simple model of global cascades on random networks. Proc. Nat. Acad Sci. USA 99, 5766–5771 (2002)MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    Wielandt H.: Finite Permutation Groups. Academic Press, New York (1964)MATHGoogle Scholar
  81. 81.
    Wu J.: Symmetric functional-differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)MATHCrossRefGoogle Scholar
  82. 82.
    Yoshida K.: The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology. Hiroshima Math. J. 12, 321–348 (1982)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Z. Balanov
    • 1
  • W. Krawcewicz
    • 1
    • 2
  • D. Rachinskii
    • 1
    • 3
  • A. Zhezherun
    • 3
  1. 1.Department of Mathematical SciencesUniversity of Texas at DallasRichardsonUSA
  2. 2.College of ScienceChina Three Gorges UniversityYichangChina
  3. 3.Department of Applied MathematicsUniversity College CorkCorkIreland

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